Quan um T anspo in
Nanowi es wi h Spin-O bi
In e ac ion: e ec o
quasi-bound s a es
Alba Pascual Gil
Di ec ed by Sebas i´an Be ge e and Vi aly Golo ach
Ma e ial Physics Depa men
PhD P og am o Physics o Nanos uc u es and
Ad anced Ma e ials
Uni e si y o he Basque Coun y (UPV/EHU)
Decembe , 2019
(cc)2020 ALBA PASCUAL GIL (cc by-nc 4.0)
A Pepa y Jos´e Luis
Ag adecimien os
P ime o quisie a ag adece a Sebas i´an Be ge e po su di ecci´on y ayuda en
es os cua o a˜nos como supe iso de mi esis. Tu pe spec i a, op imismo y
paciencia me han se ido de g an ayuda e inspi aci´on pa a segui es e camino
y c ece como pe sona. G acias po acoge me en el g upo de Mesoscopic
Physics du an e es a e apa de mi ida. As´ı mismo, quie o ag adece a Vi aly
Golo ach odo el apoyo an o acad´emico como pe sonal que me ha pe mi ido
lle a a cabo es a esis. Aspi o a ene al menos la mi ad de u gene osidad
como pe sona y u amo po la ciencia.
Quisie a ambi´en ex ende es e ag adecimien o al es o del g upo de
Mesoscopic Physics del Cen o de F´ısica de Ma e iales. G acias Da io
Be cioux po us consejos al inicio de mi es ancia y ambi´en g acias Tineke
an den Be g po u apoyo y amis ad. Finalmen e, g acias al es o de
es udian es y compae os de despacho, C is ina, Mikel, Bogusz, Xian-Peng y
Julie, po odos los a os compa idos en es e pe ´odo de mi ida.
A ni el m´as pe sonal, quie o da las g acias a mis pad es po ansmi i me
sus alo es y apoya me en odo momen o pa a pode hace siemp e lo que
yo quisie a. A mi amilia po su amo incodicional y a mi he mana Cla a
po su paciencia in ini a. A mi peque˜na amilia en Donos ia po odos esos
momen os inol idables y deba es acalo ados en las comidas y co ee b eaks.
En especial g acias a Mo i z, Pa i y Tom´aˇs, no se ´ıa hoy quien soy hoy
en d´ıa sin oso os. G acias ambi´en a Chusa y Ma eo po da me e ugio
espi i ual, a Paloma po apo a le ese oque de humo a mi ida, a Yaiza po
comple a me, y al es o de amigos que me han acompa˜nado en es e iaje.
Finalmen e, g acias a mi pa eja Juan Ca los po es a ah´ı siemp e.
2
Acknowledgemen s
Fi s o all, I would like o hank Sebas i´an Be ge e o all he guidance
du ing he pas ou yea s as my PhD supe iso . You ision, op imism
and pa ience ha e been o g ea help and an inspi a ion o g ow as a pe son.
Thank you o welcoming me in o he Mesoscopic Physics g oup du ing he
du a ion o my s ay. I would also like o hank Vi aly Golo ach o all his
help no only on an academic le el bu also on a pe sonal one. I aspi e o
ha e you gene osi y as a pe son and you passion o science.
I would like o ex end his acknowledgemen o he es o he Mesoscopip
Physics g oup om he Ma e ials Physics Cen e in Donos ia. Thank you
Da io Be cioux o all you ad ice in he ea ly s ages o my s ay he e and
Tineke an den Be g o all you suppo and iendship. Finally, hanks o
he es o s uden s and o ice ma es, C is ina, Mikel, Bogusz, Xian-Peng and
Julie, o all his ime spen oge he .
On a mo e pe sonal no e, I would like o hank my pa en s o hei
upb inging and encou agemen o pu sue my d eams. To my amily o
hei uncondi ional lo e and my sis e Cla a o he in ini e pa ience. To my
li le amily in Donos ia o e e y un o ge able momen and e e y hea ed
discussion du ing co ee b eaks. And specially, hank you Mo i z, Pa i and
Tom´aˇs, I wouldn’ be me oday wi hou you. Thank you Chusa and Ma eo
o o e ing me spi i ual e uge, o Paloma o he spo o humo in my li e, o
Yaiza o comple ing me and o he es o my iends whe e e hey a e o
accompanying me du ing his jou ney. Las bu no leas , hank you Juan
Ca los o being he e always.
3
Resumen
Es a esis iene como obje i o el es udio de anspo e en nanohilos
semiconduc o es con in e acci´on esp´ın-´o bi a e impu ezas. A d´ıa de hoy
es os nanohilos son de los ma e iales m´as e s´a iles pa a el dise˜no de
disposi i os cu´an icos. Ejemplo de ello, es el in enso es udio de los
e miones de Majo ana, que pueden de ec a se en los ex emos de
nanohilos, cuando ´es os es ´an en con ac o con un supe conduc o
[1, 2, 3, 4, 5, 6, 7]. C ucial pa a la apa ici´on de los e miones de Majo ana
es la in e acci´on esp´ın-´o bi a y el bajo ni el de deso den en los nanohilos.
De hecho, el deso den en nanohilos cu´an icos a ec a ue emen e a la
conduc ancia de los modos de Majo ana [8, 9, 10, 11, 12, 13, 14].
Po o o lado, las es uc u as semiconduc o as con in e acci´on
esp´ın-´o bi a nos lle an a ´as en el iempo has a la p opues a de Da a y
Das de un ansis o [15], que p opone el con ol sob e la in e acci´on
esp´ın-´o bi a po medio de un ga e pa a o a el esp´ın del elec ´on y as´ı
con ola el anspo e de ca ga en e dos elec odos e omagn´e icos. Los
in en os de ab ica al disposi i o han opado con a ios
p oblemas [16, 17, 18], inclu´ıda la baja e iciencia en la inyecci´on de esp´ın del
e omagne o en el semiconduc o , pe o ambi´en con la elajaci´on del esp´ın
inducida po el sca e ing del elec ´on debido al deso den en el disposi i o.
De los ejemplos an e io es se desp ende que el es udio del deso den
equie e especial a enci´on. En pa icula en espues a a la p egun a de
c´omo a ec a el deso den al anspo e de ca ga y esp´ın en un nanohilo
cuasi-unidimensional. Es a p egun a es absolu amen e no i ial. En los
nanohilos, el desplazamien o es ´a con inado en una di ecci´on y los
po ado es de ca ga solo pueden desplaza se en la di ecci´on o ogonal a la
del po encial de con inamien o. La combinaci´on del con inamien o con la
p esencia de la in e acci´on esp´ın-´o bi a de ipo Rashba induce el
acoplamien o en e subbandas. Es o a ec a ue emen e al anspo e en el
4
hilo cu´an ico, llegando po ejemplo a sup imi la modulaci´on de esp´ın pa a
alo es g andes de la in e acci´on Rashba [19, 20, 21, 22]. Po o o lado, la
p esencia de una impu eza puede lle a a la o maci´on de es ados
cuasi-ligados localizados en o no a la impu eza. En abajos p e ios se ha
demos ado que la p esencia de modos e anescen es lle a a en´omenos poco
usuales en el anspo e elec ´onico, como po ejemplo la pe ec a
ansmisi´on en el umb al ene g´e ico en el que una nue a subbanda es
accesible y comienza a p opaga . Tambi´en se ha obse ado que ce ca, pe o
po debajo de es e umb al, la apa ici´on de es ados cuasi-ligados localizados
en o no a una impu eza a ac i a es esponsable del bloqueo o al de
canales de ansmisi´on [23, 24]. La combinaci´on de ambos e ec os nunca ha
sido a ada.
En es a esis abo damos es e ema y p esen amos un es udio e´o ico
exhaus i o del anspo e elec ´onico en nanohilos cu´an icos
semiconduc o es con in e acci´on esp´ın-´o bi a en la p esencia de impu ezas.
Modelamos el nanohilo cu´an ico como un sis ema cuasi-unidimensional en
el que el mo imien o de los elec ones es ´a con inado en la di ecci´on
pe pendicula a la de p opagaci´on. La compe ici´on en e la in e acci´on
esp´ın-´o bi a, el con inamien o la e al y la impu eza hace que el p oblema
sea al amen e no- i ial. Pa a hace en e a es e p oblema usamos una
combinaci´on de ´ecnicas y ap oximaciones que nos pe mi en iden i ica
no edosas p opiedades del anspo e de ca ga y del anspo e de esp´ın.
Espec´ı icamen e, desc ibimos la conduc ancia median e el o malismo de
Landaue -B¨u ike , ex endi´endolo pa a el caso de campos dependien es del
esp´ın. Desc ibimos el anspo e a a ´es de es e o malismo en unci´on de
los coe icien es de sca e ing. Pa a calcula los coe icien es de la ma iz de
sca e ing usamos la ecuaci´on de Lippmann-Schwinge , un m´e odo
ampliamen e usado en el a amien o del sca e ing en la mec´anica
cu´an ica.
Una de las p incipales di icul ades en el a amien o de la in e acci´on
esp´ın-´o bi a en un po encial de con inamien o es la hib idaci´on de las
subbandas. Pa a supe a es e p oblema in oducimos la ans o maci´on de
Sch ie e -Wol , una ans o maci´on de gauge que elimina es a hib idaci´on
man eniendo la complicaci´on de los e ec os de Rashba en la unci´on de onda
ans o mada.
Combinando las ´ecnicas mencionadas calculamos de o ma anal´ı ica la
coduc ancia pa a el anspo e de ca ga y esp´ın en un nanohilo con
in e acci´on de ipo Rashba en p esencia de una impu eza pun ual.
5
Encon amos que la impu eza acopla es ados p opagan es y es ados
e anescen es en el nanohilo cu´an ico, induciendo es ados cuasi-ligados
localizados en o no a la impu eza. Po o a pa e, la in e acci´on
esp´ın-´o bi a de ipo Rashba pe mi e dis in os mecanismos de anspo e
elec ´onico. Como consecuencia, la conduc ancia p esen a ansmisi´on
bal´ıs ica pe ec a en la ene g´ıa umb al donde el siguien e canal se uel e
ansmisi o. Adem´as, po debajo de dicha ene g´ıa umb al la conduc ancia
p esen a una educci´on signi ica i a. Demos amos que pa a las subbandas
m´as bajas en es a ene g´ıa esonan e, odos los elec ones inyec ados en el
nanohilo cu´an ico en un es ado p epa ado de al mane a que su esp´ın se
al´ınea en cie a di ecci´on p e e en e, solo ienen una o ma de ansmi i a
a ´es de la impu eza: median e un p oceso en el que su esp´ın sal a a la
o ien aci´on opues a. No solo es ´es a la ´unica o ma de p opaga hacia el
o o lado de la impu eza, sino que la p obabilidad de que suceda es e
p oceso de in e si´on del esp´ın aumen a no ablemen e espec o a la
p obabilidad ue a de la esonancia. Es m´as, en la ene g´ıa exac a de la
esonancia, mien as la p obabilidad de ansmisi´on man eniendo la misma
o ien aci´on en el esp´ın se educe has a ce o, la p obabilidad de ansmisi´on
con in e si´on del esp´ın es m´axima.
M´as all´a del anspo e de ca ga, ambi´en de i amos exp esiones pa a las
co ien es de esp´ın en el nanohilo y de i amos una exp esi´on pa a el o que de
esp´ın-´o bi a inducido po la impu eza. Demos amos que es e o que depende
comple amen e de los p ocesos de in e si´on del esp´ın en el sca e ing. O o
esul ado cla e de es a esis es la elaci´on subyacen e en e el campo de gauge
SU(2) y la ansmisi´on con in e si´on del esp´ın.
La esis es ´a o ganizada de la siguien e mane a: el Cap´ı ulo 1 es la
in oduccion a la esis. En los Cap´ı ulos 2 y 3 ex endemos es a in oducci´on
pa a habla de concep os gene ales que son usados a lo la go del es o de la
esis. En pa icula , en el Cap´ı ulo 2 p opo cionamos un b e e epaso sob e
el a amien o e´o ico del sca e ing en nanohilos median e la ecuaci´on de
Lippmann-Schwinge . En el Cap´ı ulo 3 in oducimos la in e acci´on
esp´ın-´o bi a en sis emas de baja dimensionalidad como 2DEG y nanohilos.
En es e ´ul imo caso explicamos la di icul ad de diagonaliza el
Hamil oniano co espondien e debido a la hib idaci´on de las subbandas.
En el Cap´ı ulo 4 in oducimos el o malismo de Landaue -B¨u ike pa a
la desc ipci´on del anspo e cu´an ico. Ex endemos la de i aci´on habi ual
pa a inclui un sis ema con in e acci´on esp´ın-´o bi a de ipo Rashba y como
consecuencia el sis ema man iene no solo co ien es de ca ga sino ambi´en
6
CHAPTER 1. INTRODUCTION
s ongly a ec conduc ance o he ze o modes. [8, 9, 10, 11, 12, 13, 14].
On he o he hand, he spin-o bi in e ac ion couples e icien ly he
elec on spin o i s o bi al deg ees o eedom, making i possible o a ec
he spin by enginee ing he scala po en ial along he pa h o he elec on.
One can en ision designs in which he desi ed e ec o he spin-o bi
in e ac ion is s ongly enhanced, which can be used o imp o e de ice
unc ionali y. The idea o using he spin-o bi in e ac ion o o a e he
elec on spin goes back o he Da a-Das ansis o [15], in which he
con ol o e he spin-o bi in e ac ion was p oposed o be used o modula e
he conduc ance o a e omagne -semiconduc o - e omagne de ice.
A emp s o implemen his ansis o [16, 17, 18] aced se e al p oblems,
including he low spin injec ion e iciency om e omagne in o
semiconduc o and he de imen al e ec o he sca e ing o he elec on
on diso de , which leads o spin elaxa ion. The in e play be ween
supe conduc i i y and spin-dependen ields also plays a undamen al ole
in he eme ging ield o supe conduc ing spin onics [32, 33, 34, 35].
Beside possible applica ions o semiconduc ing sys ems wi h spin-o bi
in e ac ion, he e a e s ill undamen al ques ions ega ding he elec onic
anspo in such s uc u es ha s ill equi e a heo e ical analysis. In
nanowi es, an open ques ion is how a de ec may a ec he spin and cha ge
anspo in a quasi one dimensional wi e. The answe o his ques ion is
a om i ial in a quasi-one-dimensional quan um wi e o med by
applying a con ining po en ial o a 2DEG . On he one hand he
combina ion o he quan iza ion o mo ion along he o hogonal axis and
he p esence o an in insic Rashba spin-o bi in e ac ion gi es ise o
in e -subband mixing ha can s ongly a ec anspo p ope ies o he
nanowi e, o ins ance supp essing spin-modula ion o la ge alues o
Rashba coupling [19, 20, 21, 22]. On he o he hand he p esence o a
sca e ing cen e , as o example an impu i y, may lead o o ma ion o
quasi-bound s a es localized a ound he impu i y. P e ious s udies ha e
shown ha he p esence o e anescen modes leads o unusual p ope ies in
he anspo such as pe ec anspa ency when he Fe mi ene gy
app oaches subband minima and he blocking o channels due o
quasi-bound s a es localized a ound an a ac i e impu i y [23, 24].
Combina ion o bo h spin-o bi in e ac ion and impu i y sca e ing emains
almos unexplo ed .
In his hesis we add ess his issue and p esen a ho ough heo e ical
s udy o he elec onic anspo in semiconduc ing nanowi es wi h Rashba
11
CHAPTER 1. INTRODUCTION
spin-o bi coupling in he p esence o impu i ies. We model he nanowi e as
a quasi-one-dimensional sys em whe e he mo ion o he elec ons is
con ined in he di ec ion pe pendicula o he anspo di ec ion. The
in e play be ween he spin-o bi coupling, con inemen and impu i y
po en ial, makes he p oblem highly non- i ial. We ackle his issue
h ough a combina ion o heo e ical echniques and app oxima ions, which
allows us o iden i y s iking no el p ope ies o bo h he cha ge and spin
anspo . Speci ically, we desc ibe he conduc ance by using he well
es ablished Landaue -B¨u ike o malism, which we ex end o he case o
spin-dependen ields. Wi hin his o malism he anspo is desc ibed in
e ms o he sca e ing coe icien s. In o de o calcula e hese coe icien s
we use he Lippmann-Schwinge equa ion, a widely used me hod o ea
sca e ing in quan um mechanics [36, 37, 38, 20, 39]. One o he main
di icul ies when dealing wi h spin-o bi coupling in a con ining po en ial is
he in e mixing o subbands. In o de o o e come his p oblem we
in oduce he Sch ie e -Wol ans o ma ion wi h which we gauge away
his in e mixing while s ill accoun ing o i s e ec s.
By he combina ion o he abo e echniques we compu e he cha ge and
spin conduc ances o he Rashba nanowi e in he p esence o a poin -like
impu i y. We ind ha he impu i y couples e anescen and p opaga ing
s a es in he nanowi es, inducing quasi-bound s a es; while he Rashba
spin-o bi in e ac ion allows o di e en spin-dependen mechanisms o
elec onic anspo . As a esul , he cha ge conduc ance p esen s pe ec
ballis ic ansmission a he h eshold ene gy o a channel ha becomes
p opaga ing. In addi ion, below his h eshold ene gy he e appea s a dip
in he conduc ance as a consequence o he quasi-bound s a es s ongly
supp essing ansmission. We p o e ha o he lowes subbands a his
esonan ene gy all elec ons injec ed in a p epa ed spin-up s a e sca e
om he impu i y o a spin-down s a e. Fu he mo e, his spin- lip
mechanism is no only he only ansmission allowed a esonan ene gy bu
i is also enhanced. We de i e he exp essions o he spin cu en s in he
nanowi e and ind ou and exp ession o he spin-o bi o que induced by
he impu i y and he spin- lip mechanisms o anspo . While he e ec s
o Rashba spin-o bi coupling in he cha ge conduc ance a e qui e ele an ,
ou key esul consis s in inding he unde lying ela ion be ween he
spin- lip ansmission and he SU(2) ield.
The hesis is o ganized as ollows:
In Chap e s 2 and 3 we ex end he in oduc ion, by discussing gene al
12
CHAPTER 1. INTRODUCTION
concep s used in he es o he hesis. In pa icula , in Chap e 2 we
p o ide a b ie o e iew o sca e ing in semiconduc ing nanowi es and he
Lippmann-Schwinge equa ion, ou heo e ical ool o de e mine he
sca e ing coe icien s. In Chap e 3 we discuss he spin-o bi coupling in
low dimensional sys ems as 2DEGs and quasi 1D nanowi es. In he la e
case we explain he di icul y o diagonalizing he Hamil onian due o he
subband in e mixing.
In Chap e 4 we in oduce he Landaue -B¨u ike o malism o he
desc ip ion o quan um anspo . We ex end he cus oma y de i a ion o a
sys em wi h Rashba spin-o bi coupling. This au oma ically ex ends he
o malism o a spin-dependen si ua ion. The sys em now suppo s bo h
spin and cha ge cu en s and we in oduce he concep o spin-bias in he
leads connec ed o he nanowi e. The non-conse a ion o he spin-cu en
a bo h sides o he impu i y is in e p e ed as a spin-o bi o que a ising
om he spin- lip ansmission mechanisms induced in he impu i y by he
Rashba spin-o bi coupling. The main esul o his chap e is he
exp ession o he cu en s and o que in e ms o he sca e ing coe icien s.
In Chap e 5 we calcula e he sca e ing coe icien s explici ly o he
nanowi e. In o de o do his we pe o m a gauge ans o ma ion and de i e
an exp ession o he coe icien s accu a e up o second o de o
pe u ba ion in he spin-o bi coupling s eng h. As a i s s ep, we gauge
away he in e mixing o subbands due o Rashba spin-o bi in e ac ion by
pe o ming a Sch ie e -Wol ans o ma ion. This allows us o ob ain he
sca e ing s a es in he whole wi e by means o he Lippmann-Schwinge
equa ion ollowing he discussion o Chap e 2. In a second s ep, we
calcula e he ansmission coe icien s, which a e now spin-dependen due
o he Rashba spin-o bi coupling. We iden i y wo ypes o ansmissions:
one ha p ese es he spin o he sca e ed pa icle and one ha lips i .
In Chap e 6 we p esen he main esul s o he anspo p ope ies o
he nanowi e. We ocus on bo h, cha ge and spin cu en s. We show ha
he conduc ance p esen s s iking ea u es ela ed o he p esence o quasi-
bound s a es localized a ound he impu i ies. We show ha a non-magne ic
impu i y can lip spin as a consequence o Rashba spin-o bi coupling, and
ha he spin- lip ansmission e lec s simila esonan beha io . As a esul ,
we p o e ha a he esonan ene gy he only ansmission allowed is h ough
he spin- lip mechanism and we discuss how his measu able ansmission
SU(2) symme y ac o s. This esul pa es he way o a sensi i e in e e ence
echnique o measu e he SU(2) gauge ield in nanowi es.
13
CHAPTER 1. INTRODUCTION
Each chap e has i s own conclusion sec ion. Ne e heless we b ie ly
summa ize he whole hesis in Chap e 7.
14
Chap e 2
Theo e ical desc ip ion o
anspo in semiconduc ing
nanowi es
Semiconduc o s a e a he hea o mode n elec onics. In pa icula , hey
a e impo an building blocks o nanos uc u es wi h e sa ile applica ions
due o he accu a e con ol o elec onic anspo ia doping and ex e nal
elec ic ields [40]. Fu he mo e, he ad ances in g ow h echniques such as
molecula beam epi axy (MBE) and pa e ning echniques, allows o c ea e
high-qua li y, meaning highe elec on mobili y, he e os uc u es ha
exhibi quan um con inemen e ec s and a a ie y o quan um phenomena.
The disco e y o conduc ance quan iza ion in low-dimensional sys ems (see
Fig. 2.1)launches an in ensi e esea ch o anspo p ope ies ela ed o
he cha ge o he elec on [41]. In addi ion, he ield o spin onics ex ended
he esea ch o spin-dependen anspo phenomena, and he use o
semicoduc o s o he design, and ab ica ion o no el spin-based elec onic
de ices. The idea behind possible applica ions in his ield elies on he
con ol o he spin dynamics and elaxa ion by means o ex e nal ields.
The co ne s one o many o he ad ances in hese ields a e
wo-dimen ional elec on gas (2DEG) ypically o med a he in e ace o
III-V semiconduc o he e os uc u e which lead o he obse a ion o new
in e es ing phenomena, absen in bulk sys ems, such as Shubniko -de Haas
oscilla ions [42, 43], he in ege [44] and ac ional quan um Hall e ec
[45, 46, 46, 47] and he quan ized conduc ance [48, 49].
In a wo-dimensional elec on gas elec ons a e con ined o a na ow
15
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
Va UME 60, NUMBER 9PHYSICAL REVIEW LETTERS 29 FEaRU~RV 1988
15
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GATE VOLTAGE IV)
—12
GATE VOLTAGE (V)
FIG. 1. Poin -con ac esis ance as a unc ion o ga e ol -
age a 0.6K. Inse : Poin -con ac layou .
FIG. 2. Poin -con ac conduc ance as a unc ion o ga e
ol age, ob ained om he da a o Fig. 1a e sub ac ion o
he lead esis ance. The conduc ance shows pla eaus a mul i-
ples o e/xh.
pinched o a Vg =—
2.2V.
We measu ed he esis ance o se e al poin con ac s
as a unc ion o ga e ol age. The measu emen s we e
pe o med in ze o magne ic ield, a 0.6K. An ac lockin
echnique was used, wi h ol ages ac oss he sample kep
below kT/e, o p e en elec on hea ing. In Fig. 1 he
measu ed esis ance o apoin con ac as a unc ion o
ga e ol age is shown. Unexpec edly, pla eaus a e ound
in he esis ance. In o al, six een pla eaus a e obse ed
when he ga e ol age is a ied om —
0.6 o —
2.2V.
The measu ed esis ance consis s o he esis ance o he
poin con ac , which changes wi h ga e ol age, and a
cons an se ies esis ance om he 2DEG leads o he
poin con ac . As demons a ed in Fig. 2, aplo o he
conduc ance, calcula ed om he measu ed esis ance
a e sub ac ion o alead esis ance o 400 0, shows
clea pla eaus a in ege mul iples o e/&A. The abo e
alue o he lead esis ance is consis en wi h an es-
ima ed alue based on he lead geome y and he esis-
i i y o he 2DEG. We do no know how accu a e he
quan iza ion is. In his expe imen he de ia ions om
in ege mul iples o e/zh migh be caused by he unce -
ain y in he esis ance o he 2DEG leads. Inse ing he
poin -con ac esis ance a V~= —
0.6V(750 0) in o
Eq. (1) we ind o he wid h W,„=360nm, in eason-
able ag eemen wi h he li hog aphically de ined wid h
be ween he ga e elec odes.
The a e age conduc ance inc eases almos linea ly
wi h ga e ol age. This indica es ha he ela ion be-
ween he wid h and he ga e ol age is also almos
linea . F om he maximum wid h W,„(360nm) and
he o al numbe o obse ed s eps (16) we es ima e he
inc ease in wid h be ween wo consecu i e s eps o be 22
nm.
We p opose an explana ion o he obse ed quan iza-
ion o he conduc ance, based on he assump ion o
quan ized ans e se momen um in he con ac cons ic-
ion. In p inciple his assump ion equi es acons ic ion
much longe han wide, bu p esumably he quan iza ion
is conse ed in he sho and na ow cons ic ion o he
expe imen . The poin -con ac conduc ance G o ballis-
ic anspo is gi en by "
G=e NpW(I /2m)( [k„~ ).
The b acke s deno e an a e age o he longi udinal wa e
ec o k, o e di ec ions on he Fe mi ci cle, Np
=m/eh 2is he densi y o s a es in he wo-dimensional
elec on gas, and Wis he wid h o he cons ic ion. The
Fe mi-ci cle a e age is aken o e disc e e ans e se
wa e ec o s k» =~nz/W (n =1,2,...), so ha we can
w i e
T
&Ik. l&= Jd'k ak, )&(k —
kF) g6' k»—
7C F8',-)8' (3)
Ca ying ou he in eg a ion and subs i u ing in o Eq. (2), one ob ains he esul
N, (4)
whe e he numbe o channels (o one-dimensional subbands) N, is he la ges in ege smalle han kFW/x. Fo
849
Figu e 2.1: Quan ized conduc ance o a poin -con ac as a unc ion o ga e
ol age in a GaAs/InGaAs he e os uc u e. The conduc ance changes in
quan ized s eps o 2e2/h. This igu e aken om [48].
quan um well (QW) along he g ow h di ec ion. Elec ons can only mo e in
he plane pe pendicula o ha di ec ion. T anspo p ope ies, and band
s uc u e o 2DEGs can be modi ied by in oducing dopan s du ing g ow h,
which con ibu e wi h elec ons o holes o he QW, and by ca e ully
choosing he ma e ials in he quan um well and he ba ie s [47].
The wo ways o ealizing 2DEGs a e by band in e sion and
he e os uc u e based sys ems. In Fig. 2.2 we can see he elec os a ic
po en ial Vz(z) (along he g ow h di ec ion z) expe ienced by conduc ion
band elec ons in wo si ua ions: one shows a iangula quan um well,
which o ms, e.g., a he in e ace be ween n-doped AlGaAs and undoped
GaAs and he o he squa e quan um well, whe e a hin laye o he
semiconduc ing ma e ial suppo ing he 2DEG, he e GaAs, is sandwiched
be ween laye s o a di e en semiconduc ing ma e ial, he e AlGaAs [50, 51].
In he i s case, he Fe mi ene gy o bo h will align a he in e ace o
he semiconduc o s, whe e ansla ional in a iance is b oken, wi h elec ons
coming ou om he n-AlGaAs lea ing behind an accumula ion o holes
which leads o a bending o he conduc ion and alence band [52]. Fo la ge
enough hole concen a ion, he conduc ion band dips below he Fe mi
16
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
su ace
F
Figu e 2.2: (a) T iangula quan um well in an in e sion laye semiconduc o
he e oes uc u e. (b) Squa e quan um well in sandwich-like he e oes uc u e.
ene gy a he su ace. We can see hen, ha he elec on densi y is sha ply
peaked nea he GaAs-AlGaAs in e ace o ming a hin conduc ion laye o
hickness compa able o he Fe mi wa eleng h, ha is he wo-dimensional
elec on gas, hence he name in e sion laye . This laye is o med na u ally,
howe e , he mobili y o laye -in e sion 2DEGs is se e ely limi ed.
Fu he mo e, since he elec ons li e a his in e ace, he quali y o he
2DEG is highly dependen on de ails o ab ica ion. To achie e highe
mobili ies he quan um well has o be deepe [53, 54].
This can be done by in oducing dopan s in sandwiches o ma e ials
wi h di e ing bandgaps. When wo such enginee ed ma e ials wi h unequal
bandgaps a e b ough in o con ac , he Fe mi ene gy o he wo ma e ials
will align and can o m a quan um well. To align he chemical po en ial in
he InGaAs/InAs/InGaAs sandwich s uc u e, cha ge is ans e ed om
emo e dopan s, in oduced du ing g ow h, and in o he quan um well.
Wi h he help o ga e elec odes (ex e nal elec ic ields), o by cle e
sample ab ica ion, a la ge a ie y o po en ial ene gy s uc u es can be
achie ed in he 2DEG. This way he mo ion o elec ons can be u he
con ined wi hin he 2DEG semiconduc o he e os uc u e plane leads o
(quasi) one-dimensional quan um wi es and ze o dimensional quan um
do s. Quan um con inemen gi es ise o new and undamen ally impo an
physics phenomenon, i is he e o e in e es ing o s udy quan um anspo
h ough hese quan um con ined mesoscopic sys ems [55].
The main ocus in his hesis is he heo e ical s udy o anspo h ough
a quan um nanowi e in he p esence o an impu i y o de ec . The e o e we
summa ize in his chap e he main heo e ical ools o i s desc ip ion.
17
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
2.1 Theo e ical app oach o sca e ing in
nanowi es
Quan um wi es ha e been p oposed as basic elemen s in he design o many
quan um de ices. Because o hei size he elec onic anspo ely on a
ull quan um mechanical app oach a he han a classical one. The mos
impo an pa ame e s (o leng h scales) ha desc ibe a semiconduc o a e
he phase cohe ence leng h `φ, Fe mi wa eleng h λF, and he mean ee pa h
`eo elec ons. In a mesoscopic de ice he `φis much la ge han he physical
dimensions (leng h Land wid h W) o he de ice, while he λFis compa able
o hese dimensions. I `eis much la ge han Land W he de ice is in he
ballis ic egime in which elec ons p opaga e h ough he de ice wi hou
being sca e ed, ei he elas ically o inelas ically, by impu i ies o phonons
espec i ely.
In his hesis we a e in e es ed in anspo h ough a nanowi e in he
p esence o an impu i y o de ec . In pa icula we desc ibe how he elec onic
anspo is a ec ed by he p esence o he impu i y, o in o he wo ds how
he nanowi e conduc ance depends on impu i y and in insic p ope ies o
he wi e. Because i is essen ial o ou nex analysis, we in oduce he e
he Lippmann-Schwinge equa ion, which we will use o he desc ip ion o
quan um sca e ing .
We s a discussing his app oach in a gene al 3D si ua ion and hen
we ocus on sca e ing on a del a-po en ial in a pu ely 1D sys em. We will
discuss he limi a ions o he Bo n app oxima ion. Finally we ocus on a
mo e ealis ic nanowi e desc ibed by a ans e se con ining po en ial
The main goal in sca e ing heo y is o ob ain he wa e unc ions
desc ibing he sca e ing pa icle gi en a p ope bounda y condi ions
imposed, by he incoming pa icle. We can hen begin by cons uc ing he
solu ion o he Sch ¨odinge equa ion mee ing hese wo c i e ia in o mal
e ms.
2.1.1 The Lippmann-Schwinge Equa ion: heo e ical
desc ip ion
Following Re .[56], we conside a sys em desc ibed by he ollowing
Hamil onian
H=H0+V , (2.1)
18
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
wi h H0=p2
2m∗
edesc ibes he ee elec ons wi h mass m∗
eand V ep esen ing
he sca e ing po en ial.
The goal is o ind a solu ion o he s a es |ψio he Sch dinge equa ion:
H|ψi=E|ψi,(2.2)
such ha in he limi ing o a anishing po en ial V→0 he solu ion eco e ed
would be ha o he unpe u bed sys em H0|φi=E|φi, i.e. |ψi→|φi.
Fo mally one can w i e
|ψi=V
E−H0|ψi.(2.3)
Then one can w i e he solu ion o Eq. (2.2) as a sum o he pa icula
solu ion |φiand he homogeneous solu ion in Eq.(2.3) as ollows,
|ψi=1
E−H0
V|ψi+|φi.(2.4)
Some complica ions a ise om he singula na u e o [E−H0]−1as he
con inuous spec um o H0will include E. This p oblem can be
ci cum en ed by subs i u ing E→E±ias a way o encode he bounda y
condi ions o in eg a ion, so one may w i e he so-called
Lippmann-Schwinge equa ion as ollows,
|ψ(±)i=|φi+1
E−H0±iV|ψ(±)i,(2.5)
The physical meaning o (±) will be discussed la e by e alua ing |ψ(±)ia
long dis ances. Fo he momen , we w i e he Lippmann-Schwinge equa ion
in he coo dina e basis,
h |ψ(±)i=h |φi+Zd 0h |1
E−H0±i| 0ih 0|V|ψ(±)i.(2.6)
In o de o sol e his in eg al equa ion one mus i s e alua e he ke nel
de ined by,
G±( , 0) = h |1
E−H0±i| 0ih 0|,(2.7)
which is no hing mo e han he G een’s unc ion o he Helmhol z equa ion,
∇2+k2G±( , 0) = 2m∗
e
~2δ( , 0).(2.8)
19
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
The di icul y o mos sca e ing p oblems lies in inding he p ope G een’s
unc ion ha sol es Eq. (2.8) o he sys em. Fo he momen , in ou
de i a ion we may no w i e explici ly G±( , ‘0) so ha Eq. (2.6) eads,
h |ψ(±)i=h |φi+Zd 0G±( , 0)h 0|V|ψ(±)i.(2.9)
No ice ha he wa e unc ion h |ψ(±)iin he p esence o he sca e e is
w i en as a sum o he inciden wa e h |φiand a e m ha ep esen s he
sca e ing in e ac ion. In mos physical sys ems one wo ks wi h he posi i e
solu ion o he G een’s unc ion G+( , 0) as i sa is ies he so called
ou going bounda y condi ions (as opposed o he nega i e solu ion
G−( , 0) co esponding o he less in ui i e incoming bounda y
condi ions). This means ha G+( , 0) ga an ees an ou going low om 0
o choosing E−H0+ iin Eq. (6.4), while G−(x, x0) on he o he hand
leads o an incoming cu en om o 0choosing E−H0−i. F om he e
on, we assume he posi i e case and d op he (±) sign e e ence in ou
desc ip ion o no a ion simplici y.
Now, in o de o e alua e he speci ic beha io o h |ψ(±)imo e explici ly
le us conside a local po en ial, ha is a po en ial diagonal in he coo dina e
ep esen a ion. The po en ial Vis conside e o be local i i can be w i en
as
h 0|V| 00i=V( 0)δ( 0− 000),(2.10)
and as a esul ,
h 0|V|ψ(±)i=Zd 00h 0|V| 00ih 00|ψ(±)i
=V( 0)h 0|ψ(±)i.(2.11)
I we de ine he inciden wa e unc ion o be a plane wa e φ( ) = h |φi, hen
he equa ion Eq. (2.9) simpli ies o,
ψ( ) = φ( ) + Zd 0G( , 0)V( 0)ψ( 0),(2.12)
gi ing he sca e ing s a es o an incoming pa icle e alua ed a posi ion x.
Fo a ini e ange po en ial, he sca e ing s a e inside he suppo egion
will ha e a con ibu ion limi ed o his space. So in e ec he Lippmann-
Schwinge equa ion p o ides a way o s udy sca e ing p ocesses as a esul
20
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
41 EVANESCENT MODES AND SCATTERING IN QUASI-ONE-. .. 10 363
wi h inc easing s eng h o he sca e e . This is indeed
ue, al hough no shown in he igu es. In Fig. 4 he in-
e subband ansmission is small because he po en ial is
ela i ely weak. A he onse o he second subband in
Fig. 4(a} only abou 6% o he inciden ca ie s a e con-
e ed in o he second no mal mode h ough T,2, and
4—
5% a e con e ed in o he hi d no mal mode ia T,3
a he bo om o he hi d subband. Figu e 4(b) gi es
only be ween 1%%A and 2%%uo con e sion om he second o
he hi d mode a he bo om o he hi d subband ia
T23~
We can unde s and some ea u es o Fig. 4by a guing
om he Fe mi "golden- ule" sca e ing a e. To do his
we do no conside he in asubband ansmission T&&,
T~q, o T33 as hey a e simply he esul o le o e pa i-
cles which did no sca e and can be ob ained om he
equi emen s o cu en conse a ion. Conside i s he
in e subband ansmission T&2, T&3, and T23. The in e -
subband ansmission has amaximum nea he onse o a
subband and decays like he in e se squa e oo o ene gy
away om he maximum. This can be unde s ood om a
Fe mi's "golden- ule" iewpoin , whe e he p obabili y o
sca e ing is p opo ional o he inal densi y o s a es in
he subband which decays like I/~E. In Appendix B
we show ha he dominan e m in he in e subband
sca e ing p obabili y is indeed gi en by an exp ession
simila o he golden ule. The in e subband ansmis-
sion and e ec ion coe icien T,2=R,zin Fig. 4(a) also
sho~s in e es ing beha io a ound he bo om o he
hi d subband, s aying ze o on bo h sides o he subband
minima. The e is no sca e ing ou o mode one in o
mode wo a he bo om o he hi d subband. We ha e
ye o ind agood explana ion o his lack o mode con-
e sion o e lec ion a he subband minima. Howe e ,
he o e all shapes o he ansmission and e lec ion
coe icien s a e s ill well unde s ood by golden- ule a gu-
men s.
Gi en he golden- ule-like shapes o he in e subband
ansmission and e lec ion coe icien s and he in asub-
band e lec ion, we can a gue o he shape o he in-
asubband ansmission. Le us do so o T». Because
pa icles mus be conse ed so ha 1=T» +T,2
+R,2+R», and since R» =0 on bo h sides o he sub-
band minima, he d op in T» a e eaching pe ec
ansmission a he second subband mus he equal o
T,z+R» =2T&2, o jus wice he in e subband ansmis-
sion coe icien . This is shown in Fig. 4(a). Simila ly, he
discon inui y in T» in Fig. 4(a} a he minima o he
hi d subband is jus wice T,3.
Nex , le us examine he sca e ing coe icien s o an
a ac i e po en ial. Figu e 5shows a5- unc ion sca e -
e o compa able s eng h o he one in Fig. 4, bu when
IIII
020 40 60 80 100
Ene gy {meY}
FIG. 6. Two-p obe conduc ance h ough a5- unc ion de ec
in he quasi-one-dimensional wi e in uni s o 2e /h. The solid
line co esponds o he epulsi e sca e e om Fig. 4, while he
dashed line gi es he conduc ance o he a ac i e sca e e
om Fig. S. When he elec on ene gy aligns wi h asubband
minimum, he conduc ance h ough he de ec is equal o he
ballis ic conduc ance. A hese special ene gies he wi e is pe -
ec ly anspa en as i no sca e e we e p esen . The e is only
asmall di e ence be ween he conduc ance o he weak epul-
si e sca e e and he ideal ballis ic conduc ance h oughou he
en i e ange o elec on ene gies. Fo he a ac i e sca e e ,
he new dips in conduc ance co espond o quasi-bound-s a es
de eloping in he wi e. The dis ance in ene gy om hese dips
o he subband minimum is he quasi-bound-s a e ene gy. No e
also ha , e en hough he epulsi e sca e e is s onge , he
conduc ance o he a ac i e sca e e is much smalle due o
he p esence o he quasi-bound-s a e.
I2.0-
Cd
V
o1.0-
V
020
I I
40 60 80 100
Ene gy {meY}
FIG. 7. Two-p obe conduc ance in uni s o 2e /h o an a -
ac i e sca e e ha ing y=—
8 eV cm (solid line), y=—
9
eV cm (do ed line), and y=—
20 eV cm (dashed line). Begin-
ning wi h he do ed line om Fig. 6showing he weakes a -
ac i e sca e e ha ing y= —
6 eVc n, he o e all conduc-
ance le el dec eases and he new dips co esponding o he
quasi-bound-s a es mo e lowe in ene gy as he sca e e is
made mo e a ac i e. As he sca e e becomes so a ac i e
ha he quasi-bound-s a es mo e below he bo om o he nex
lowes subband, he new dips i s disappea and he conduc-
ance hen inc eases as he sca e e is made s onge . This
unusual e ec occu s because he bound s a es ha e now mo ed
below he ene gy ange in which hey can block conduc ion.
Figu e 2.3: Conduc ance h ough a del a-impu i y in he quasi-1D nanowi e
in uni s o 2e2/h. Fo he a ac i e sca e e 0<0 (dashed line), he
conduc ance a he h eshold ene gy is he ballis ic conduc ance and below
hese h eshold ene gies conduc ance dips appea in he co esponding o
quasi-bound s a es p esen in he nanowi e. Fo he epusi e sca e e 0>0
(solid line), he esonan dip is no p esen . This igu e aken om [23].
27
CHAPTER 2. THEORETICAL DESCRIPTION OF TRANSPORT IN
SEMICONDUCTING NANOWIRES
63, 64, 65], double-del a sca e e s[66], ini e-size sca e e s [67, 68, 69], and
e en in o magne ic impu i ies [70], in he p esence o an e e nal magne ic
ield [71] and o ime-dependen po en ials [72, 73, 68]. In all hese wo ks,
he esonan cha ac e is ics o he ansmission we e discussed in ela ion o
he p esence o quasi-bound s a es in he sys em. Howe e , o ou knowledge
he sca e ing om an impu i y in he nanowi e in he p esence o Rashba
spin-o bi coupling emains unexplo ed.
2.2 Conclusion
In his chap e we in oduce he Lippmann-Schwinge equa ion which we
will use o he heo e ical desc ip ion o quan um sca e ing o a
semiconduc ing nanowi e. In pa icula we desc ibe how he elec onic
anspo is a ec ed by he p esence o he impu i y in he nanowi e. We
i s desc ibe he app oach in a gene al sys em and la e ocus on sca e ing
on a del a-po en ial in a pu ely 1D sys em, discussing he limi a ions o he
Bo n app oxima ion. Finally we ocus on a mo e ealis ic nanowi e
desc ibed by a ans e se con ining po en ial, discussing he eme gence o
esonan beha iou in he ansmission as a consequence o quasi-bound
s a es p esen in he nanowi e. This e ec a ises om he localized
impu i y coupling he e anescen and p opaga ing modes o he nanowi e.
Ou in e es is in he possible e ec s a ising om he in e play be ween
he Rashba in e ac ion and quasi-bound s a es. Fo his eason, Chap e
3 in oduces he key ing edien in ou s udy, namely he Rashba spin-o bi
in e ac ion in quan um nanowi es.
28
Chap e 3
Spin O bi Coupling in
semiconduc o s
The ield o spin onics aims o c ea e de ices ha ake ad an ages o bo h,
he spin and cha ge deg ees-o - eedom o elec ons. In pa icula ,
semiconduc ing spin onics acili a e he s udy o he undamen al concep s
in he ield hanks o he easy in eg a ion wi h nowadays semiconduc o
elec onics. One o he key ing edien s ga ne ing a en ion in his ield is
he spin-o bi coupling, specially some o ms o symme y-dependen
spin-o bi coupling ealized in semiconduc ing he e os uc u es hos ing a
wo-dimensional elec on gas (2DEG). Such is he case o he Rashba
spin-o bi in e ac ion.
In his Chap e we p o ide a b ie in oduc ion o spin-o bi in e ac ion
o he Rashba ype. We discuss he spec al p ope ies o low dimensional
s cu u es wi h Rashba spin-o bi coupling and p o ide a b ie o e iew o
wo possible applica ions o ma e ials wi h Rashba spin-o bi coupling,
namely he Da a-Das spin- ansis o and he de ec ion o Majo ana Bound
S a es. Ou in e es in Rashba spin-o bi coupling is ela ed o
spin-dependen anspo in quan um nanowi es. Fo his eason, we i s
in oduce he Hamil onian model o a 2DEG and discuss i s spec um and
symme ies. By u he con ining he 2DEG, one can c ea es a quasi one
dimensional guide, o quan um nanowi e . The con inemen po en ial add
complexi y o he p oblem, and he Hamil onian is no longe anali ically
sol able wi hou app oxima ions. Indeed, he subbands a e de o med and
a oided c ossings appea in he ene gy dispe sion, as we desc ibe in his
chap e . La e , in Chap e 5, we will ea Rashba spin-o bi in e ac ion in
29
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
a pe u ba i e way. The e o e in he p esen chap e we p esen he exac
solu ion o he momen um in he ene gy dispe sion whe e he subbands
c oss ( he only poin ha p esen s spin degene acy) which we will use as
he unpe u bed solu ion.
3.1 In oduc ion o Spin-O bi Coupling
The spin-o bi coupling (SOC) is a widely s udied e ec ha desc ibes he
in e ac ion be ween he spin o a pa icle wi h i s mo ion in he p esence o
an elec ic ield. And i can be desc ibed by ollowing Hamil onian [56],
HSO =~
4m2c2ˆ
~σ (∇V×p),(3.1)
whe e ˆ
~σ = (ˆσx,ˆσy,ˆσz) is he ec o o Pauli ma ices, mis he es mass o he
elec on, V( ) is he elec os a ic po en ial in which he elec on p opaga es
wi h momen um p. Fo example, in a omic physics V( ) is he Coulomb
po en ial o he a omic co e.
In semiconduc o physics V( ) is he po en ial o a c ys alline la ice
ha a ises om he hyb idiza ion o he elec on o bi als o neighbo ing
a oms. The spec al p ope ies o hese elec ons a e cha ac e ized by he
band ene gy En(k) and a ec ed by he spin-o bi coupling. The e ec s o
spin-o bi coupling in InAs, GaAs, InSb o o he ma e ials ha a e commonly
used in he ealiza ion o nanowi es, whe e he ene gy o he op alence band
is s ongly spli ed in subbands depending on spin. [74, 75, 76].
Fu he mo e, he lack o cen o-symme y in he zinc-blende s uc u e
o III-V c ys als and he con inemen o 2DEGs allows o signi ican
in e sion asymme y spin-o bi coupling e ec s in he la ice po en ial,
li ing he spin degene acy by spli ing he ene gy bands in he absence o a
magne ic ield. The e ec s o his ype o spin-o bi coupling can be be e
unde s ood by explo ing he ela ion be ween symme y and band spli ing,
speci ically ime- e e sal symme y (TRS) and spa ial in e sion symme y
(SIS).
Spa ial In e sion and Time Re e sal Symme ies
The ela ion be ween symme y and he spli ing o he bands is
undamen al o he unde s anding o some ypes o spin-o bi coupling
30
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
Re e sed P ese ed
p→ −p q →q
B→ −B E →E
σ→ −σ p2/2m→p2/2m
Table 3.1: Obse ables p ese ed and e e sed unde he ime e e sal
ans o ma ion.
[77]. The i s symme y o ele ance is he ime e e sal symme y. When
a sys em unde goes a ime e e sal ans o ma ion T: → − cha ac e ized
by he ime e e sal ope a o T, some obse ables a e p ese ed while
o he s a e e e sed. Some o hese obse ables a e p esen ed in able 3.1.
Then,unde he e e sal o ime: because he angula momen um is
e e sed
L
→ −
L
and he so is he spin σ→ −σ, he spin-o bi is p ese ed
L
·σ→
L
·σand he momen um
k
→ −
k
. I a sys em is symme ic unde
ime e e sal (and he spin is hal -in ege ), he K ame s heo em implies
En(σ, k) = En(−σ, −k) (3.2)
o any band ene gy o a gi en spin σand o a gi en momen um
k
,
co esponds a ene gy degene a e band wi h opposi e spin −σand opposi e
momen um −
k
.
The o he impo an symme y is he spa ial in e sion symme y. Unde
space e e sal R:
→ −
, while
L
→
L
and σ→ −σand consequen ly he
spin-o bi
L
·σ→ −
L
·σand momen um
k
→ −
k
. Then, in he case o a
sys em wi h spa ial in e sion symme y,
En(σ, k) = En(σ, −k) (3.3)
meaning ha o any band wi h gi en spin σand momen um
k
, he e is
ano he degene a e band wi h same spin σand opposi e momen um −
k
.
And i he sys em p esen s bo h ime e e sal and spa ial in e sion
symme ies, hen
En(σ, k) = En(−σ, k).(3.4)
Thus, in a sys em wi h wo spin eigens a es ↑and ↓ ha p esen s bo h space
in e sion symme y (SIS) and ime e e sal symme y (TRS) he ene gy
dispe sion o he wo subbands o e laps. Howe e , o sys ems wi h TRS
31
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
E
(a)
k
↓ ↑
E
(b)
k
↓↑
E
(c)
k
↓↑
Figu e 3.1: (a) Degene a e ene gy dispe sion o a sys em wi h TRS and
SIS.(b) Gaped spec um o a sys em whe e TRS is b oken.(c) Shi ed ene gy
dispe sion o a sys em whe e SIS is b oken and spin-degene acy is li ed.
bu b oken SIS he spin-↑and spin-↓subbands ha e di e en ene gy a a
gi en momen um k o he same spin so ha ,
En(σ, k)6=En(−σ, k) (3.5)
as is he case o sys ems wi h spin-o bi coupling in non-cen osymme ic
ma e ials (see Fig.3.1(c)). Fu he mo e, i ime e e sal symme y is b oken
he K ame s degene acy in Eq.(3.2) is li ed and
En(σ, k)6=En(−σ, −k) (3.6)
case when an ex e nal magne ic ield is applied o he sys em (see Fig.3.1(b)).
Thus a po en ial ha b eaks spa ial in e sion symme y li s
spin-degene acy as s a ed in Eq.(3.5) while a po en ial ha b eaks ime
e e sal symme y li s spin-degene acy and K ame s degene acy as seen in
Eq.(3.6).
Symme y dependen spin-o bi coupling
As desc ibed by Eq.(3.1) he main sou ces o SOC a e elec ic ields,
o igina ing om asymme ies o he c ys alline po en ial h ough i s
g adien ∇V. The e o e, i is an in insic e ec , s ongly depending on he
ma e ial and i s s uc u e. As we al eady discussed in he case o
zincblende III-V he e os uc u es such as GaAs, AlGaAs, InAs, e c., hese
asymme ies b eak down he spa ial in e sion spin-spli ing he spec um
wi h wo possible o igins,
1. The i s one is bulk in e sion asymme y (BIA), i.e., con a y o o he
c ys alline s uc u es such as ha o silicon, he zinblende s uc u e
32
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
lacks an in e sion cen e . This asymme y is ixed o a gi en sample, is
in insic o he sys em and i is no possible o manipula e i ex e nally.
The spin-o bi coupling caused by his in e sion asymme y is known
as D esselhaus in e ac ion [78].
2. The second one is only possible in low dimensional sys ems whe e he
mo ion o elec ons is con ined o wo dimension (2DEG), o example
in quan um wells, whe e he e is a lack o in e sion symme y in he
g ow h di ec ion. This is he s uc u al in e sion asymme y (SIA), and
he impo ance o his mechanism lies in he ac ha he asymme y in
he con inemen po en ial can be a ied by elec os a ic means, allowing
o une he SOC s eng h by an ex e nal ga e ol age. The spin-o bi
in e ac ion co esponding o his asymme y is called Rashba spin-o bi
coupling (RSOC)[79].
The ela i e impo ance be ween bo h spin-o bi in e ac ions,
D esselhaus and Rashba, a ies depending on he band s uc u e o he
ma e ial, he elec on densi y and he geome y o he sample unde
in es iga ion. In na ow-gap III-V quan um wells, howe e , he Rashba
SOC is gene ally much la ge han he D esselhaus,as well as being mo e
in e es ing due o i s unabili y. As a consequence, in his hesis he ocus
will be on he Rashba in e ac ion, neglec ing he D esselhaus e m.
3.2 Applica ions o Rashba Spin-O bi
Coupling
In 1990 he i s applica ion o RSOC was p oposed as wha is known as
he Da a-Das ansis o o spin-Field E ec T ansis o (spin-FET)[15] bu
i was no ealized un il la e [80, 81]. This oy-model was de eloped as an
analog o he elec o-op ic modula o and is based on he spin p ecession
induced by he Rashba e ec .
I ollows om he gene al exp ession o he spin-o bi coupling in
Eq.(3.1), ha he Rashba SOC gi es ise o an in e nal magne ic ield
BRSOC and i can be w i en as BRSOC =α(Ez) (k×z), i.e., he
magni ude o he ield is p opo ional o he momen um kand a
ol age-dependen pa ame e α, and i is poin ing in he di ec ion
33
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
Semiconduc o 2DEG
Ga e
Vol age
Figu e 3.2: Skech o a Da a-Das spin ansis o , wi h 2DEG sandwiched in
be ween wo e omagne s. The injec ed spin can be con oled by uning he
RSOC, which in u ned is con olled by a ga e ol age. I he alignmen o
he elec on spins, as hey each he d ain, is pa allel o his e omagne ,
hen he ansis o will egis e a non-ze o cu en . On he con a y, i like
in his igu e he magne iza ion o he d ain e omagne is an ipa allel o
he elec on spins, he ansis o will egis e a ze o-cu en .
pe pendicula o bo h kand z(wi h zbeing he g ow h di ec ion o he
quan um well).
In he absence o an ex e nally applied magne ic ield, he spin will
p ecess a ound his e ec i e magne ic ield BRSOC in a simila way as he
La mo -p ecession a ound an ex e nal magne ic ield. The p ecession
equency depends on he magni ude o he in e nal magne ic ield |BRSOC|,
and hence can be uned by applying a ga e ol age [82, 83, 84, 85, 86]. This
p ope y has led o he p oposal o a Da a-Das ” oy-model” [15], also
known as he Da a-Das spin- ansis o .
Da a and Das conside a ballis ic anspo channel wi h Rashba SO
coupling in-be ween e omagne ic leads ac ing as spin pola ize s(see Fig.
3.2). When a spin is injec ed om one o he leads, i p ecesses a ound he
Rashba ield BRSOC un il he spin a i es a he o he e omagne ic lead
( he d ain). The elec on ansmission p obabili y in o he d ain depends on
he ela i e alignmen o i s spin wi h he magne iza ion o he d ain ( his
being ixed). Since he equency o he p ecession o he spin du ing he
a el o he d ain can be con olled ia ga e ol age, so can he sou ce-
o-d ain cu en (o conduc ance). The impo ance o his oy-model o
he ield o spin onics consis s no on i s physical ealiza ion bu on he
scien i ic discussion spa ked a ound i abou he ole o Rashba SOC in he
spin dynamics o 2DEG, i s in e play wi h D esselhaus SOC and Zeeman. In
34
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
e u n, his esea ch has lead o he disco e y o new spin- ela ed phenomena
and hei applicabili y in new de ices.
NATURE PHYSICS DOI: 10.1038/NPHYS1915 ARTICLES
s-wa e supe conduc o
Semiconduc ing wi e
B
x
y
z
E
k
μ
B, 1
A, 1 B, 2
A, 2 A, 3 B, 3 B, N
A, N
a
bc
γγ γ γγγ γγ
Figu e 1 |Majo ana e mions appea a he ends o a 1D ‘spinless’ p-wa e
supe conduc o , which can be expe imen ally ealized in semiconduc ing
wi es21,22.a, Pic o ial ep esen a ion o he g ound s a e o equa ion (1) in
he limi µ=0, =|1|. Each spinless e mion in he chain is decomposed in
e ms o wo Majo ana e mions γA,xand γB,x. Majo anas γB,xand γA,x+1
combine o o m an o dina y, ini e-ene gy e mion, lea ing wo ze o-ene gy
end Majo anas γA,1and γB,Nas shown23.b, A spin–o bi -coupled
semiconduc ing wi e deposi ed on an s-wa e supe conduc o can be d i en
in o a opological supe conduc ing s a e exhibi ing such end Majo ana
modes by applying an ex e nal magne ic ield21,22.c, Band s uc u e o he
semiconduc ing wi e when B=0 (dashed lines) and B6=0 (solid lines).
When µlies in he band gap gene a ed by he ield, pai ing inhe i ed om
he p oxima e supe conduc o d i es he wi e in o he opological s a e.
cha ac e is ics o Majo ana e mions— hey a e hei own
an ipa icle and cons i u e ‘hal ’ o an o dina y e mion. In his
limi he Hamil onian becomes
H=−i
N−1
X
x=1
γB,xγA,x+1
Consequen ly, γB,xand γA,x+1combine o o m an o dina y e mion
dx=(γA,x+1+iγB,x)/2, which cos s ene gy 2 , e lec ing he wi e’s
bulk gap. Conspicuously absen om H, howe e , a e γA,1and γB,N,
which ep esen end-Majo ana modes. These can be combined in o
an o dina y (al hough highly non-local) ze o-ene gy e mion dend =
(γA,1+iγB,N)/2. Thus he e a e wo degene a e g ound s a es which
se e as opologically p o ec ed qubi s a es: |0iand |1i=dend†|0i,
whe e dend|0i=0. Figu e 1a illus a es his physics pic o ially.
Away om his limi he Majo ana end s a es no longe e ain
his simple o m, bu su i e p o ided he bulk gap emains ini e23.
This occu s when |µ|<2 , whe e a pa ially illed band pai s. The
bulk gap closes when |µ|=2 . Fo la ge |µ|, pai ing occu s in a
ully occupied o acan band, and a i ial supe conduc ing s a e
wi hou Majo anas eme ges.
Realizing Ki ae ’s opological supe conduc ing s a e expe imen-
ally equi es a ‘spinless’ sys em ( ha is, wi h one pai o Fe mi
poin s) ha p-wa e pai s a he Fe mi ene gy. Bo h c i e ia can
be sa is ied in a spin–o bi -coupled semiconduc ing wi e deposi ed
on an s-wa e supe conduc o by applying a magne ic ield21,22 (see
Fig. 1b). The simples Hamil onian desc ibing such a wi e eads
H=Zdxψx†−¯
h2∂x2
2m−µ−i¯
huˆ
e·σ∂x
−gµBBz
2σzψx+(|1|eiϕψ↓xψ↑x+h.c.)(3)
The ope a o ψαxco esponds o elec ons wi h spin α, e ec i e
mass m, and chemical po en ial µ. (We supp ess he spin indices
excep in he pai ing e m.) In he hi d e m, udeno es he
spin–o bi 31,32 s eng h, and σ=(σx,σ y,σz) is a ec o o Pauli
ma ices. This coupling a ou s aligning spins along o agains he
uni ec o ˆ
e, which we assume lies in he (x,y) plane. The ou h
e m ep esen s he Zeeman coupling due o he magne ic ield
Bz<0. No e ha spin–o bi enhancemen can lead33 o g2.
Finally, he las e m e lec s he spin-single pai ing inhe i ed om
he supe conduc o by means o he p oximi y e ec .
To unde s and he physics o equa ion (3), conside i s
Bz=1=0. The dashed lines in Fig. 1c illus a e he band
s uc u e he e—clea ly no ‘spinless’ egime is possible. In oducing
a magne ic ield gene a es a band gap ∝|Bz|a ze o momen um, as
he solid line in Fig. 1c depic s. When µlies in his gap he sys em
exhibi s a single pai o Fe mi poin s as desi ed. Tu ning on 1
weakly compa ed o he gap hen e ec i ely p-wa e pai s e mions
in he lowe band wi h momen um kand −k, d i ing he wi e
in o Ki ae ’s opological phase21,22. (Single pai ing in equa ion (3)
gene a es p-wa e pai ing because spin–o bi coupling a ou s
opposi e spins o kand −ks a es.) Quan i a i ely, ealizing he
opological phase equi es21,22 |1|<gµB|Bz|/2, which we he ea e
assume holds. The opposi e limi |1|>gµB|Bz|/2 e ec i ely
iola es he ‘spinless’ c i e ion because pai ing s ongly in e mixes
s a es om he uppe band, p oducing an o dina y supe conduc o
wi hou Majo ana modes.
In he opological phase, he connec ion o equa ion (1) becomes
mo e explici when gµB|Bz| mu2,|1|whe e he spins nea ly
pola ize. One can hen p ojec equa ion (3) on o a simple one-
band p oblem by w i ing ψ↑x∼(u(ey+iex)/gµB|Bz|)∂x9xand
ψ↓x∼9x, wi h 9x he lowe -band e mion ope a o . To leading
o de , one ob ains
He ∼Zdx9x†−¯
h2∂x2
2m−µe 9x
+|1e |eiϕe 9x∂x9x+h.c.(4)
whe eµe =µ+gµB|Bz|/2and he e ec i e p-wa epai ield eads
|1e |eiϕe ≈u|1|
gµB|Bz|eiϕ(ey+iex) (5)
The dependence o ϕe on ˆ
ewill be impo an below when we
conside ne wo ks o wi es. Equa ion (4) cons i u es an e ec i e
low-ene gy Hamil onian o Ki ae ’s model in equa ion (1) in he
low-densi y limi . F om his pe spec i e, he exis ence o end-
Majo anas in he wi e becomes mani es . We exploi his co espon-
dence below when add essing uni e sal p ope ies such as b aiding
s a is ics, which mus be sha ed by he opological phases desc ibed
by equa ion (3) and he simple la ice model, equa ion (1).
We now seek a p ac ical me hod o manipula e Majo ana
e mionsin he wi e.Asmo i a ion, conside applyingaga e ol age
o adjus µuni o mly ac oss he wi e. The exci a ion gap ob ained
om equa ion (3) a k=0 a ies wi h µas
Egap(k=0) =
gµB|Bz|
2−p|1|2+µ2
Fo |µ|< µc=√(gµBBz/2)2−|1|2 he opological phase wi h end
Majo anas eme ges, whe eas o |µ|> µca opologically i ial
phase appea s. A uni o m ga e ol age hus allows he c ea ion o
emo al o he Majo ana e mions. Howe e , when |µ|=µc he
bulk gap closes, and he exci a ion spec um a small momen um
beha es as Egap(k)≈¯
h |k|, wi h eloci y =2u|1|/(gµB|Bz|). The
gap closu e is clea ly undesi able, as we would like o manipula e
Majo ana e mions wi hou gene a ing u he quasipa icles.
This p oblem can be ci cum en ed by employing a ‘keyboa d’
o locally unable ga es as in Fig. 2, each impac ing µo e a ini e
NATURE PHYSICS |VOL 7 |MAY 2011 |www.na u e.com/na u ephysics 413
Figu e 3.3: (a) Ske ch o Majo ana Fe mions appea ing a he ends o a
nanowi e. Each spinless e mion in he chain is o med by he o e lap o
wo Majo ana e mions γA,x and γB,x. Majo anas γB,x and γA,x+1 o m
an o dina y e mion wi h ini e ene gy, lea ing wo uncombined Majo ana
Fe mions a he ends o he nanowi e. (b) Se -up o he obse a ion o
MF: a semiconduc ing nanowi e wi h spin-o bi coupling si s on op o a s-
wa e supe conduc o while an ex e nal magne ic ield Bis applied. (c) In
he absence o a magne ic ield (dashed lines), he ene gy spec um is spin-
spli , bu i an ex e nal magne ic ield is applied pe pendicula o he HSO a
helical gap opens (solid lines) and supe conduc i i y by p oximi y can d i e
he nanowi e o a opological s a e. Figu e aken om [87].
Mo e ecen ly he e has been a e i al o in e es in s udying SOC in
semiconduc ing hyb id s uc u es due o he possibili y o inding Majo ana
ze o modes hos ed in Rashba nanowi es in con ac wi h supe conduc ing
elec odes, which a e possible candida es o opological quan um
compu a ion due o hei non-Abelian s a is ics [88, 89, 90, 87]. The basic
idea is ha such a s uc u e can become a opological supe conduc o
35
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
unde he igh ci cums ances and suppo wo non-local Majo ana Bound
S a es a he ends o he nanowi e (see Fig. 3.3(a)). The s ong SOC
p esen in he nanowi e shi s he wo pa abolic bands depending on hei
spin pola iza ion and applying an ex e nal magne ic ield pe pendicula o
he SO ield b eaks he TRS o he sys em opening a gap a he c ossing
poin o he pa abolas (k= 0), as seen in Fig. 3.3(c). I he Fe mi ene gy µ
is inside he opened gap he degene acy is wo- old ins ead o ou - old. The
p oximi y o a s-wa e supe conduc o induces pai ing in he nanowi e
be ween elec on s a es o opposi e momen um and opposi e spins and
induces a supe conduc ing gap, ∆. Combining his wo- old degene acy
wi h an induced gap c ea es a opological supe conduc ing phase o
BZ>p∆2+µ2li ing elec on-hole symme y and Majo anas a ise as
ze o-ene gy (i.e. mid-gap) bound s a es, one a each end o he wi e
[1, 2, 3, 4, 5, 6]. A isualiza ion o such a se -up can be seen in Fig. 3.3(b).
3.3 The Rashba model o 2DEG
In his sec ion we ocus on how Rashba spin-o bi coupling a ec s he spec al
p ope ies o a ee elec on in a 2DEG, be o e going in o a desc ip ion o a
quan um nanowi e whe e u he con inemen is applied o he 2D sys em o
ob ain a quasi-1D sys em [91].
The e ec i e Hamil onian o an elec on mo ing in a 2DEG sys em in
he (x, y)−plane in he p esence o he Rashba spin-o bi coupling and wi h
e ec i e elec on mass meis gi en by,
H0=p2
2m∗+α
~(σ×p)z,(3.7)
wi h eigen alues
E±(k) = ~2k2
2m∗±αk =~2
2m∗(k±kR)2−∆R,(3.8)
whe e k=pk2
x+k2
yis he momen um, kR=αm∗
~2is he Rashba spin-o bi
coupling cons an wi h momen um dimensions and ∆R=αm∗
~2. The las
e m o Eq.(3.8) esul s in a downwa d shi o he bands ha eno malizes
he chemical po en ial, al ough i is o en neglec ed as i is second o de in
α.
36
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
Each block can be sol ed by shi ing pyas ollows,
ψ±(y) = e±iy/λSO Φn(y),(3.22)
whe e Φn(y) can be shown o sa is y he equa ion o he quan um ha monic
oscilla o in Eq.(2.26) and λSO =~/m∗
eα.
Bo h blocks ha e iden ical eigen alues and hei wa e unc ions a e ela ed
o each o he by a gauge ans o m. The solu ion o he ini ial p oblem
becomes
ψn,±(y, s) = e±iy/λSO Φn(y)χ±(s),(3.23)
wi h degene a e eigen alues
En,±≡En=~ω0n+1
2−m∗
eα2
2.(3.24)
No e ha he s a es ψnσ in Eq.(3.23) obey he o hono maliza ion
condi ion hψn0σ0|ψnσi=δn0nδσ0σwhe e he scala p oduc is aken in bo h
he y-coo dina e and he spin spaces,
hψn0σ0|ψnσi:= X
sZ+∞
−∞
dyψ∗
n0σ0(y, s)ψnσ(y, s).(3.25)
Howe e , wi hou summa ion o e he spin deg ee o eedom he s a es ψn0σ0
and ψnσ o n06=na e o hogonal only p o ided σ0=σ,
Z+∞
−∞
dyψ∗
n0σ(y, s0)ψnσ(y, s)∝δn0n.(3.26)
This is due o he phase ac o e±iy/λSO d opping ou only when same spin
s a es a e in ol ed. To emphasize ha he wa e unc ion in Eq.(3.23) does
no sepa a e in o a p oduc o a y-coo dina e componen and a spin
componen , we w i e he s a es as
ψnσ(y, s) = eiˆσxy/λSO Φn(y)χσ(s).(3.27)
A p oduc o wo s a es wi hou summa ion o e he spin indices educes o
he di ec p oduc o he ope a o s eiˆσxy/λSO aken om each o he s a es
43
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
in Eq.(3.27). I is con enien o ep esen such a di ec p oduc simply by
supplying an index o he Pauli ma ix,
e−iˆσxy/λSO ⊗eiˆσxy/λSO →ei(ˆσa−ˆσb)y/λSO ,(3.28)
whe e ˆσa
xand ˆσb
xha e sepa a e Hilbe spaces o he ime being, un il we
con ac he spin indices. The quan i y o in e es is, he e o e,
Z+∞
−∞
dyΦ∗
n0(y)Φn(y)ei(ˆσa
x−ˆσb
x)y/λSO ,(3.29)
which educes o he ollowing Fou ie ans o m
Fn0n(q) = Z+∞
−∞
dyΦ∗
n0(y)Φn(y)eiqy.(3.30)
No e ha Fn0n(q)=[Fnn0(−q)]∗and also ha Fn0n(0) = δn0n. Ac ually, we
will need Fn0n(q) e alua ed a q=±2/λSO.
The o m- ac o Fn0n(q) can be calcula ed o he case o ha monic
con inemen wi h he unc ions Φn(y) as gi en abo e. Since Φn(y) a e
chosen o be eal, we ha e Fn0n(q) = Fnn0(q), which subsequen ly leads o
he ela ion Fn0n(−q) = [Fn0n(q)]∗. Then, wi hou loss o gene ali y, we ake
n0≥nand ob ain
Fn0n(q) = 2n0n!
2nn0!Ln0−n
nq2λ2
y
2
×iqλy
2n0−n
exp −q2λ2
y
4,(3.31)
whe e Lα
n(ξ) is he Lague e polynomial,
Lα
n(ξ) = 1
n!eξξ−α∂n
∂ξn(e−ξξn+α).(3.32)
This limi case o kx= 0 se es as he unpe u bed solu ion o build upon
o he ollowing subsec ion.
3.4.2 Pe u ba i e solu ion a ound kx≈0
In o de o build a solu ion a ound kx= 0 we ake he unpe u bed
Hamil onian o be ha o Eq.(3.18) so ha he pe u bed sys em is gi en
by
H=H0+α~kx.(3.33)
44
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
whe e he e ms p opo ional o αkxcan be ea ed by pe u ba ion heo y.
Le us conside alues o kxwhich a e small enough, such ha he ollowing
egime holds
α~kxFnn0(q0)En−En0, n 6=n0.(3.34)
This condi ion oughly e e s o ”s aying away om he a oided c ossings”
and is equi alen o λy/λSO 1. This small pa ame e is e y impo an
as i appea s again in Chap e 5 in he con ex o pe u ba ion heo y bu
o he Sch ie e -Wol ans o ma ion. In his case, we ea he e m
α~kxσyas pe u ba ion, whe eas he e m −αpyσxis ea ed exac ly.
Howe e , in Chap e 5 he opposi e is ue:α~kxσyis ea ed exac ly, while
he e m −αpyσxis conside ed a pe u ba ion. The in e es in he
calcula ion p esen ed in he cu en subsec ion is o ind possible
con ibu ions o o de α2a ising om he e m −αpyσxalone. The eason
behing his is because hey may p esen co ec ions o he second o de
(α2) in ou calcula ion in Chap e 5.
One could expec ha in o de o de e mine his, i is su icien o conside
he poin kx= 0, which is exac ly sol able. Howe e , ha poin is degene a e
and we ha e o conside i s icini y o unde s and how he s a es p opaga e
and wha a e hei anspo p ope ies when sca e ing o an impu i y.
Fo his eason, we conside he ze o h-o de o pe u ba ion heo y in
he small pa ame e in Eq. (3.34). This co esponds o he degene a e
pe u ba ion heo y a ound he poin kx= 0 o each subband nsepa a ely.
While his app oach is alid o a s ong spin-o bi in e ac ion and a e y
small kx, we a e in e es ed he e in answe ing he ques ion abou he ole o
he second-o de co ec ions due o −αpyσx. In ma ix o m, he diagonal
(n0=n) pa o he Hamil onian o he sys em is gi en by,
ˆ
H1D
n=En−iαn~kx
iαn~kxEn,(3.35)
whe e he basis is gi en as be o e by he s a es in Eq. (3.23). We deno ed
αn=αFn(q0) wi h q0= 2/λSO.The o m- ac o Fn(q)≡ Fnn(q) is eal and
simpli ies o
Fn(q) = Lnq2λ2
2exp −q2λ2
4.(3.36)
Any co ec ion a ising om he o m ac o is is q2
0∝α2and by mul iplying
i by he α~kxo he pe u ba ion heo y in Eq.(3.35) i goes wi h α3and
45
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
hence is beyond he accu acy o any calcula ion done in his wo k. Howe e ,
we ca e abou any α2co ec ion a ising om he s a es and he e is also an
α2o e all ene gy shi , see Eq. (3.24).
The eigens a e o Eq. (3.35) co esponding o he ene gy
En,+=En+~kxαn,(3.37)
is cons uc ed ou o he s a es in Eq. (3.23)
χ+=1
21
1eiy/λSO +i
2−1
1e−iy/λSO .(3.38)
And he eigens a e co esponding o he ene gy
En,−=En−~kxαn,(3.39)
is cons uc ed as
χ+=1
21
1eiy/λSO −i
2−1
1e−iy/λSO .(3.40)
They bo h a e u he mul iplied by he same Φn(y) and by eikxx, since hese
o bi al componen s o he wa e unc ion a e in common o he subband n.
As expec ed, o his ze o h o de o pe u ba ion heo y in α~kx, he s a es
a e no a ec ed a all by he pa ame e αn. I en e s only in he ene gy and
oge he wi h he cons an e m ~k2
x/2m∗
ewill de e mine he di ision in o le
and igh mo e s.
The eigens a es o Eq. (3.35) can also be w i en in a compac o m. I
we mul iply bo h s a es by a phase ac o eiπ/4, hen we ob ain
χ+=
cos π
4+y
λSO
isin π
4+y
λSO
,
χ−=
isin π
4+y
λSO
cos π
4+y
λSO
.(3.41)
These s a es educe a y= 0 o
χ+=1
√21
i, χ−=1
√2i
1,(3.42)
46
CHAPTER 3. SPIN ORBIT COUPLING IN SEMICONDUCTORS
which a e eigens a es o σy. Howe e , a y6= 0, hey a e no longe eigens a es
o σy.
One can e i y ha he wo s a es in Eq. (3.41) o igina e om he non-
abelian gauge ac o
eiσxπ
4+y
λSO ,(3.43)
which mul iplies he usual up and down s a es o he σzPauli ma ix. As
a esul o his, he co ec ions o he s a es o o de α2a e no a ec ing
he sca e ing po en ial, because he abo e gauge ac o commu es wi h he
sca e ing po en ial. This is a ele an esul o all analy ical calcula ions
done in his hesis, as we now can ensu e he accu acy o he calcula ion
o he pe u ba i e me hods, up o he second o de α2, use in subsequen
chap e s.
3.5 Conclusions
In summa y, in his chap e we in oduce he Rashba spin-o bi coupling
going o e some o i s applica ions in he ield o spin onics and in he
de ec ion o Majo ana Bound S a es. We b ie ly discuss he spec al
p ope ies o a 2DEG wi h Rashba spin-o bi coupling be o e e iewing he
complexi ies in ol ed in he analy ical solu ion o he model Hamil onian
o a quan um nanowi e whe e he 2DEG is u he con ined. As a esul o
his con inemen , no only is he ene gy spec um o he sys em s ongly
a ec ed bu also he pola iza ion o he spin. The combined e ec o RSOC
and con inemen gi es aise o an i-c ossings be ween b anches o opposi e
spin, de o ming he spec um. Mo eo e , we discussed he subbband mixing
a he o igin o his phenomena which leads o coupling be ween
p opaga ing and e anescen s a es in he quan um nanowi e. As a esul , a
p ope calcula ion needs o ake in o accoun enough subbands. Finally, we
p esen an exac solu ion o he kx= 0 ha we use as he esul o he
unpe u bed p oblem o ob ain he solu ion o he p oblem in he icini y
o he poin kx≈0. This calcula ion allows us o ensu e he accu acy o ou
pe u ba i e app oach up o α2in Chap e 5.
47
Chap e 4
The Landaue -Bu ike
desc ip ion o T anspo
The Landaue -B¨u ike o malism is widely used as a me hod o s udy cha ge
anspo in mesoscopic sys ems [93, 94, 95]. I p o ides a e y in ui i e
desc ip ion o mac oscopic e ec s in e ms o sca e ing p ope ies, ha may
be ela ed o mic oscopic de ails o a sys em. In his chap e we gene alize
he Landaue -B¨u ike o malism o include he e ec s o Rashba spin-o bi
coupling. Besides he impo ance o such gene aliza ion, ou esul s will help
us o de i e ou esul s on anspo p ope ies o a nanowi e wi h Rashba
spin-o bi coupling and a single poin -like impu i y in he nex chap e s.
In he nex sec ions we p esen he Landaue -B¨u ike o malism o
desc ibe bo h he cha ge and spin conduc ance. In his way we p o ide a
simple desc ip ion o spin-dependen anspo ha allows us o sepa a e
he spin-bias and ol age-bias con ibu ions o he spin cu en . The
Landaue -B¨u ike o malism app oach o deal wi h spin cu en s has been
ea ed nume ically o discussed in some models in he p esence o Rashba
spin-o bi coupling o ei he he Sin Hall E ec o h ee- e minal
spin- il e s [96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106]. Ve y ecen ly
an ex ension o he o malism was de eloped o be e unde s and he
o igin and symme ies in ol ed in spin cu en s in magne ic mul i-laye ed
sys ems [107]. In his chap e we de i e an analy ical exp ession o he
spin cu en in e ms o he spin-dependen ansmission coe icien s and
discuss u he he implica ions o sca e ing a an impu i y on he
anspo . We iden i y a spin- o que as a consequence o spin- lip
ansmission mechanism media ed by such an impu i y (see subsec ion
48
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
(a)
sca e e
𝒂
𝑳↑ 𝒂
𝑹↑
𝒃
𝑳↑ 𝒃
𝑹↑
𝒂
𝑳↓ 𝒂
𝑹↓
𝒃
𝑳↓ 𝒃
𝑹↓
(b)
sca e e
𝑎
𝑎
𝑎
𝑎
𝑏
𝑏
𝑏
𝑏
Figu e 4.1: (a) Two- e minal spin-dependen geome y. (b) Fou - e minal
geome y.
4.3.1). In addi ion, we ind a conec ion be ween ou esul s and he concep
o he spin-mixing conduc ance in oduced in Re .[108].
4.1 The Landaue -B¨u ike app oach o
quan um anspo
The goal o he Landaue -B¨u ike app oach is o w i e exp essions o he
cu en in e ms o ansmission p obabili ies be ween di e en e minals
connec ed by a sca e ing egion. He e we ollow we Re e ence [109], and
gene alize he de i a ion o he o malism o he case o spin-dependen
anspo . Indeed, we a e conside ing wi es wi h Rashba spin-o bi
in e ac ion and he e o e he sca e ing ampli udes depends on he spin.
In Fig.4.1 (a) we show he ypical wo- e minal sys em. The sca e ing
egion, in ou case a nanowi e, is connec ed o wo ideal leads ha we e e
as le (L) and igh (R). We ea each spin species as independen
channels labeled by he index σ=↑,↓, see ske ch in Fig.4.1 (a). The leads
a e cha ac e ized by empe a u e Tασ and chemical po en ial µασ, wi h
α=L, R.
I is wo h no icing ha he wo- e minal spin-dependen geome y is
equi alen o he ou - e minal geome y ske ched in 4.1 (b), whe e he
empe a u es o he e minals a e T1,2=TL↑,L↓and chemical po en ials
49
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
µ1,2=µL↑,L↓, and simila ly T3,4=TR↑,R↓and µ3,4=µR↑,R↓. Thus, ou
analysis o 2- e minal se up o spin-dependen channels can be mapped o
a 4- e minal si ua ion wi h independen channels.
I is assumed ha he leads a e a local equilib ium and he e o e he
elec onic dis ibu ion unc ions in each lead is gi en by he Fe mi
dis ibu ion unc ion:
α,σ (E) = e(E−µα)/kBTα+ 1−1, α =L, R ;σ=↑,↓(4.1)
(see Fig.4.1). I is impo an o no e ha we a e conside ing he con ac
leads o be wide compa ed o he c osssec ion o he quasi-1D nanowi e, so
ha as a as he ese oi s a e conce ned, he nanowi e ep esen s only a
small pe u ba ion, and hus i is alid o desc ibe he local p ope ies in
e ms o an equilib ium s a e. E en hough he dynamics o he sca e ing
p oblem a e desc ibed in e ms o a Hamil onian, he p oblem conside ed is
i e e sible[109]. This means ha he p ocesses o a pa icle en e ing o
exi ing he nanowi e a e unco ela ed e en s; he ese oi s a e ully
de e mined by hei espec i e Fe mi dis ibu ions, and ac as pe ec
sou ces and sinks o he pa icles independen ly o he ene gy o he
pa icle en e ing o lea ing he nanowi e.
Be ween he leads we conside a ballis ic nanowi e wi h Rashba
spin-o bi coupling and a local impu i y po en ial ha will ac as a sou ce
o sca e ing inside he nanowi e. Fa om he impu i y we assume ha
ans e se mo ion longi udinal mo ion o pa icles a e sepa able. As
desc ibed in Sec ion 2.1.3 he mo ion om le o igh con ac
(longi udinal mo ion) is no -con ined and he sys em is cha ac e ized by he
conse ed wa e- ec o kn, whe e ndeno es he index numbe o ans e se
channels in oduced by he quan iza ion ac oss he leads in he ans e se
di ec ion co esponding o ans e se ene gies EL,R;n,σ, which can be
di e en o he le and igh leads. We deno e wi h NL,R (E) he numbe
o incoming channels in he le and igh lead, espec i ely.
We now in oduce he c ea ion and annihila ion ope a o s deno ed by α,
nand σ.The ope a o s ˆa†
αnσ (E) and ˆaαnσ (E) c ea e and annihila e elec ons
espec i ely, wi h o al ene gy Ein he ans e se channel nin he αlead,
which a e inciden upon he sample. Simila ly, he c ea ion ˆ
b†
αnσ (E) and
annihila ion ˆ
bαnσ (E) ope a o s e e o elec ons in he ou going s a es. They
50
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
obey an icommu a ion ela ions:
ˆa†
Lnσ (E) ˆaαn0σ0(E0) + ˆaαn0σ0(E0) ˆa†
αnσ (E) = δαβδnn0δσσ0δ(E−E0),(4.2a)
ˆaαnσ (E) ˆaαn0σ0(E0) + ˆaαn0σ0(E0) ˆaαnσ (E) = 0 ,(4.2b)
ˆa†
αnσ (E) ˆa†
αn0σ0(E0) + ˆa†
αn0σ0(E0) ˆa†
αnσ (E) = 0 .(4.2c)
We in oduce c ea ion and annihila ion ope a o s, ˆ
b†
αnσ (E) and ˆ
b†
αnσ (E),
and hei an icommu a ion ela ions in ou going s a es in he same way as
incoming s a es in eqs. (4.2a) o (4.2c).
The ope a o s ˆaαnσ (E) and ˆ
bαnσ (E) a e ela ed ough he sca e ing
ma ix Sas ollows,
ˆ
bL1↑
ˆ
bL1↓
...
ˆ
bLN↑
ˆ
bLN↓
ˆ
bR1↑
ˆ
bR1↓
...
ˆ
bRN↑
ˆ
bRN↓
=S
ˆaL1↑
ˆaL1↓
...
ˆaLN↑
ˆaLN↓
ˆaR1↑
ˆaR1↓
...
ˆaRN↑
ˆaRN↓
.(4.3)
We can w i e a simila equa ion o Eq. (4.3) o he he mi ian conjuga ed
ma ix S† ela ing he c ea ion ope a o s ˆa†
αnσ (E) and ˆ
b†
αnσ (E).
The ma ix Shas dimensions (NL+NR)×(NL+NR). I s elemen s a e
ene gy-dependen , and i has he ollowing block s uc u e
S= σσ0 0
σσ0
σσ0 0
σσ0.(4.4)
He e he diagonal blocks σσ0and 0
σσ0desc ibe elec on e lec ion o he le
and o he igh ese oi , espec i ely. The o -diagonal blocks σσ0and 0
σσ0
co espond o he elec on ansmission h ough he sample om he le
o he igh ese oi and om he igh o he le ese oi , espec i ely.
As discussed in Sec ion 2.1.3, he lux conse a ion in he sca e ing p ocess
implies he uni a i y o ma ix S. In addi ion, in he p esence o ime- e e sal
symme y as discussed in Sec ion ?? he sca e ing ma ix is also symme ic.
Ou goal in he ollowing sec ions is o desc ibe he anspo h ough a
nanowi e wi h an impu i y. Speci ically, we de i e an exp ession o he o al
51
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
cu en ope a o ˆ
IL( , ) in he le lead a away om he localized impu i y,
and ob ain an exp ession o he cha ge and spin-dependen conduc ances.
In he Landaue -B¨u ike o malism he cu en ope a o is exp essed in e ms
o he c ea ion and annihila ion ope a o s.
4.2 Cha ge conduc ance
The conse a ion o cha ge implies he con inui y equa ion in quan um
mechanics o he cha ge densi y ρ= eˆ
Ψ†ˆ
Ψ
dρ
d +∇·j= 0 .(4.5)
F om his exp ession and eh Sch ¨odinge equa ion we can de i e an
exp ession o he cha ge cu en densi y j. To do so we s a by
di e en ia ing wi h espec o ime he exp ession o he cha ge densi y
dρ
d = e "dˆ
Ψ†
d ˆ
Ψ + ˆ
Ψ†dˆ
Ψ
d #,(4.6)
whe e ˆ
Ψ = ˆ
Ψ1,ˆ
Ψ2Tis a wo-componen spino and ˆ
Ψ†i s he mi ian
conjuga e. We now make use o he Sh ¨odinge equa ion i~dˆ
Ψ
d =Hˆ
Ψ and
i s adjoin , whe e he Hamil onian o he sys em is gi en by,
H=−~2∇2
2m∗
e−i~α(∂xˆσy−∂yˆσx) + Vcon (y).(4.7)
On he one hand, he kine ic e m and he con inemen po en ial in Eq.4.7
leads us o w i e he kine ic con ibu ion o he e olu ion o he cha ge
densi y in Eq.(4.6) as
dρK
d =e
i~ˆ
Ψ†−−~2
2m∇2ˆ
Ψ−−−~2
2m∇2ˆ
Ψ†ˆ
Ψ + ˆ
Ψ†Vcon ˆ
Ψ−Vcon ˆ
Ψ†ˆ
Ψ
=−e~
2mi∇hˆ
Ψ†(∇ˆ
Ψ) −(∇ˆ
Ψ†)ˆ
Ψi,(4.8)
whe eas he con ibu ion om he spin-o bi coupling o he Hamil onian o
Eq.(4.6), usually e e ed as he anomalous e m o he cu en , leads on he
52
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
conduc ance as G=I/V wi h cu en Igi en by Eq.(4.29), esul ing in
G=1
V
e
2π~ZdE T † [ (E−µL)− (E−µR)]
=1
V
e
2π~ZdE T † [−(µL−µR) 0(E)]
=e2
2π~ZdE T † δ(E−EF).
In he las s ep in Eq.(??) we assume in he ze o- empe a u e limi he Fe mi
dis ibu ion unc ion is a s ep unc ion whose de i a i e becomes a del a in
ene gy − 0(E) = δ(E−EF). Finally we ob ain:
G=e2
2π~T †(EF) (EF).(4.30)
Eq. (4.30) is he well-known Landaue -B¨u ike exp ession. I es ablishes he
connec ion be ween he sca e ing ma ix and he conduc ance o he sys em.
We mus no ice ha independen ly o he choice o basis, he conduc ance
can be exp essed in e ms o ansmission p obabili ies Tn o each channel,
as he exp ession †(EF) (EF) is diagonalizable and hence T † =PnTn.
Fu he mo e, ano he e sion o Eq. (4.30) allows us o w i e he conduc ance
in e ms o he ansmission p obabili ies o elec ons lea ing he le lead
L om a channel nand wi h spin σ o a i e o he mchannel in he igh
lead Rwi h spin σ0,
G=e2
2π~X
mσ0,nσ | mσ0,nσ|2.(4.31)
Once we ha e e iewed he way o w i e he cha ge conduc ance in he
Landaue -B¨u ike o malism we will gene alize i in he nex sec ion o
he case o spin-dependen obse ables such as he spin-cu en and
spin-pola ized conduc ance.
4.3 Spin cu en along he nanowi e
As we men ion in he p e ious sec ion, in Chap e 5 we ob ain he solu ions
o he Hamil onian desc ibed by Eq.(4.7) ia a gauge ans oma ion ha
p ese es he dynamics o he sys em. In his ans o med sys em, e ec i ely
desc ibed by he Hamil onian in Eq. (4.14) we can de ine a spin-densi y (no
59
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
equi alen o he ” ue spin”) he y-componen . Since we expec ha away
om he impu i y o he spin o be conse ed we can use he con inui y
equa ion. This way, we a e able o de i e exp essions o he spin cu en s
in bo h leads.
In analogy o p e ious subsec ion, we s a i s de i ing he
co esponding conse a ion equa ion om he spin densi y Sy= eˆ
Ψ†ˆσyˆ
Ψ by
di e en ia ing wi h espec o ime,
dSy
d = e "dˆ
Ψ†
d ˆσyˆ
Ψ + ˆ
Ψ†ˆσy
dˆ
Ψ
d #.(4.32)
Fo he kine ic e m and con inemen desc ibed by ou p oblem Hamil onian
in Eq.(4.14) we can w i e
(dSy)K
d =e
i~ˆ
Ψ†ˆσy−−~2
2m∇2ˆ
Ψ−−−~2
2m∇2ˆ
Ψ†ˆσyˆ
Ψ
+ˆ
Ψ†ˆσyVcon ˆ
Ψ−Vcon ˆ
Ψ†ˆσyˆ
Ψi
=−e~
2mi∇hˆ
Ψ†ˆσy(∇ˆ
Ψ) −(∇ˆ
Ψ†)ˆσyˆ
Ψi.(4.33)
And o he addi ional spin-o bi o anomalous e m he con ibu ion is gi en
by,
d(Sy)SO
d =e
i~ˆ
Ψ†ˆσyHSO ˆ
Ψ−HSO ˆ
Ψ†ˆσyˆ
Ψ
=e
i~(ˆ
Ψ∗
1,ˆ
Ψ∗
2)ˆσyαx~−∂xˆ
Ψ2
∂xˆ
Ψ1−αx~(−∂xˆ
Ψ∗
2, ∂xˆ
Ψ∗
1)ˆσyˆ
Ψ1
ˆ
Ψ2
=eαx
i(ˆ
Ψ∗
1,ˆ
Ψ∗
2)−i∂xˆ
Ψ1
−i∂xˆ
Ψ2−(−∂xˆ
Ψ∗
2, ∂xˆ
Ψ∗
1)−iˆ
Ψ2
iˆ
Ψ1
=−αe∂xhˆ
Ψ†ˆ
Ψi.(4.34)
Taking in o accoun bo h kine ic and spin-o bi con ibu ions in Eq.(4.33)
and Eq.(4.34) espec i ely, we can w i e he o al e olu ion o he spin-
densi y pola ized along y-axis as ollows
dSy
d =−e~
2mi∂xhˆ
Ψ†ˆσy(∂xˆ
Ψ) −(∂xˆ
Ψ†)ˆσyˆ
Ψi−eαx∂xˆ
Ψ†ˆ
Ψ.(4.35)
60
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
All he e ms in Eq.(4.35) can be ga he ed unde he same pa ial de i a i e
∂xand he e o e w i en in he o m o dSy
d +∂xjy
x= 0, ha is o say we can
de ine a conse ed spin cu en along x wi h spin-pola iza ion along ysuch
ha ,
jy
x= (jy
x)K+ (jy
x)SO
=e~
2mihˆ
Ψ†ˆσy(∂xˆ
Ψ) −(∂xˆσyˆ
Ψ†)ˆ
Ψi−eαxˆ
Ψ†ˆ
Ψ.(4.36)
Then, in he amewo k o he second quan iza ion one can w i e he cu en
ope a o as an in eg al o Eq.(4.36) in e ms o he ield ope a o s ˆ
Ψ wi h a
kine ic con ibu ion gi en by
(ˆ
Iy
L)K(x, ) = e~
2mZdEdE0X
nσ
ei(E−E0) /~
2π~√ Ln0σ0 Lnσ
×nˆa†
Lnσ(E) (kn(E)σ+σkn0(E0)−2kR) ˆaLnσ(E0)e−i(kn(E)−kn0(E0))x
−ˆ
b†
Lnσ(E) (kn(E)σ+σkn0(E0)+2kR)ˆ
bLnσ(E0)ei(kn(E)−kn0(E0))x
+ˆa†
Lnσ(E) (kn(E)σ−σkn0(E0)−2kR)ˆ
bLnσ(E0)e−i(kn(E)+kn0(E0))x
−ˆ
b†
Lnσ(E) (kn(E)σ−σkn0(E0)+2kR) ˆaLnσ(E0)ei(kn(E)+kn0(E0))xo.
Again, he exp ession in Eq.(4.37) can be signi ican ly simpli ied by aking
in o accoun ha alues o Eand E0a e close o each o he and ha he
wa e ec o s kn(E) and eloci ies a y slowly wi h ene gy a ound he Fe mi
ene gy. This way,
(ˆ
Iy
L)K(x, ) = e
4πm ZdEdE0X
nσ
ei(E−E0) /~
Ln0σ0
×2nˆa†
Lnσ(E) (kn(E)σ−kR) ˆaLnσ(E0)−ˆ
b†
Lnσ(E) (kn(E)σ+kR)ˆ
bLnσ(E0)
−2kRˆa†
Lnσ(E)ˆ
bLnσ(E0)e−2ikn(E)x+ˆ
b†
Lnσ(E)ˆaLnσ(E0)e2ikn(E)xo.
(4.37)
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CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
And o he spin-o bi con ibu ion Eq.(4.36),
(ˆ
Iy
L)SO(x, ) = eαx
2π~ZdEdE0X
nσ
ei(E−E0) /~
Ln0σ0
×nˆa†
Lnσ(E)ˆaLnσ(E0) + ˆ
b†
Lnσ(E)ˆ
bLnσ(E0)
+ˆa†
Lnσ(E)ˆ
bLnσ(E0)e−2ikn(E)x+ˆ
b†
Lnσ(E)ˆaLnσ(E0)e2ikn(E)xo.
(4.38)
Finally he exp ession o he spin cu en pola ized along he y-axis in he
le lead is gi en by he sum o bo h kine ic Eq. (??) and spin-o bi
con ibu ions Eq. (4.38), so ha
ˆ
Iy
L( ) = ˆ
IK
L+ˆ
ISO
L
=e
2π~ZdEdE0ei(E−E0) /~X
nσ nˆa†
Lnσ(E)σˆaLnσ(E0)−ˆ
b†
Lnσ(E)σˆ
bLnσ(E0)o.
(4.39)
Now we will ocus on he e m in be ween he b acke s whe e he sum o e
he σindex is going o in luence he inal esul as we will see. Reo ganizing
a bi he di e en sums i is possible o ew i e he exp ession o he spin-
pola ized cu en in Eq. (4.39) as gi en by he ollowing exp ession only in
e ms o he c ea ion and annihila ion ope a o s o he incoming basis,
ˆ
Iy
L( ) = e
2π~X
αβ X
m0mX
s0sZdEdE0ei(E−E0) /~ˆa†
αnσ(E)Bm0s0,ms
αβ (L;E, E0)ˆaβnσ(E0),
(4.40)
he e again, αand β ake he ese oi alues Lo Rand
Bm0s0,ms
αβ (L;E, E0) = δm0mδs0sδαLδβL s−Pnσ S†
αm0s0,Lnσ(E)σSLnσ,βms(E0). In
o de o de i e he a e age spin-pola ized cu en , we need o know ha
he p oduc o he c ea ion and annihila ion ope a o s o a elec on Fe mi
gas a he mal equilib ium is
Dˆa†
αm0s0(E)ˆaβms(E0)E=δm0mδs0sδαβδ(E−E0) αs0(E).(4.41)
We ha e added a spin index in he Fe mi dis ibu ion,
αs0(E) = e(E−µαs0)/kBTαs0+ 1−1, o desc ibe e en ually di e en spin
62
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
chemical po en ial and spin empe a u e in each lead. This means we ha e
an addi ional index wi h espec o Eq.(4.25). Simila ly, in he a e aging
p ocess we no ice ha only α=β e ms a e going o su i e and ha
SLnσ,Lms = nσ,ms and SLnσ,Rms = 0
nσ,ms . Then, using he p e iously
de i ed uni a y iden i ies and his o he one Ps †
s,σ σs + †
s,σ σs= 1 we
can ew i e he in eg and in Eq. (4.39) using he Eq.(4.41),as ollows
X
αβ X
m0mX
s0sDˆa†
αms(E)Bm0s0,ms
αβ (L;E, E0)ˆaβms(E0)E
=X
m0σ0 σ0−X
mσ
†
m0σ0,mσσ mσ,m0s0! Lσ0(E)δ(E−E0)
+X
m0σ0X
mσ 0†
m0σ0,mσσ 0
mσ,m0σ0 Rσ0(E)δ(E−E0).(4.42)
Subs i uing Eq. (4.42) in o Eq. (4.39) we ob ain he exp ession o he a e age
spin-cu en :
Dˆ
Iy
L( )E=e
2π~ZdE "X
m0σ0 σ0−X
mσ
†
m0σ0,mσσ mσ,m0s0! Lσ0(E)
+X
m0σ0X
mσ 0†
m0σ0,mσσ 0
mσ,m0σ0 Rσ0(E)#.(4.43)
Due o he spin-dependence o he Fe mi dis ibu ion we can w i e explici ly
he spin cu en o Eq. (4.43) as ollows
Dˆ
Iy
L( )E=e
2π~ZdE T↑ L↑(E)−T↓ L↓(E)−T0
↑ R↑(E) + T0
↓ R↓(E)
+ †
↓↑ ↑↓ + †
↑↓ ↓↑( L↑(E)− L↓(E))
+ 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑( R↑(E)− R↓(E))i.(4.44)
whe e T↑= †
↑↑ ↑↑ + †
↓↑ ↑↓ is he ansmission ampli ude wi h spin-up on
he le lead. Simila ly, T↓= †
↓↓ ↓↓ + †
↑↓ ↓↑ o he ansmission ampli ude
wi h spin-down. In o de o keep ack o he o igin o each ansmission
ampli udes we wo k wi h T0
↑,↓= 0†
↑↑ 0
↑↑+ 0†
↓↑ 0
↑↓, ha is o say, he ansmission
ampli udein he le lead o elec ons inciden om he igh , al hough due
63
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
o he uni a i y o he sca e ing ma ix hey a e equi alen o T↑,↓. In he
absence o spin bias, i.e.( L,R↑= L,R↓), Eq. (4.44) is educed o:
Dˆ
Iy
L( )E=e
2π~ZdE [T↑−T↓] ( L(E)− R(E)) ,
whe e we used T↑,↓=T0
↑,↓.
In a gene al case, when no assump ion is made ega ding he chemical
po en ials, we can w i e in linea esponse
ZdET↑(E) ( L↑(E)− R↑(E)) = T↑µ↑
L−µ↑
R,
ZdET↓(E) ( L↓(E)− R↓(E)) = T↓µ↓
L−µ↓
R.
(4.45)
and simila ly,
ZdE †
↓↑ ↑↓ + †
↑↓ ↓↑( L↑(E)− L↓(E)) = †
↓↑ ↑↓ + †
↑↓ ↓↑µ↑
L−µ↓
L,
ZdE 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑( R↑(E)− R↓(E)) = 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑µ↑
R−µ↓
R.
(4.46)
These di e ences in chemical po en ial a e µ↑,↓
L−µR↑,↓=eV ↑,↓
L→Rand
o he spin-biases µ↑
L,R −µ↓
L,R= eVy
L,R as we pola ize he spin along he
y-di ec ion. Subs i u ing Eqs.(??)-(4.46) in o Eq.(4.44) ha inally we can
exp ess he spin cu en pola ized alon y-di ec ion in he le lead as
hIs
Li=e2
2π~hT↑V↑
L→R−T↓V↓
L→R+ †
↓↑ ↑↓ + †
↑↓ ↓↑Vy
L+ 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑Vy
Ri.
(4.47)
Wi h his exp ession we close he subsec ion gi ing a way o p obe he spin
cu en in he le lead in o de o ind he spin-dependen conduc ance o
di e en spin and cha ge biases applied. The same calcula ion o he igh
lead is s aigh o wa d and is used in he nex subsec ion as we discuss he
appea ance o he spin o que ela ed o he spin- lip p ocesses in anspo
calcula ions.
64
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
4.3.1 Spin To que in he nanowi e
In he p e ious sec ion we de i ed he spin cu en on he le lead, a
away om he impu i y, due o he p esence o Rashba spin-o bi coupling.
Howe e , close o he impu i y we can no longe assume con inui y o he
spin-densi y and Eq.(4.32) p esen s a sou ce e m,
∂Sy
∂ +∂xjy
x=T(x).(4.48)
he igh hand side is a o que e m T(x) = T0δ(x−ximp) due o he p esence
o he impu i y. By in eg a ing he exp ession o he spin-cu en o he
s a ic p oblem (∂Sy
∂ = 0) a ound he posi ion o he impu i y,
T0=Zximp+
ximp−
(∂xjy
x)dx =−hIy
Ri−hIy
Li.(4.49)
he exp ession o he spin cu en on he igh side o he impu i y ximp +
co esponds o a cu en a eling om le o igh , implying a nega i e sign.
Then, we need o calcula e he exp ession o he a e age o he spin-
cu en on he igh lead om Eq.(4.40) whe e he indexes o he Llead
ha e been subs i u ed wi h Lindexes. This implies a change o he sca e ing
coe icien s in ol ed ( ins ead o 0and 0ins ead o ),
hIs
Ri=e2
2π~hT↑V↑
R→L−T↓V↓
R→L+ 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑Vy
R+ †
↓↑ ↑↓ + †
↑↓ ↓↑Vy
Li,
(4.50)
whe e V↑,↓
R→L=−V↑,↓
L→R. So ha he Eq.(4.49) o he o que in combina ion
wi h he esul s o Eq.(4.47) and Eq.(4.50) becomes,
T0=−e2
2π~h †
↓↑ ↑↓ + †
↑↓ ↓↑ + †
↓↑ ↑↓ + †
↑↓ ↓↑Vy
L
+ 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑ + 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑Vy
Ri.(4.51)
As a esul e idence by Eq.(4.51), applying a spin-bias on ei he he igh
o le lead gene a es a spin-o bi o que exp essed he e as a unc ion o he
sca e ing coe icien s ela ed o spin- lip p ocesses media ed by he impu i y.
In he esul sec ion o Chap e 6 we discuss ully he impo ance o his
esul .
65
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
4.3.2 Rela ion he spin-mixing conduc ance
The mixing conduc ance is a concep o g ea impo ance o anspo
be ween noncollinea e omagne s and is esponsible o he spin o a ion
a ound he magne iza ion axis o he e omagne [108]. This quan i y is
i n oduced as a way o highligh he ac ha sca e ing mixes he wo
componen s in a spino and one mus no see spin species as independen .
In o de o show his, hey p opose a oy model whe e an elec on incides in
he sca e e om he igh , ψie−iikxxand is e lec ed on o he igh side o
he sca e e as ψ e−iikxx. Bo h he inciden and he inal s a e a e ela ed
h ough he sca e ing ma ix and con ibu e o he spin cu en as shown
in he Landaue -B¨u ike exp ession o Eq.(4.39),
j(S)
α=~
2 xψ∗
iˆσαψi−ψ∗
ˆσαψ ,(4.52)
wi h inal s a es gi en by
ψ = 0
0 ⊗ψi,(4.53)
subs i u ing he spino s pola ized in he x, y-di ec ions and in eg a ing o e
ene gy we ob ain he spin cu en s:
IS
x≈(Re G↑↓Vx
R+ Im G↑↓Vy
R) (4.54)
IS
y≈(Re G↑↓Vy
R+ Im G↑↓Vx
R),(4.55)
whe e he complex conduc ance G↑↓ is gi en by,
G↑↓ =G01− ∗
⊗.(4.56)
We wan o es ablish a compa ison be ween he y-componen o he spin
cu en in Eq.(4.55) o ou p e ious esul o he spin cu en on he igh
side o he sca e e as desc ibed by Eq.(4.50). Fi s , in his oy model
he e appea s o be spin bias only on he igh side o he sca e e , hence
Vy
L= 0 in Eq.(4.50). In he oy model he only p ocess conside ed is he
e lec ion, esul in in T↑=T↓= 0, one mus emembe ha hese
ansmission ampli udes include bo h spin-conse ed and spin- lip
ansmission p ocesses. By his accoun , he exp ession in Eq.(4.50)
66
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
becomes o he oy-model whe e a spin bias is applied on he igh side o
he sca e e
hIs
Ri=e2
2π~ 0†
↓↑ 0
↑↓ + 0†
↑↓ 0
↓↑Vy
R.(4.57)
Naza oz w i es his sca e ing ma ix in he basis o he ˆσzma ix,
ˆ
S=|↑zih↑z|ˆ
S+|↓zih↓z|ˆ
S⊗,(4.58)
bu when we calcula e ou S-ma ix in Chap e 5 i will be in he basis o
ˆσy. So, by w i ing he sca e ing ma ix in Eq.(4.58) in ou basis,
ˆ
S=1
2h(ˆ
S+ˆ
S⊗)(|↑yih↑y|+|↓yih↓y|)+(ˆ
S−ˆ
S⊗)(|↑yih↓y|+|↓yih↑y|)i.
(4.59)
F om Eq.4.59 we can w i e he e lexion ma ix as,
ˆ 0=1
2 + ⊗ − ⊗
− ⊗ + ⊗.(4.60)
The ou -o -diagonal elemen s in Eq.(4.60) a e he 0
↑↓ and 0
↓↑ in Eq. (4.57).
This equa ion w i en in he language o and ⊗becomes,
hIs
Ri=e2
2π~
1
42( − ⊗)†( − ⊗)Vy
R
=e2
2π~
1
2h †
+ †
⊗ ⊗− †
⊗− †
⊗ iVy
R
=e2
2π~
1
4h2−2 Re( †
⊗ )iVy
R
=e2
2π~Re(1 − †
⊗ )Vy
R.(4.61)
I we conside no spin-bias o he pola iza ion along x-di ec ion, Vx
R= 0,
i is p e y clea ha we eco e Eq.(4.55) in Eq. (4.61). We see hen ha
e en in he absence o he adi ional conduc ance T↑=T↓= 0, we eco e
some mixing conduc ance on he igh side o he impu i y. This means ha
he spin cu en lows e en in he absence o cha ge anspo .
4.4 Conclusions
In summa y, in his chap e we ex end he Landaue -B¨u ike o malis o
include he e ec o he spin-o bi coupling o he desc ip ion o he
67
CHAPTER 4. THE LANDAUER-BUTTIKER DESCRIPTION OF
TRANSPORT
anspo p ope ies o a nanowi e. We de i e an exp ession o he spin
cu en pola ized along y-di ec ion in he le lead o be a away om he
impu i y. This exp ession, Eq.(4.47), is he highligh o his chap e . I
allows us o exp ess such a spin cu en as a unc ion o he sca e ing
coe icien s o he S-ma ix making i easy o ack he con ibu ions o he
spin cu en om he di e en ol age and spin-biases. In combina ion wi h
he exp ession o he spin cu en on he igh lead, Eq.(4.50), allows us o
desc ibe a o que ha a ises a he impu i y posi ion, as a consequence o
he spin- lip anspo mechanisms esul ing om he Rashba spin-o bi
coupling. This mechanism will ha e impo an consequence on he
conduc ance as i is discussed in subsequen chap e s. Fu he mo e, we a e
able o make a connec ion be ween ou exp essions and hose ob ained in
he con ex o he spin-mixing conduc ance in magne ic hyb id s uc u es.
So a we ha e exp essed all anspo p ope ies in e ms o he sca e ing
coe icien s. In he nex chap e we de e mine such coe icien s o he
sca e ing om an impu i y in a nanowi e in he p esence o Rashba
spin-o bi coupling.
68
CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
e m. A he same ime, he sine e m d ops ou , since i has only o -
diagonal ma ix elemen s, i.e. i admixes exci ed subbands. We expand ha
la e Hamil onian up o o de α2,
Hwi e =p2
y
2m∗
e
+m∗
eω2
0
2y2+αx~kxσz
−2m∗
eαxαykxσyy+~2k2
x
2m∗
e−m∗
eα2
y
2,(5.25)
and e-w i e i as ollows
Hwi e =p2
y
2m∗
e
+m∗
eω2
0
2y−2αxαy
ω2
0
kxσy2
+αx~kxσz
+~2k2
x
2m∗
e−m∗
eα2
y
2−2m∗
eα2
xα2
y
ω2
0
k2
x.(5.26)
The shi in he ha monic oscilla o cen e can be gauged away in a simila
way as he linea in momen um e ms o he spin-o bi in e ac ion, which
also could be in e p e ed as a shi o he kine ic ene gy cen al posi ion as
a unc ion o momen um. The nex ans o ma ion has he o m
˜
ψkxn(y) = e−iσypyy0/~¯
ψkxn(y),(5.27)
whe e py=−i~∂yand
y0=2αxαy
ω2
0
kx.(5.28)
A e his ans o ma ion he Hamil onian eads
Hwi e =p2
y
2m∗
e
+m∗
eω2
0
2y2
+αx~kxσzcos 2pyy0
~−σxsin 2pyy0
~
+~2k2
x
2m∗
e−m∗
eα2
y
2−2m∗
eα2
xα2
y
ω2
0
k2
x.(5.29)
The las e m is o ou h o de in αand can be omi ed, because we a e
accu a e only o he second o de . Simila ly he cosine and sine con ain y0
75
CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
which is p opo ional o α2and he e is ano he αin on o he whole e m.
As a esul , we ob ain, up o o de α2, he inal e ec i e Hamil onian
Hwi e =p2
y
2m∗
e
+m∗
eω2
0
2y2+αx~kxσz
+~2k2
x
2m∗
e−m∗
eα2
y
2.(5.30)
This Hamil onian is a he simple. I can be sol ed analy ically and he
co esponding G een’s unc ion can be w i en s aigh away, as we will see
in he nex sec ions. The eigen unc ions can be he w i en by summa izing
up he abo e ans o ma ions:
Ψkxn(x, y) = eiσxm∗
eαy
~y+π
4e−iσy2αxαy
~2ω2
0
pxpy1
√LeikxxΦn(y),(5.31)
whe e we es o ed ~kx→pxin he second ac o . We can also es o e
~kx→pxin Eq. (5.30), since he Hamil onian is diagonal in he quan um
numbe kx. The la e al wa e unc ions Φn(y) o a ha monic con inemen a e
gi en by Eq.(2.30) as discussed in Chap e 2.
5.2 Sca e ing s a es
We now ollow he p ocedu e desc ibed in Chap e 2 o ob ain he
sca e ing s a es o he e ec i e sys em o a mul iband quasi-1D nanowi e
in he p esence o Rashba in e ac ion. We ocus i s on he G een’s
unc ions co esponding o he e ec i e Hamil onian Eq.(5.30). The
G een’s unc ion is a sum o G een’s unc ions o he independen
sub-bands,
G( , 0) = X
n
Φn(y)Φ∗
n(y0)Gn(x, x0).(5.32)
This exp ession implies sepa a ion o a iables o he channel wi hou
impu i y. In ou case, we ake ou he spin deg ee o eedom in o a ma ix
s uc u e,
ˆ
G( , 0) = X
nσ
Φn(y)Φ∗
n(y0)|σihσ|Gnσ(x, x0).(5.33)
The G een’s unc ion o each indi idual channel is gi en by
Gnσy(x, x0) = 2m∗
e
~2
i
2kn
eikn|x−x0|e−iσy
m∗
eαx
~(x−x0),(5.34)
76
CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
whe e
kn=1
~
u
u
2m∗
e"E−n+m∗
eα2
x+α2
y
2#.(5.35)
Fo wha ollows i is con enien o w i e hese G een’s unc ions as ollows,
Gnσy(x, x0) = gnσyunσy(x)u∗
nσy(x0), x > x0,
nσy(x) ∗
nσy(x0), x < x0,(5.36)
whe e
unσy(x) = m∗
e
~kn
eiknxe−iσy
m∗
eαx
~x,(5.37)
nσy(x) = m∗
e
~kn
e−iknxe−iσy
m∗
eαx
~x.(5.38)
These unc ions a e no malized o ca y uni lux densi y. Wi h such a
no maliza ion, we ha e he common ac o o be
gnσy=i
~.(5.39)
The s a e unσy(x) is he ou going s a e on he igh side o he impu i y(x>x0
o x→+∞). I is a igh mo e and we shall choose his s a e also as an
inciden s a e om he le , ΦL(x) = unσy(x). Simila ly, he s a e nσy(x) is
he ou going s a e on he le side o he sou ce (x<x0o x→ −∞). I is a
le mo e and we shall choose his s a e also as an inciden s a e om he
igh , ΦR(x) = nσy(x).
The Lippmann-Schwinge equa ion a e he Sch ie e -Wol
ans o ma ion eads:
Ψ( ) = Φ( )−Zˆ
G( , 0)eMVimp( 0)e−MΨ( 0)d2 0,(5.40)
whe e we explici ly w i e he ans o med ˆ
Vimp( 0). No ice ha in he basis
Eq. (5.36), he incoming s a e Φ( ) can be w i en as
ΦL
nσ(x, y) = unσ(x)Φn(y)|σi,(5.41)
o inciden om he le , and simila ly
ΦR
nσ(x, y) = nσ(x)Φn(y)|σi,(5.42)
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CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
o inciden om he igh . Because o he small suppo o Vimp( 0), he
exponen s ge p ojec ed on a s a e wi h 0= 0, and e ec i ely beha es as a
del a-like unc ion. A e in oducing he dimensionless p ojec o
| 0ih 0|,(5.43)
we ob ain om Eq. (5.40) in he limi o a poin -like sca e e ,
Ψ( ) = Φ( )− 0eM( 0)ˆ
G( , 0)| 0ih 0|e−M( 0)Ψ( 0).(5.44)
By expanding he exponen ial unc ions and using Eq. (5.14) we ob ain:
Ψ( ) = Φ( )− 0
׈
G( , 0) + 2iαxαy
ω2
0h∂x0∂y0ˆ
G( , 0)iˆσz
×Ψ( 0)−2iαxαy
ω2
0
ˆσz[∂x0∂y0Ψ( 0)].(5.45)
In o de o de e mine he wa e unc ion e−SΨ( ) a posi ion = 0, we ac
wi h e−Son he Lippmann-Schwinge equa ion
e−SΨ( ) = e−SΦ( )−Ze−Sˆ
G( , 0)eSVimp( 0)e−SΨ( 0)d2 0,(5.46)
and se = 0:
1
+Ze−Sˆ
G( 0, 0)eMVimp( 0)d2 0e−SΨ( 0) = e−MΦ( 0).(5.47)
The con ibu ion o e anescen s a es o he sum o e nin he G een’s
unc ion di e ges i we simply se 0= 0 o a δ-like Vimp( 0). Howe e , we
can do ha o he p opaga ing s a es and his is he eason why one can
ake he e m e−MΨ ou o he in eg and.
The bound s a es a e de e mined by he condi ion:
de
1
+Ze−Sˆ
G( 0, 0)eSVimp( 0)d2 0= 0 .(5.48)
The exp ession e−Sˆ
G( 0, 0)eS ep esen s he exac G een’s unc ion o he
channel (wi hou impu i y). Howe e ou ea men is pe u ba i e in he
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CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
spin-o bi in e ac ion, and he o e:
e−Sˆ
G( , 0)eS≈ˆ
G( , 0)
+2iαxαy
ω2
0h∂x0∂y0ˆ
G( , 0)iˆσz
−2iαxαy
ω2
0
ˆσzh∂x∂yˆ
G( , 0)i+. . . .
(5.49)
To a ain uni a i y o he sca e ing ma ix, addi ional (o e coun ing) e ms
migh be necessa y, as explained in he nex sec ion.
No ice ha he G een’s unc ion is diagonal in he basis o σy,
|+i=1
√21
i,
|−i =1
√21
−i,(5.50)
and he e o e σz eads.
ˆσz=|+ih−|+|−ih+|.(5.51)
In his basis he G een’s unc ion is a diagonal 2 ×2 ma ix
ˆ
G( , 0) = G++ 0
0G−− ,(5.52)
whe e G±± is he G een’s unc ion p ojec ed on o he s a e wi h σy=±1.
We cons uc symme ic and an isymme ic combina ions wi h espec o he
change o sign o x−x0,
Gs=G++ +G−−
2=X
nσ
Φn(y)Φ∗
n(y0)i m∗
e
~2kn
eikn|x−x0|cos m∗
eαx
~(x−x0),
(5.53)
Ga=G++ −G−−
2=X
nσ
Φn(y)Φ∗
n(y0)m∗
e
~2kn
eikn|x−x0|sin m∗
eαx
~(x−x0).(5.54)
Because o he ansla ional in a iance o e x, he single band G een’s
unc ion obeys
∂xGnσ(x, x0) = −∂x0Gnσ(x, x0).(5.55)
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CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
This is also alid o he ull G een’s unc ion ˆ
G( , 0), bu only wi h espec
o xand x0. The e o e, we can w i e Eq. 5.49 as
e−Sˆ
G( , 0)eS≈ˆ
G( , 0)
−2iαxαy
ω2
0h∂x∂y0ˆ
G( , 0)iˆσz
−2iαxαy
ω2
0
ˆσzh∂x∂yˆ
G( , 0)i,(5.56)
o in a ma ix o m
e−Sˆ
G( , 0)eS≈ G++ −2iαxαy
ω2
0∂x(∂y0G++ +∂yG−−)
−2iαxαy
ω2
0∂x(∂y0G−− +∂yG++)G−− !.
(5.57)
The sca e e po en ial is assumed o be symme ic. In pa icula Vimp( )
has mi o symme y wi h espec o x→ −x, whe e xis measu ed o his
pu pose wi h espec o x0. The e o e, in Eq. (5.48) only he symme ic pa
o Gcon ibu es o in eg als o he o m
ZG++( 0, 0)Vimp( 0)d2 0.(5.58)
No ice ha because Ghas a block s uc u e in spin space, we can conside
each block con ibu ion o Eq. (5.48) independen ly. On he o he hand he
an i-symme ic pa o Gcon ibu es o he in eg als o he o m
Z[∂xG++( , 0)]| = 0Vimp( 0)d2 0,(5.59)
only he an i-symme ic pa o Gen e s. As a esul , he ma ix in Eq. (5.57)
which en e s in Eq. (5.48) can be w i en as
Gs−2iαxαy
ω2
0∂x(∂y0−∂y)Ga
2iαxαy
ω2
0∂x(∂y0−∂y)GaGs!.(5.60)
We in oduce he sho no a ions:
⟪Gs⟫=1
0ZGs( 0, 0)Vimp( 0)d2 0,
⟪∂2Ga⟫=1
0Z[∂x(∂y0−∂y)Ga( , 0)]| = 0Vimp( 0)d2 0.
(5.61)
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CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
and ew i e Eq. (5.48) as
de h
1
+ 0⟪ˆ
G⟫i= 0 ,(5.62)
whe e
⟪ˆ
G⟫= ⟪Gs⟫−2iαxαy
ω2
0⟪∂2Ga⟫
2iαxαy
ω2
0⟪∂2Ga⟫ ⟪Gs⟫!.(5.63)
O explici ly
(1 + 0⟪Gs⟫)2−2αxαy 0
ω2
02
⟪∂2Ga⟫2= 0 .(5.64)
This is he condi ion o bound s a es, which can be w i en also as
1 + 0⟪Gs⟫±2αxαy
ω2
0
⟪∂2Ga⟫= 0 .(5.65)
Acco ding o he K ame s heo em, he wo solu ions ob ained om his
equa ion ( o he ±sign) mus be degene a e and he e o e, ⟪∂2Ga⟫has o
anish a he leading o de o ou app oxima ion (a igo ous p oo is p esen ed
in Appendix A. Such ha Eq. (5.47) esul s in
e−SΨ( 0) = 1
1 + 0⟪Gs⟫e−SΦ( 0).(5.66)
Inse ing his esul in o Eq. (5.45) we inally ob ain
Ψ( ) = Φ( )− 0
1 + 0⟪Gs⟫
׈
G( , 0) + 2iαxαy
ω2
0h∂x0∂y0ˆ
G( , 0)iˆσz
×Φ( 0)−2iαxαy
ω2
0
ˆσz[∂x0∂y0Φ( 0)].(5.67)
The sca e ing ma ix
we can now calcula e he sca e ing ma ix. Fo his, we send a sa e ΦL
nσ( )
inciden om he le and look a x→+∞. The G een’s unc ion a x→+∞
is ˆ
G( , 0) = X
nσ
Φn(y)Φ∗
n(y0)|σihσ|gnσunσ(x)u∗
nσ(x0).(5.68)
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CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
The ansmission ampli ude o sca e om le o igh is ound om
RL
mσ0,nσ =δΨ( )
δΦL
mσ0( )Φ( )→ΦL
nσ( )
.(5.69)
The inal exp ession o he ansmission is he ob ained om Eqs(5.67-5.69).
In he absence o sca e e , he ansmission ampli ude is uni y
RL
mσ0,nσ =δmnδσ0σ.(5.70)
In he p esence o a sca e e , an addi ional e m appea s
RL
mσ0,nσ =δmσ0,nσ +ARL
mσ0,nσ ,(5.71)
whe e ARL
mσ0,nσ ≡Amσ0,nσ is he o wa d sca e ing ampli ude. I is con enien
o w i e Amσ0,nσ as a 2 ×2 block-ma ix in he spin space. In ac we can
w i e ˆ
Am,n =−iˆ
Amˆ
Bn,(5.72)
wi h ˆ
Amand ˆ
Bngi en by
ˆ
Am=−i 0ˆgmΦ∗
mˆu†
m+2iαxαy
ω2
0
Φ0∗
mˆu0†
mˆσz,
ˆ
Bn=1
1 + 0⟪Gs⟫Φnˆun−2iαxαy
ω2
0
ˆσzΦ0
nˆu0
n.(5.73)
He e, all unc ions a e e alua ed a he posi ion o he sca e e . We ha e
also in oduced such ma ices:
ˆun(x) = m∗
e
~kn
eiknxe−iˆσy
m∗
eαx
~x,
ˆu†
n(x) = m∗
e
~kn
e−iknxeiˆσy
m∗
eαx
~x,
ˆ n(x) = m∗
e
~kn
e−iknxe−iˆσy
m∗
eαx
~x,
ˆ †
n(x) = m∗
e
~kn
eiknxeiˆσy
m∗
eαx
~x.(5.74)
Simila ly, o ob ain he e lec ion ampli ude we calcula e
RL
mσ0,nσ =δΨ( )
δΦR
mσ0( )Φ( )→ΦL
nσ( )
,(5.75)
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CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
whe e we adop ed he basis o he ou going s a es on he le o be ΦR
mσ0( ),
i.e. o be he inciden s a es om he igh . I is impo an ha he G een’s
unc ion is now aken o x→ −∞, which eads
ˆ
G( , 0) = X
nσ
Φn(y)Φ∗
n(y0)|σihσ|gnσ nσ(x) ∗
nσ(x0).(5.76)
In a simila way we ob ain
ˆ m,n =−iˆ
Cmˆ
Bn,(5.77)
whe e
ˆ
Cm=−i 0ˆgmΦ∗
mˆ †
m+2iαxαy
ω2
0
Φ0∗
mˆ 0†
mˆσz,(5.78)
and ˆ
Bnis gi en by Eq. (5.73).
5.3 B ie discussion o he uni a y o he S-
ma ix
The p oblem wi h ou app oach o ob ain he sca e ing coe icien s is ha
being a pe u ba i e me hod he uni a i y o he sca e ing ma ix is a ec ed
and ˆ
S†ˆ
S6= 1 bu some o he he mi ian ma ix ˆ
A, such ha ˆ
S†ˆ
S=ˆ
A. One
can de ice a me hod o eco e uni a i y. Being He mi ian, he ma ix ˆ
Ahas
eal eigen alues and is diagonalizable ia a uni a y ans o ma ion,
ˆ
A=ˆ
U†ˆ
Adiag ˆ
U , (5.79)
whe e he eigen alues o ˆ
Aa e he diagonal elemen s o ˆ
Adiag and he
eigen ec o s o ˆ
Aa e he colums o ˆ
U. Now his means
ˆ
S†ˆ
S=ˆ
U†ˆ
Adiag ˆ
U=⇒ˆ
Uˆ
S†ˆ
U†ˆ
Uˆ
Sˆ
U†=ˆ
Adiag .(5.80)
As he ma ix ˆ
Adiag is diagonal wi h eal alues, i is possible o w i e i as he
squa e o qˆ
Adiag in o de o in e hem on he le hand side o Eq. (5.80)
and ew i e he new uni a y sca e ing ma ix,
ˆ
S0=ˆ
Uˆ
Sˆ
U†ˆ
Adiag−1/2.(5.81)
83
CHAPTER 5. SCATTERING MATRIX COEFFICIENTS IN A
NANOWIRE
This implies eno maliza ion o he sca e ing coe icien s which can be
ecalcula ed om he abo e exp ession. In he nex chap e when we
p esen he main esul s we impose uni a i y o he calcula ion o all
obse ables.
5.4 Conclusions
In his chap e we p esen a de ail de i a ion o he sca e ing coe icien s
o a sho - ange, del a- like impu i y in a nanowi e wi h Rashba spin-o bi
coupling whe e elec ons mo ion is con ined in he ydi ec ion. We do so by
p obing he sca e ing s a es a he ex emes o he nanowi e ia he
Lippmann-Schwinge equa ion. The in e subband mixing a ising om he
in e play be ween Rashba spin-o bi coupling and he ha monic
con inemen complica es he analy ical solu ion o he Lippmann-Schwinge
equa ion. Ou way o deal wi h hese di icul ies is by pe o ming a
Sch ie e -Wol ans o ma ion. In his manne , we gauge away he
in e subband mixing, up o second o de in pe u ba ion heo y. As a
esul , he G een’s unc ion in he Lippmann-Schwinge can be ob ained
s aigh o wa dly. On he o he hand he complexi y is abso bed in o he
impu i y po en ial o he eigen unc ions o he sys em. Consequen ly, he
impu i y, ha is assumed om he beginning o be a scala , acqui es a
spin-s uc u e. F om he o m o he sca e ing coe icien s we al eady
unde s and ha by sca e ing a he he impu i y he elec on spin may
lip. Such spin- lip p ocesses will ha e impo an consequence on he
anspo p ope ies o he wi e, as discussed in he nex chap e , whe e we
will use he esul s o he p ese and p e ious chap e s. We close he
chap e desc ibing a me hod o eco e uni a i y o he S-ma ix, a e he
pe u ba i e app oach used in he calcula ion.
84
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
(a)
10 5 0 5 10
²
0.0
0.2
0.4
0.6
0.8
1.0
(
²
)
(b)
10 5 0 5 10
²
0.0
0.2
0.4
0.6
0.8
1.0
(
²
)
(c)
10 5 0 5 10
²
0.0
0.2
0.4
0.6
0.8
1.0
(
²
)
Figu e 6.3: Fano line-shape () as a unc ion o he dimensionless ene gy
pa ame e o : (a) q→ ∞, whe e he ansmission occu s h ough he
disc e e s a e, (b) q= 1 and he ansi ion h ough he disc e e he disc e e
and he con inuum o s a es is o equal s eng h wi h minimum a Emin =
ER−Γ/2qand maximum a Emax =ER+ Γ/2q, and (c) q= 0 o he
esonan symme ic line-shape.
quasi-bound s a es was i s in oduced by Bagwell [23]. This discussion
was la e picked up by [24] and ela ed o he sum o e e anescen modes
h ough he G een’s unc ion. As he e anescen mode is associa ed wi h a
decaying leng h(co esponding o he e anescen κn), hese s a es a e no
p ope ly bound as opposed o he s able bound-s a e o a del a-sca e e in
Eq.(2.23)), as discussed by o he au ho s [57, 58, 59].
A his poin , i is wo h emphasizing he ad an ages o using he
Lipmann-Schwinge app oach. As no ed by Re . [65], many o he app oach
he sys em by sol ing he Sch ¨odinge equa ion h ough ma ching he
wa e unc ions o he modes on bo h sides o he impu i y po en ial and
ob aining an in ini e se o coupled equa ions[23, 117, 118]. In o de o
sol e such a p oblem one needs o unca e he sys em o equa ions, and
co espondingly escale he coupling cons an s. In con as , by using he
Lippmann-Schwinge equa ion he sys em is analy ically sol able, as all he
in o ma ion o he coupling is encoded in he G een’s unc ion o he ba e
nanowi e.
Quasi-bound s a es as Fano esonances
As poin ed ou be o e, he p esence o quasi-bound s a es is due o he
coupling be ween a disc e e s a e and he con inuum o subbands a ailable
in he nanowi e. This is no hing bu a Fano esonance, desc ibed by he
asymme ic Fano line-shape [119],
() = (+q)2
1 + 2,(6.9)
91
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
whe e = (E−ER)/Γ is he dimenssionless ene gy measu ed om he
esonance, Γ is he esonance wid h and qis he asymme y pa ame e
in oduced by Fano in his o iginal pape [120]. By equa ion Eq.(6.9) we see
ha he minimum min = 0 occu s a =−qand he maximum
max = 1 + q2a = 1/q.
In he limi |q| → ∞, he ansi ion occu s h ough a disc e e s a e as
he ansi ion ough he con inuum becomes e y weak. This esul s in a
Lo en zian peak o he o m ()→1/(1 + 2) (see Fig. 6.3 (a)). I he
asymme y pa ame e is close o uni y q→1, ansi ion h ough bo h he
disc e e and he con inuum and Eq.(6.9) leads o cu es o he ype shown
in in 6.3 (b). Finally, o he case q→0 he Fano esonance becomes
()→q2/(1 + q2) o ming a dip a E=ERwi h a symme ical lineshape
(see Fig. 6.3 (c)). This las case is unique o Fano esonance and is some imes
e e ed in he li e a u e as an i- esonance [119].
We can now i ou esul o he conduc ance as exp essed in Eq. (6.3)
o Eq. (6.9) and ob ain, he ollowing exp essions o he Fano pa ame e s,
ER=En− 2
0Im2(G)−1
2
0Im2(G)+12 2
0Φ4
n
2,(6.10)
q=±2 0Im (G)
| 2
0Im2(G)−1|,(6.11)
Γ = ± 3
0Im (G) Φ4
n| 2
0Im2(G)−1|
2
0Im2(G)+1 .(6.12)
The esonance ene gy in Eq. (6.10) coincides, up o a co ec ion ac o , wi h
Eq. (6.7). In he limi o weak impu i y such ac o equals uni y and we
eco e ha esul . This is equi alen o a Lo en zian when q→ ∞.
The possibili y o des uc i e in e e ence leading o asymme ic line-
shapes due o diso de has been widely s udied in quasi-one-dimensional
wa eguides [72, 58, 67, 121, 122]. Bu he inclusion o RSOC as a sou ce o
Fano esonances is e en mo e in e es ing, ei he in nanowi es [92, 123, 124].
6.1.2 E ec o Rashba spin-o bi coupling in he
conduc ance
So a we ha e desc ibed he e ec o del a-like impu i y in a simple
nanowi e. In his sec ion we s udy how he Rashba spin-o bi coupling
92
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
0.0 0.5 1.0 1.5 2.0
E
/ ω
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
G
/
G
0
(a)
1.375 1.380 1.385 1.390 1.395 1.400 1.405 1.410
E
/ ω
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
G
/
G
0
(b)
Figu e 6.4: (a) Conduc ance in he p esence o a sca e e o s engh
0=−0.9 and RSOC αx=αy= 0.2 (solid ed line). The ballis ic
conduc ance is shown in black dashed lines. A he h eshold ene gy below
1.5~ω, he ansmission is pe ec . Close and below he h eshold he
conduc ance exhibi s a dip ela ed o he quasi-bound s a e o ming in he
nanowi e as explained in he main ex . (b) A zoom in o he quasi-bound
s a e esonance. One clea ly see ha he ansmission is no ully supp essed.
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CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
a ec s he nanowi e conduc ance. The exp ession o he conduc ance is no
longe as simple as he one desc ibe in Eq. (6.3). As men ioned in he
in oduc ion o his sec ion, in he p esence Rashba spin-o bi coupling
anspo p ope ies a e spin-dependen . The combina ion o he Rashba
spin-o bi coupling wi h he del a-like impu i y leads o s iking anspo
p ope ies, bo h in he cha ge and spin conduc ances, due o spin- lip
e en s discussed below. A i s glance, he cha ge conduc ance shown in
Fig.6.4(a) shows simila esonan ea u es as hose in a wi e wi hou
Rashba spin-o bi coupling (see p e ious sec ion and Fig.6.1): a pe ec
ballis ic ansmission a he h eshold and he dips jus below he
h eshold. Howe e , a close look shows impo an di e ences. One is he
shi o lowe ene gies o h eshold ene gy gi en by
n=~ω0(n−1/2) −m∗
e(α2
x+α2
y)/2) as a consequence o he sinking in he
subband ene gy dispe sion. As men ioned in he in oduc ion o his
chap e , he p esence o Rashba spin-o bi coupling allows o di e en
ansmission mechanisms media ed by he impu i y embedded in he
nanowi e. As a esul , anmission o an incoming elec on can occu
conse ing he spin-alignmen wi h p obabili y | ↑↑|2 o s a es p epa ed
wi h spin-up (| ↓↓|2 o s a es p epa ed wi h spin-down) o wi h a lip in he
alignmen o he spin wi h p obabili y | ↑↓|2 o s a es p epa ed wi h
spin-up(| ↓↑|2). Al hough spin- lip ansmission is much smalle han
spin-conse ed ansmission as i depends on 2αxαy/ω2
0as shown in
Eq. (6.1), i is no negligible. The ele ance o his ac o will become clea
in Sec ion 6.2, whe e we s udy he spin anspo and i s ela ion wi h a
SU(2) ield. As in he case whe e Rashba spin-o bi coupling is absen , he
esonan beha iou o he conduc ance is dic a ed by he denomina o in
Eq. (6.1). When he ene gy app oaches he h eshold o he opening o he
nex subband, he dominan na u e o Re (G)→ ∞ in he denomina o
esul s in a ze o-p obabili y o he spin- lip ansmission. This is he same
cha ac e is ic ha esponsible o he pe ec ansmission o he
spin-conse ed componen . Consequen ly, he conduc ance a he h eshold
ene gy is 2Ndue o he li ing o spin-degene acy (see Fig.6.4(a)).
The main e ec o Rashba spin-o bi coupling is he absence o he ull
sup ession o he conduc ance a he esonan ene gy, as shown in de ail in
Fig.6.4(b). To cla i y he o igin o his, we e e again o Eq. (6.1) in o de
94
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
0.4 0.6 0.8 1.0 1.2 1.4 1.6
E
/ ω
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
G/G
0
ballis ic conduc ance
0
.
0
λy
0
.
5
λy
1
.
0
λy
1
.
5
λy
(a)
1.30 1.35 1.40 1.45 1.50
E
/ ω
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
G/G
0
ballis ic conduc ance
0
.
0
λy
0
.
5
λy
1
.
0
λy
1
.
5
λy
(b)
Figu e 6.5: (a) Conduc ance o a sca e e o s engh 0=−0.9. The
di e en colou co espond o di e en posi ion o he impu i y yimp. The
posi ion is gi en in uni s o he con inemen leng h λy=p~/m∗
eω0. The
chosen RSOC is αx=αy= 0.2. (b) Zoom in o he esonance o he same
plo s.
o w i e he ollowing exp ession o he ansmission o a spin-up s a e,
T0↑∼N−Im2(G)
Im2(G)+2αxαy
ω2
02(2Φ0Φ0
0)2/k0
Im2(G),(6.13)
95
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
I becomes clea om Eq. (6.13) ha while he spin-conse ed con ibu ion
o he ansmission will be ully supp essed when he e is only one
p opaga ing band, he e is a ini e con ibu ion o he spin- lip
ansmission. We can conclude hen, ha he e ec o he quasi-bound
s a e in he cha ge conduc ance is s ongly spin-dependen in he p esence
o RSOC in e ac ion. Speci ically , a he esonan ene gy only spin- lip
ansmission is allowed while spin-conse ed ansmission is comple ely
supp essed. In a nex sec ion we discuss his in mo e de ail and he
consequences on he spin-dependen anspo . Bu be o e ha we p esen
a sys ema ic s udy o he cha ge conduc ance as a unc ion o he impu i y
s eng h and la e al posi ion.
Dependence on he impu i y posi ion and s eng h
As b ie ly men ioned abo e, due o he ansla ional symme y along x-axis,
he impu i y posi ion ximp is i ele an . Howe e , he la e al posi ion yimp is
impo an . As he e en la e al modes Φn(yimp) anish a he cen e o he
ha monic oscilla o , esul ing in he decoupling o he subbands as he
imagina y pa o he G een’s unc ion in he denomina o o Eq. (6.1)
anishes Re (G)=Φ2
n(0)/κ0= 0. This is a consequence o he symme ic
s a es o he quan um ha monic oscilla o o e en modes, n= 0,2,4, .... As
a consequence he e is no p opaga ing weigh o he impu i y po en ial o
compensa e. The lack o coupling be ween subbands supp esses he dip
esonance. In mos o he plo s, when no explici ly said, i is assumed ha
he impu i y is loca ed a yimp 6= 0, in o de o s udy he dip.
In Fig.6.5(a) we p esen he conduc ance as a unc ion o di e en la e al
posi ions o he impu i y. One can see he supp ession o he esonan dip o
yimp = 0, which only appea s when he ans e se posi ion o he impu i y
b eaks he mi o symme y. We de ine he binding ene gy o he quasi-
bound s a e as he ene gy di e ence be ween he h eshold and he esonan
ene gy EQBS =E h −ER. Focusing in o ene gies close o he esonance (see
Fig. 6.5(b)), we can app ecia e ha i s EQBS inc eases wi h he dis ance
om he cen e up o a ce ain maximum alue a a ound yimp = 1.0λy, and
hen dec eases o la ge alues.
I is also wo h no icing ha he u he away he impu i y is om he
cen e , he highe conduc ance minima is, implying he e is a highe spin- lip
ansmission p obabili y. This becomes clea om Eq. (6.13), as he spin- lip
ansmission depends on he de i a i e o he wa e- unc ion a he impu i y
96
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
posi ion Φ0
0(yimp). Simila ly, in Fig. 6.6(a) we s udy he dependence o he
quasi-bound s a e esonan ene gy wi h he impu i y s eng h o a ixed
posi ion, yimp =λy. We can see in Fig. 6.6 (b) ha while he binding
ene gy o he quasi-bound s a e, EQBS, s ongly inc eases wi h he s eng h
o he impu i y po en ial 0, he conduc ance minima weakly depends on
0. This can be unde s ood om Eq. (6.13) since bo h spin-conse ed and
spin- lip ansmission p obabili ies depend on he impu i y s eng h in he
same way.
E ec i e 1D po en ial
We ha e also de i ed an e ec i e one-dimensional model (see Appendix C
o de ails). Wi hin such e ec i e model, he e ec o all highe subbands is
p ojec ed on o he a single band. This esul s in he ollowing ansmission
coe icien ,
RL
00,σ0σ=δσ0σ−i
m∗
e
~2 0
1+i 0m∗
e
~2
Φ2
0
k0hΦ∗
0Φ0
k0+2αxαy
ω2
0(2k0Φ∗
0Φ0
0)ei2(k0+kR)ximp
k0ˆσzi.(6.14)
Compa ison be ween Eq. (6.14) and Eq. (6.1) e eals simila i ies. Howe e , in
he s ic 1D si ua ion he sum o e all he e anescen modes is no aken in o
accoun . This sum, which appea in he denomina o o Eq. (6.1) o he quasi
1D case, is absence in Eq. (6.14). As discussed in Sec ion 6.1.1, e anescen
modes a e equi ed o o m a quasi-bound s a e. As o he nume a o in
Eq. (6.1), we eco e he p e ious esul o Eq. (6.14), bu o cou se o a
single p opaga ing band pe spin species (m=n= 0) and ene gy E0.
The absence o he quasi-bound s a e in he 1D model p e en he
esonan beha iou o be obse ed, blue cu e in Fig.6.7. Thus he esul
o he conduc ance ob ained om Eq. (6.1) is a good app oxima ion o
he ene gies close o he bo om o he p opaga ing band. Howe e , he e is
a way o eco e he esul s ob ained in he quasi 1D case om he pu e
1D, by adding ”by hand” in he denomina o o he second e m in
Eq.(6.14), he con ibu ion o he e anescen modes. By doing his one
ob ains an excellen ag eemen be ween he ull solu ion and he one
ob ained om he 1D model, as shown in Fig.6.8.
97
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
0.4 0.6 0.8 1.0 1.2 1.4 1.6
E
/ ω
0.0
0.5
1.0
1.5
2.0
2.5
G/G
0
ballis ic conduc ance
−
0
.
9
−
0
.
7
−
0
.
5
−
0
.
3
−
0
.
(a)
1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50
E
/ ω
0.0
0.1
0.2
0.3
0.4
0.5
G/G
0
ballis ic conduc ance
−
0
.
9
−
0
.
7
−
0
.
5
−
0
.
3
−
0
.
(b)
Figu e 6.6: The dependence o he conduc ance on he impu i y s eng h
o a nanowi e wi h RSOC: (a) Conduc ance o di e en s eng h 0o he
a ac i e sca e . We ha e chosen RSOC αx=αy= 0.2, and yimp = 1.0λy.
(b) Zoom o he esonance o he same alues.
6.2 Spin-dependen anspo p ope ies
We unde s and om p e ious sec ions he ele an ole o Rashba spin-o bi
coupling on he cha ge anspo p ope ies o a nanowi e wi h an impu i y.
In p e ious sec ions, we iden i ied wo di e en sca e ing mechanisms
98
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
0.4 0.6 0.8 1.0 1.2 1.4 1.6
E
(
ω
)
0.0
0.5
1.0
1.5
2.0
Conduc ance
Figu e 6.7: Conduc ance o an e ec i e 1D model (blue) s conduc ance
o he ull quasi-1D model ( ed) in uni s o G0 o an impu i y po en ial
o s eng h 0=−0.9 a posi ion yimp = 1.0λy espec o he cen e o he
nanowi e wi h Rashba pa ame e s αx=αy= 0.2.
0.4 0.6 0.8 1.0 1.2 1.4 1.6
E
(
ω
)
0.0
0.5
1.0
1.5
2.0
0.6 0.8 1.0 1.2 1.4 1.6
E
(
ω
)
0.00
0.02
0.04
0.06
0.08
0.10
(a) Conduc ance (b) T
↑ ↓
Figu e 6.8: (a)Conduc ance and (b) spin- lip ansmission o he quasi-1D
model (solid ed line) and o he e ec i e 1D model (dashed blue line) a e
adding by hand he con ibu ion om he e anescen modes.
99
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
media ed ia he impu i y po en ial, which a e dis inguished by whe he
he spin is conse ed o lipped a e sca e ing. The in e play be ween
hese wo mechanism mani es on a modi ica ion o he quasi-bound s a es
esonance in he cha ge conduc ance. Speci ically, as shown in Fig. 6.4 and
Eq. (6.13), a he esonan ene gy he dip in he conduc ance does no
each ze o as was he case in o anspo in he absence o Rashba
spin-o bi in e ac ion. Because his e ec is due o spin-dependen
p ocesses, one expec s ha i has consequences on he spin anspo i sel .
The e o e in his sec ion we ocus on he spin-dependen anspo .
In Fig. 6.9(a) we plo he spin- lip ansmission as a unc ion o he
injec ion (Fe mi) ene gy. E en hough i is small in compa ison o he o al
conduc ance, spin- lip anspo is no negligible. We can obse e wo
ea u es ha a e closely ela ed o he esonan cha ac e is ics discussed in
Sec ion 6.1.2. To begin wi h, he spin- lip ansmission is exac ly ze o a
he h eshold ene gy whe e he nex p opaga ing subband in opened. This
is in ag eemen wi h ou conclusions in Sec ion om Eq. (6.1), namely: he
dominan na u e in he denomina o o he eal pa o he G een’s unc ion
Re (G)→ ∞ a he h eshold ene gy esul s in pe ec anspo o he
spin-conse ed ansmission while he spin- lip anspo is comple ely
supp essed. On he o he hand, below he h eshold ene gy he spin- lip
ansmission p esen s a signi ican enhancemen . In Fig.6.9(b) we plo he
a io be ween spin- lip ansmission p obabili y and o al ansmission
p obabili y T↑↓/(T↑↑ +T↑↓) o an incoming spin-up s a e. We can obse e
ha a ce ain ene gy below he h eshold he only anspo allowed is
spin- lip anspo .
100
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
(2αxαyσz) ha appea s in Eq. (6.21). The physical in e p e a ion o his
displacemen is ha depending on he o ien a ion o he spin-o bi ield
FSO ∼(2αxαyσz) he pa icle will ”see” he cen e o he con inemen in
one di ec ion o he opposi e.
In Fig.6.13 we illus a e he beha iou o he SU(2) ield in he nanowi e
as an elec on p opaga es along i . One see ha an elec on desc ibing a
cyclic mo ion on a closed pa h in he 2DEG (xy−plane) gi es ise o a spin-
o bi al SU(2) ield FSO, while in (b) he ield FSO changes sign when he
pa h is a e sed in he opposi e di ec ion. Fig.6.13(c) ske ches an elec on
mo ing uni o mly in he nanowi e wi h speed n(E) along x. The ajec o y
unde goes an oscilla o y mo ion along ywi h equency ω0, as go e ned by
size quan iza ion due o he po en ial V(y). The a ea swep by he adius-
ec o o he elec on du ing i s mo ion is oscilla ing abou ze o a e age alue,
esul ing in a consis en ly posi i e (nega i e) FSO o he lowe (uppe ) hal
o he wi e. The luc ua ing ield FSO, despi e being ze o on a e age, couples
he elec on spin o he cen al posi ion y0o he elec on wa e unc ion
ha is displaced wi h espec o o he cen e o he wi e acco ding o he
o ien a ion o he FSO ield. We can see he eme gence o he ∼(2αxαy) wi h
o igin in he SU(2) symme y in Eq. (6.1). This equa ion di ec ly ela es he
spin- lip ansmission p obabili y o he e ec o SU(2) gauge.
In Fig.6.14 we ske ch he di e en anspo p ocesses ha may occu o
an elec on a eling in he x-di ec ion. I he elec on is p epa ed wi h spin-
up, and i s ajec o y is in he lowe hal o he nanowi e plane ha esul s in
a posi i e SU(2) ield FSO, hen he elec on will ”see” he impu i y po en ial
as i i we e displaced om i s posi ion by a quan i y y0≈2αxαykx/ω2
0. On
he o he hand, i he elec on is desc ibing a ajec o y in he uppe hal
o he nanowi e plane ha esul s in a nega i e SU(2) ield FSO, hen he
elec on will ”see” he impu i y po en ial as i i we e displaced om i s
posi ion by a quan i y −y0. Upon in e e ence o bo h possible anmission
pa hs, we pick up a phase ϕ esul ing in a il o he spin. In o he wo ds, a
he esonan ene gy whe e he only allowed anmission is h ough spin- lip
a measu emen o he conduc ance would se e o p obe he SU(2) gauge
ield.
107
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
( )
±y0
S
S
φ
R
e-iφ/2
e
iφ/2
x
y
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Figu e 6.14: T anspo mechanisms o an incoming spin-up s a e. Due o
he SU(2) he e is a shi in he posi ion o he impu i y ha depends on
spin, as a consequence he spin picks up a phase ansla ing in a il wi h
espec o he quan iza ion axis.
6.3 Conclusions
To summa ize, in his he chap e we use he app oaches de eloped in
p e ious chap e s o he hesis o s udy he anspo p ope ies o a
nanowi e wi h RSOC and a del a-like impu i y. We i s discuss he dip in
he conduc ance ha appea s in he absence o Rashba spin-o bi
in e ac ion and i s ela ion wi h he Fano esonance. In Subsec ion 6.1.2 we
discuss he e ec o RSOC on he anspo and ound a spin- lip
ansmission con ibu ion as a consequence o he in e -band mixing
media ed ia an a ac i e impu i y. As a consequence, we show ha he
p esence o a quasi-bound s a e blocks he ansmission channel ha
p ese es spin, while i enhances he spin- lip ansmission. In addi ion, we
make a sys ema ic s udy o he quasi-bound s a e esonan ene gy wi h
espec o he la e al posi ion and he s eng h o he impu i y and
eco e ing a esul om he Chap e 4 ha depends en i ely on he
spin- lip ansmission, namely he o que. While he e a e e ec s o Rashba
spin-o bi in e ac ion o ele ance in he cha ge conduc ance, ou key esul
consis s in inding he unde lying ela ion be ween he spin- lip
ansmission and he SU(2) ield. A he esonan ene gy, he only
108
CHAPTER 6. QUASI-BOUND STATES IN A NANOWIRE: EFFECTS
OF THE RASHBA SPIN-ORBIT COUPLING
ansmission allowed is h ough spin- lip pe mi ing us o connec a
physical quan i y wi h he co esponding SU(2) symme y.
109
Chap e 7
Conclusions
The goal o his hesis is o o mula e a heo e ical model o s udy
analy ically elec onic anspo in quan um nanowi es in he p esence os
Rashba spin-o bi in e ac ion. The opic may ha e di ec impac on he
ield o semiconduc ing and supe conduc ing spin onics, and Majo ana
e mions. In gene al e ms we ha e shown ha he Rashba in e ac ion in
combina ion wi h an impu i y and con inemen po en ial a ec s d as ically
he cha ge and spin anspo . A ce ain ene gies we ound a hi he o
unknown spin- lip ansmission which is ela ed o he appea ance o a
spin-o bi o que. Ou heo e ical esul s can be used in mul iple ways o
u he s udies on anspo p ope ies o con ined sys ems in he p esence
o spin-dependen ields and impu i ies.
In he in oduc o y Chap e s 1 and 2, we desc ibe he mo i a ion
behind ou wo k and ou goal o s udying he p ope ies o sca e ing om
an impu i y in a quasi-1D semiconduc ing nanowi e wi h in insic
spin-o bi coupling o Rashba ype. We in oduce he Lippmann-Schwinge
equa ion, a use ul me hod o he heo e ical desc ip ion o quan um
sca e ing. In pa icula , we use his me hod o desc ibe how he elec onic
anspo is a ec ed by he p esence o he impu i y in he nanowi e. We
i s desc ibe he app oach in a gene al 3D sys em and la e ocus on
sca e ing on a del a-po en ial in a pu ely 1D sys em, discussing he
limi a ions o he Bo n app oxima ion. Finally we ocus on a mo e ealis ic
nanowi e desc ibed by a ans e se con ining po en ial, discussing he
eme gence o esonan beha io in he ansmission as a consequence o
quasi-bound s a es p esen in he nanowi e. This e ec a ises om he
localized impu i y coupling he e anescen and p opaga ing modes o he
110
CHAPTER 7. CONCLUSIONS
nanowi e. We ocus he possible e ec s a ising om he in e play be ween
he Rashba in e ac ion and he quasi-bound s a es.
Because he Rashba spin-o bi in e ac ion plays a cen al ole in his
hesis, in Chap e 3 we p o ide an in oduc ion in his opic. Speci ically,
we b ie ly discuss he spec al p ope ies o a 2DEG wi h Rashba spin-o bi
coupling be o e e iewing he complexi ies in ol ed in he analy ical solu ion
o he model Hamil onian o a quan um nanowi e whe e he 2DEG is u he
con ined. As a esul o his con inemen , he ene gy spec um is s ongly
a ec ed,as well as he spin pola iza ion. The combined e ec o RSOC and
con inemen gi es aise o an i-c ossings be ween b anches o opposi e spin
and di e en band index, de o ming he spec um.These a e a consequence
o he subband mixing ha couples p opaga ing and e anescen s a es. We
conclude ha o add ess his p oblem enough subbands ha e o been aken
in o accoun . A he end o he chap e , we p esen an exac solu ion o
he kx= 0 ha we use la e as he ze o h o de solu ion in ou pe u ba ion
a ound he poin kx= 0. This calcula ion allows us o ensu e he accu acy
o ou pe u ba i e app oach up o α2in Chap e 5.
In Chap e 4 we ex end he widely used Landaue -B¨u ike o malism o
include he e ec o he spin-o bi coupling o he desc ip ion o he
anspo p ope ies o he nanowi e. We de i e an exp ession o he spin
cu en pola ized along y-di ec ion in he le lead conside ed o be a away
om he impu i y. This exp ession allow us o exp ess such spin cu en as
a unc ion o he sca e ing coe icien s o he S-ma ix. Wi hin his
ep esen a ion i is easy o ack he con ibu ions om he ol age and
spin-biases. This esul in combina ion wi h he exp ession o he spin
cu en allowed us o desc ibe he o que ha a ises a he impu i y
posi ion, as a consequence o he spin- lip anspo mechanisms esul ing
om he Rashba spin-o bi coupling. These mechanisms ha e impo an
consequences on he conduc ance as discussed in subsequen chap e s.
Fu he mo e, we make a connec ion be ween ou exp essions and hose
ob ained in he con ex o he spin-mixing conduc ance in magne ic hyb id
s uc u es.
In Chap e 5 we p esen a de ailed de i a ion o he sca e ing
coe icien s o a sho - ange, del a- like impu i y in a nanowi e wi h
Rashba spin-o bi coupling whe e elec ons mo ion is con ined in he y
di ec ion. We do his wi h he help o he Lippmann-Schwinge equa ion
in oduced in Chap e 2. The in e subband mixing, a ising om he
in e play be ween Rashba spin-o bi coupling and he ha monic
111
CHAPTER 7. CONCLUSIONS
con inemen , complica es he analy ical solu ion o he equa ion. Howe e ,
we deal wi h hese di icul ies by pe o ming a Sch ie e -Wol
ans o ma ion. In his manne , we gauge away he in e subband mixing,
up o second o de in pe u ba ion heo y. As a esul , he G een’s unc ion
in he Lippmann-Schwinge can be ob ained s aigh o wa dly. On he
o he hand, he complexi y is abso bed in o he impu i y po en ial o he
eigen unc ions o he sys em. Consequen ly, he impu i y, ha is assumed
om he beginning o be a scala , acqui es a spin-s uc u e. F om he o m
o he sca e ing coe icien s we al eady unde s and ha by sca e ing a
he he impu i y he elec on spin may lip. Such spin- lip p ocesses ha e
impo an consequences on he anspo p ope ies o he wi e, as
discussed in he nex chap e , whe e we use he esul s o Chap e 5 and
p e ious chap e s. We close he Chap e by desc ibing a me hod o eco e
uni a i y o he S-ma ix wi hin he pe u ba i e app oach used in he
calcula ion.
Finally, we p esen he anspo esul s in Chap e 6. Speci ically, we
use he app oaches de eloped in p e ious chap e s o s udy he anspo
p ope ies o a nanowi e wi h RSOC and a del a-like impu i y. We i s
discussed he dip in he conduc ance ha appea s by sca e ing a a
del a-like impu i y in a nanowi e in he absence o Rashba spin-o bi
in e ac ion and i s ela ion wi h he Fano esonance. In Subsec ion 6.1.2 we
discuss he e ec o RSOC on he anspo and ound a spin- lip
ansmission con ibu ion as a consequence o he in e -band mixing
media ed ia an a ac i e impu i y. We show ha he p esence o a
quasi-bound s a e blocks he ansmission channel ha p ese es spin,
while i enhances he spin- lip ansmission. In addi ion, we make a
sys ema ic s udy o he quasi-bound s a e esonan ene gy wi h espec o
he la e al posi ion and he s eng h o he impu i y. Ano he key esul is
he unde lying ela ion be ween he spin- lip ansmission and he SU(2)
ield. A he esonan ene gy, he only ansmission allowed is h ough
spin- lip pe mi ing us o connec a physical quan i y wi h he
co esponding SU(2) symme y.
Besides he e ec s discussed in his hesis, he me hods de eloped he e
can be used in u u e esea ch. We en ision a possible applica ion o he
esul s om Chap e 6, namely he enhancemen o spin- lip ansmission,
in he spi i o he Da a-Das spin- ansis o in oduced in Chap e 3.
Indeed, one can design a de ice based on a nanowi e in which one can
ex e nally une he s eng h o he impu i y po en ial and so, con ol he
112
CHAPTER 7. CONCLUSIONS
spin- lip p obabili y o injec ed elec ons om a e omagne ic lead. By
uning he chemical po en ial a he esonan ene gy, we can ensu e ha
he spin o ansmi ed elec ons is lipped. This applica ion is in he spi i
o p e ious wo ks ha ex end he spin il e model o ake ad an age o
Fano esonances in quan um do s [130] o side ings coupled o nanowi es
[131]. Mo eo e , he use hese esonances has been al eady p oposed in spin
in e sion de ices based in semiconduc ing la ices wi h spin
o bi -in e ac ion and magne ic ields [132]. Fu he mo e, we can hink o
o he ways o ex end he s udy o diso de in nanowi es. Fo example,
placing a second de ec and s udying he possible spin-dependen
ansmission. In p inciple ou me hods, based on he sca e ing ma ix, can
be ex ended s aigh o wa dly o wo and mo e impu i ies.
One u he pe spec i e o he p esen wo k is i s ex ension o include
supe conduc i i y and a Zeeman ield and see wha a e he e ec s in he
con ex o Majo ana physics. Taking hese wo ing edien s in o accoun will
be he nex s ep in he heo e ical app oach o he sca e ing p oblem
p oposed in his hesis. Adding Zeeman o he Hamil onian desc ibed in
Eq.(5.1) would imply u he wo k since he G eens unc ion will be
spin-dependen e en in he absence o he impu i y and addi ional e ms in
he Sch i e -Wol ans o ma ion will appea . Mo eo e , in oducing
supe conduc i i y equi e enla gemen o he space o include he Nambu
s uc u e.
Ano he possible ex ension o ou esul s, is he s udy o he Josephson
cu en in a nanowi e a ached o wo supe conduc ing ese oi s. The
Josephson cu en is an equilib ium cu en ha can be de e mined om
he knowledge o he subgap spec um, And ee bound sa es . One can
add ess he ques ion how he quasi-bound s a es a ec s such spec um and
hence he Josephson cu en . Acco ding o Beennakke heo y[133], he
anspo p ope ies o such junc ion can be ully de e mine by he
knowledge o he sca e ing ma ix sca e ing ma ix, ha we know om
ou analysis. Mo eo e , i he supe conduc ing leads consis o
supe conduc o s wi h a spin-spli spec um, induced by he p oximi y o a
e omagne ic insula ing ilm [134], one can s udy how he Josephson
cu en depends on he mu ual di ec ion o he magne iza ions in he
spin-spli supe conduc o s. By uning he nanowi e in o he conduc ance
dip, we know om ou esul s, ha ansmission occu s oge he wi h
spin- lip. This sugges ha he Josephson cu en will be la ge when he
spin-spli supe conduc ing leads a e in an an ipa allel con igu a ion. This
113
CHAPTER 7. CONCLUSIONS
idea can be ex ended o mul i e minal Josephson junc ions whe e di e en
opological s a es can be a i icial c ea ed [135].
114
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128
Appendix A
1D G een’s unc ion
Fo he esolu ion o he Lippmann-Schwinge equa ion o a 1D sys em,
Eq.(2.16), we will need he G een’s unc ion o he Helmhol z equa ion,
∇2+k2G±(x,x0) = 2m∗
e
~2δ(x,x0).(A.1)
This equi es he e alua ion o ,
G±(x,x0) = hx|1
E−H0±i|x0i.(A.2)
Then,
G±(x,x0) = Z∞
−∞
dp
2π~hx|1
E−H0±i|pihp|x0i
=Z∞
−∞
dK
2π
eipx/~
p2
2m∗
e−(E±i)e−ipx0/~
=1
2π
2m∗
e
~2Z∞
−∞
dKeiK(x−x0)
K2−2m∗
e
~2(E±i),(A.3)
wi h poles gi en by
K=±k 1±i2m
~2k2≃ ±k1±im∗
e
~k.(A.4)
The p oblem hen can be sol ed using Cauchy’s in eg al o mula,
IC
(z)dz = 2πi×(sum o esidues enclosed by C) ,(A.5)
129
APPENDIX A. 1D GREEN’S FUNCTION
whe e Cis he con ou de ining he pa h o in eg a ion, aken
coun e -clockwise o he uppe hal -plane and clockwise o he lowe
hal -plane. Fo E≥0 and aking in o accoun Eq.(A.5) o he poles in Eq.
(A.4) in Eq.(A.3), hen
G±(x,x0) = 2m∗
e
~2
i
2k{∓e±ik(x−x0)|x>x0∓e∓ik(x−x0)|x<x0}
=∓2m∗
e
~2
i
2ke±ik|x−x0|.(A.6)
The ±in he G een’s unc ion G±(x,x0) co esponds o he incoming (−)
o ou going (+) bounda y condi ions, which means ah o he posi i e
exponen we close in he uppe hal -plane and include he pole (+k+ i)
and o he nega i e exponen we close in he lowe hal -pane and include
(−k−i).
130
Appendix B
Ensu ing K ame s e e si ili y
The in oduc ion o he no a ions ⟪Gs⟫and ⟪∂2Ga⟫in Chap e 5, led us o
he ollowing exp ession o he bound-s a es,
1 + 0⟪Gs⟫±2αxαy
ω2
0
⟪∂2Ga⟫= 0 .(B.1)
Acco ding o he K ame s heo em, he wo solu ions ob ained om his
equa ion ( o he ±sign) mus be degene a e. The e o e, we s ongly suspec
ha ⟪∂2Ga⟫has o anish.
In Chap e 5 we in oduce he sho no a ions:
⟪Gs⟫=1
0ZGs( 0, 0)Vimp( 0)d2 0,(B.2)
⟪∂2Ga⟫=1
0Z[∂x(∂y0−∂y)Ga( , 0)]| = 0Vimp( 0)d2 0.(B.3)
(B.4)
Le us model he sca e e po en ial as
Vimp(x, y) = V0e−(x−x0)2
2σ2e−(y−y0)2
2σ2
= 0δσ(x−x0)δσ(y−y0),(B.5)
whe e 0= 2πσ2V0is he s engh o he impu i y, V0is he heigh and we
de ine he δ-like po en ial as
δσ(x) := 1
√2πσe−x2
2σ2.(B.6)
131
APPENDIX B. ENSURING KRAMERS REVERSIVILITY
The suppo o he in eg al, Eq.(B.2), is cen e ed a ound 0= 0wi hin a
ci cle o adius a∼σ.
The ollowing in eg al is use ul
Z+∞
−∞
dx0eikn|x0−x0|δσ(x0−x0) = e−1
2k2
nσ21 + e iknσ
√2.(B.7)
Since e (0) = 0 and he exponen ial ends o 1, we ha e his in eg al o be
1 in he limi σ→0. Ano he use ul in eg al is
Z+∞
−∞
dy0Φ∗
n(y0)δσ(y0−y0) = 1
√2nn!√πλ qλ2
λ2+σ2λ2−σ2
λ2+σ2n/2
×Hny0λ
√λ4−σ4exp h−y2
0
2(λ2+σ2)i.(B.8)
This in eg al in he limi σ→0 is equal o he ans e sal wa e unc ion a
he impu i y posi ion Φn(y0). As a esul , om Eqs.(B.7) and (B.8) in he
poin -like limi , Eq. (B.2) becomes,
⟪Gs⟫= Φ∗
n(y0) Φn(y0).(B.9)
Fo he calcula ion o Eq.(B.3) i is su icien o ocus on he y0in eg al.
Taking in o accoun ha he pa ial de i a i e o Gaas de ined in (5.54) is,
∂y0Ga=1
p2nn!√piλ 2n
λHn−1(y0/λ)−y0
λ2Hn(y0/λ)e−y02/λ2Φn(y),
(B.10)
we can w i e he ollowing in eg al o he y0dependence o Eq.(B.3),
Z∞
−∞
dy0(∂y0−∂y)Gaδσ(y−y0) = e−y2
0/2(λ2+σ2)Φn(y0)
p2nn!√piλ
×(2n
λ2+σ2λ2−σ2
λ2+σ2n−1/2
Hn−1λy0
√λ4−σ4
−σ2
λ2+σ2λ2−σ2
λ2+σ2n−1/2
Hn−1λy0
√λ4−σ4
−y0
1
λ2+σ2 λ2
λ2+σ2λ2−σ2
λ2+σ2n/2
Hnλy0
√λ4−σ4)
(B.11)
132
