Uni e sali y o he Gunn e ec : Sel -sus ained oscilla ions media ed by soli a y wa es
L. L. Bonilla
Escuela Poli e
´cnica Supe io , Uni e sidad Ca los III de Mad id, Bu a que 15, 28911 Legane
´s, Spain
I. R. Can alapied a
Depa amen de Fı
´sica Aplicada, Uni e si a Poli e
´cnica de Ca alunya, G ego io Ma an
˜
o
´n 44, 08028 Ba celona, Spain
~Recei ed 11 July 1996; e ised manusc ip ecei ed 5 No embe 1996!
The Gunn e ec consis s o ime-pe iodic oscilla ions o he cu en lowing h ough an ex e nal pu ely
esis i e ci cui media ed by soli a y wa e dynamics o he elec ic ield on an a ached app op ia e semicon-
duc o . By means o an asymp o ic analysis, i is a gued ha Gunn-like beha io occu s in speci ic classes o
model equa ions. As an illus a ion, an example ela ed o he cons ained Cahn-Allen equa ion is analyzed.
@S1063-651X~97!15508-8#
PACS numbe ~s!: 03.40.K , 05.60.1w, 07.50.Ek
In semiconduc o s whe e he local cu en densi y as a
unc ion o he local elec ic ield is Nshaped, he Gunn
e ec is a ubiqui ous phenomenon @1–5#. The Gunn e ec
@6#consis s o ime-pe iodic oscilla ions o he elec ic cu -
en lowing h ough an ex e nal pu ely esis i e ci cui a -
ached o a semiconduc o sample subjec o dc ol age bias.
The cu en oscilla ions co espond o he gene a ion, one-
dimensional mo ion, and annihila ion o soli a y wa es o he
elec ic ield inside he semiconduc o . Besides his, he on-
se o he Gunn e ec can be qui e in e es ing, as he cu en
may display in e mi ency accompanied by spa io empo al
s uc u es o he elec ic ield inside he semiconduc o @7#.
Recen ly he onse o he Gunn ins abili y was analyzed by
singula pe u ba ion me hods which p o ide he go e ning
ampli ude equa ion o long semiconduc o s @8#. Gunn-like
phenomena may also explain he expe imen ally obse ed
sel -sus ained oscilla ions o he cu en in doped weakly
coupled supe la ices @9#whose dominan anspo mecha-
nism is esonan unneling be ween adjacen quan um wells
@10#. In hese cases, he oscilla ions a e due o ecycling o
elec ic- ield wa e on s ~cha ge monopoles!ins ead o soli-
a y wa es @10#. The di e ence in he ype o wa es may be
acked o he bounda y condi ion a he injec ing con ac
@11,12#. Gunn-like phenomena ha e also been nume ically
obse ed in a d i en di usi e la ice-gas model o hopping
conduc i i y @13#.
A na u al ques ion ha comes o mind in ela ion wi h
hese phenomena conce ns hei uni e sali y: Gi en ha he
Gunn ins abili y appea s in widely di e en semiconduc o
sys ems and models, wha a e he ea u es a gi en model has
o ha e in o de o p esen he Gunn ins abili y? No ice ha
he Gunn e ec is in p inciple a nonequilib ium phenomenon
which may happen a om any bi u ca ion poin s. Thus he
ques ion o i s uni e sali y may no be ela ed o linea iza ion
abou ixed poin s o a eno maliza ion ans o ma ion. Ne -
e heless an asymp o ic analysis allows us o unde s and
deeply he Gunn e ec and o y o gi e a p ecise meaning
o he no ion o uni e sali y a om equilib ium. This pape
ies o gi e an answe o he uni e sali y ques ion, and i
also pu s he Gunn ins abili y in o pe spec i e by compa ing
i o phenomena occu ing in o he pa e n o ming sys ems
@14#.
F om he s udy o he Gunn ins abili y in semiconduc o
models, we can ex ac he ollowing common ea u es ha
seem o be necessa y o i s occu ence.
~1!The model should be able o suppo soli a y wa es
mo ing in a p i ileged di ec ion on a la ge enough spa ial
suppo .
~2!I should include an in eg al ~o e space!cons ain .
~3!I should ha e app op ia e bounda y condi ions ~Di-
ichle , Neumann, mixed, e c.!which ende uns able he s a-
iona y solu ions o ce ain alues o he in eg al cons ain .
We shall illus a e hese poin s by cons uc ing a simple
model ha displays he Gunn ins abili y:
]
u
]
1K
]
u
]
x5
]
2u
]
x21J2g~u!,~1!
1
L
E
0
Ludx5
.~2!
In hese equa ions he unknowns a e u(x, ) and J( ), wi h
.0 and 0,x,L;g(u) is a unc ion ha ing a local maxi-
mum gM5g(uM) ollowed by a local minimum
gm5g(um) o u.0(0,u
M
,u
m
), while Kand
a e non-
nega i e pa ame e s. Equa ions ~1!and ~2!a e o be sol ed
wi h an app op ia e ini ial condi ion o u(x,0)>0 and Di-
ichle bounda y condi ions:
u~0, !5u~L, !5
J~ !,gm
um
,1
,gM
uM.~3!
In semiconduc o models u,J, and
Lco espond o he
elec ic ield, o al cu en densi y and dc ol age bias, e-
spec i ely. Bounda y condi ions ~3!co espond o Ohm’s
law ela ing he elec ic ield and he cu en a he injec ing
and ecei ing con ac s. ~We assume ha bo h con ac s ha e
iden ical esis i i y
.0 o simplici y.!O he bounda y
condi ions ~ ixed u, mixed bounda y condi ions!do no
quali a i ely change he cha ac e o he solu ions @1,11#.
The model ep esen ed by Eqs. ~1!and ~2!wi h K50 and
ze o- lux bounda y condi ions ins ead o Eq. ~3!is known as
he cons ained Cahn-Allen equa ion, and i was ecen ly in-
PHYSICAL REVIEW E SEPTEMBER 1997VOLUME 56, NUMBER 3
56
1063-651X/97/56~3!/3628~5!/$10.00 3628 © 1997 The Ame ican Physical Socie y
oduced by Rubins ein and S e nbe g as a nonlocal eac ion-
di usion model o nuclea ion akin o he mass-conse ing
ou h-o de Cahn-Hillia d equa ion @15,16#. Equa ion ~1!
wi h a ixed cons an Jand K50 is he well-known bis able
Fishe -Kolmogo o -Pe o skiı
ˇ-Piskuno ~FKPP!equa ion,
which includes among i s possible solu ions a a ie y o a -
eling on s and pulses ~soli a y wa es!mo ing on an in ini e
one-dimensional spa ial suppo @17,14#. The pulses o he
FKPP equa ion a e uns able solu ions: hey ei he sh ink o
expand when an in ini esimal dis u bance is added @17#. The
global in eg al cons ain ~2!and Di ichle bounda y condi-
ions ~3!con e he FKPP equa ion in o a model e y simi-
la o he ypical semiconduc o ones: he cons ained Cahn-
Allen equa ion. This model does no p esen he Gunn
ins abili y i K50 because he x↔2xsymme y implies no
p e e ed di ec ion o mo ion o a eling wa es. A la ge
enough nonze o con ec i e e m K.0 b eaks he x↔2x
symme y, and i a o s wa es mo ing om le o igh . The
esul ing model sa is ies condi ions ~1!–~3!abo e, and i dis-
plays he Gunn e ec ; see Fig. 1. I may be obse ed ha he
p esen model is also ela ed o K oeme ’s model o he
Gunn e ec in n ype GaAs @2#: we jus change he con ec-
ion coe icien o a cons an Kin Ampe
` e’s law and se he
di usi i y equal o 1 in he dimensionless K oeme ’s model
s udied in Re . @12#. These changes exclude he s aigh o -
wa d ex ension o ou p e ious asymp o ic analysis, as we
canno use he shock wa es and pa icula solu ions speci ic
o K oeme ’s model o desc ibe he Gunn e ec @12#.
To unde s and hese esul s, we shall assume ha
e
51/L!1. Then i is con enien o ew i e Eqs. ~1!and ~2!
in e ms o he ‘‘slow’’ a iables s5
e
and y5
e
x. The e-
sul is
e
]
u
]
s1
e
K
]
u
]
y2
e
2
]
2u
]
y25J2g~u!,~4!
E
0
1udy5
.~5!
In he limi
e
→0 he solu ions o his sys em a e piecewise
cons an : on mos o he yin e al uis equal o one o
ano he o he ze os o g(u)2J, sepa a ed by ansi ion lay-
e s ha connec hem. A y50 and 1 he e a e bounda y
laye s ~quasis a iona y mos o he ime!, which we will call
injec ing and ecei ing laye s, espec i ely. Le us assume
ha uM,
,um, and deno e by u1(J),u2(J),u3(J) he
h ee ze os o g(u)2J. Le he ini ial p o ile u(y,0) sa is y-
ing Eq. ~5!be a squa e bump u5u3(J) o
Y1(0),y,Y2(0) andu5u1(J) elsewhe e, plus e ms o o -
de
e
, as in he ime ma ked by Eq. ~1!in Fig. 1~b!. Loca ed
a y5Y1and y5Y2,Y1,Y2, he e a e sha p wa e on s o
wid h O(
e
) connec ing u5u1(J) and u5u3(J). This ini ial
p o ile will na u ally e ol e in o he Gunn e ec as ime
goes on ~see below!. The ini ial alue o J ollows om Eq.
~5!:
5u1~J!1@u3~J!2u1~J!#~Y22Y1!1O~
e
!.~6!
The bounda y laye s and he on s connec ing u1(J) and
u3(J) a e buil om ajec o ies o he phase plane:
du
d
j
5 ,d
d
j
5
m
1g~u!2J,~7!
whe e
j
5
e
21@y2Yi(s)#,c5dYi/ds, and
m
5K2c. The
bounda y laye s a e sepa a ices connec ing he e ical line
u5
Jin he phase plane (u, ) o he saddles (u1,0) o
(u3,0) o c50: u(x)→ui(J)asx→`and u(x)→ui(J)as
(x2L)→2`~i51 and 3!a e he ma ching condi ions. Fo
each ixed alue o Jbe ween gmand gMwe can ind a
unique alue c1(J) such ha u(2`)5u1(J) and
u(`)5u3(J)@co esponding o a he e oclinic o bi connec -
ing (u1,0) o (u3,0) wi h .0#and a unique alue c2(J)
such ha u(2`)5u3(J) and u(`)5u1(J)@a he e oclinic
o bi connec ing (u3,0) o (u1,0) wi h ,0#. The unc ions
c6(J) a e depic ed in Fig. 2. They in e sec when J5J*
gi en by
J*51
u32u1
E
u1
u3g~u!du,c65K.~8!
S a ing a s50, he on s Yi(s) mo e wi h speeds
dY1
ds 5c1~J!,dY2
ds 5c2~J!,~9!
FIG. 1. ~a!The unc ion J( ) o g(u)5100(u20.2)(u
20.4)(u20.6). Pa ame e alues a e K52, L5100,
51.5, and
50.32. ~b!The co esponding p o iles o u(x, ) e alua ed a he
imes ma ked in pa ~a!o his igu e.
56 3629UNIVERSALITY OF THE GUNN EFFECT: SELF-...
whe eas hei posi ions a e ela ed o he bias
h ough Eq.
~6!. We ind an equa ion o Jby di e en ia ing Eq. ~6!and
hen inse ing Eq. ~9!in o he esul :
dJ
ds5A~J!@c1~J!2c2~J!#,~10!
A5~u32u1!2
2u1
g3
81u32
g1
8
.0, ~11!
whe e gi
8(J)[g8(ui), and we used ha g„ui(J)…5Jimplies
]
ui/
]
J51/g8(ui). This is a simple equa ion o J, demon-
s a ing ha J ends o J*exponen ially as . No ice ha his
is a e y simple explana ion o he well-known obse a ion
ha a pulse de ached om he bounda ies mo es a cons an
speed and J, gi en by he equal a ea ule ~8!,@1#.
A e a ce ain ime, he wa e on Y2 eaches 1, and we
ha e a new s age go e ned by Eq. ~6!wi h Y251 and Y1
gi en by Eq. ~9!. The equa ion o Jbecomes
dJ/ds5Ac1.0, and i s solu ion inc eases @compa e Jand u
a ime ~2!in Fig. 1#un il i su passes he alue Jcsuch ha
u2(J)5
J.~A Jc,@
]
u/
]
x#x50changes sign and he qua-
sis a iona y injec ing laye becomes uns able!@12#. Le s1be
he ea lies ime a which J5Jc. A e s5s1, he p o ile o
uchanges wi hin he bounda y laye a y50: his injec ing
laye becomes uns able, and i sheds a new wa e du ing a
as s age desc ibed by he ime scale
5(s2s1)/
e
. To ind
wha happens nex we need o pe o m a mo e complica ed
analysis keeping O(
e
) e ms in he ou e ~bulk!expansion o
uand J, and jus he leading-o de e m in all inne expan-
sions ~bounda y laye s and wa e on s!. This calcula ion has
been pe o med in de ail o a semiconduc o model @18#.I
can be shown ha he shedding
o a new wa e om he injec ing laye is go e ned by he
ollowing semi-in ini e p oblem o x.0, 2`,
,`:
u(x,
)~ a om he old wa e dying a y51!sol es Eq. ~1!
and u(0,
)5
J(
;
e
), wi h J(
;
e
)5Jc1
e
J(1)(
),
J~1!~
!5h8~
!1
a
h~
!2
g
E
2`
e2
b
~
2 !h~ !d ,~12!
h~
!5~u32u1!c1~
2
0!2
E
0
`
@u~x,
!2u1#dx ~13!
~in his equa ion all unc ions o Ja e calcula ed a J5Jc;
0is a cons an and
a
,
b
, and
g
a e posi i e pa ame e s!@18#,
and he ollowing ma ching condi ion on an app op ia e
o e lap domain: u(x,
)2u0„x;J(s)…!1, as
→2`,
s→s12. He e u0„x;J(s)…is he quasis a iona y injec ing
laye solu ion o Eq. ~7!wi h
m
5Ksuch ha
u0„0;J(s)…5
J(s) and u0„`;J(s)…5u1(J(s)) o s,s1,
J(s1)5Jc. The unc ion h(
) is he a ea los due o he
mo ion o he old on du ing he ime
minus he ins an a-
neous excess a ea unde he injec ing laye .
The solu ion o he p e ious semi-in ini e p oblem e eals
he o ma ion, g ow h, and mo ion o a new pulse in he
injec ing laye , d i en by h(
) h ough he e ec i e excess
cu en ~12!. This p ocess ends when he new pulse is
bounded by wo well- o med wa e on s ~de ached om he
injec ing laye !which a e loca ed a Y3and Y4,Y3,Y4@see
he up o ile a ime ~3!in Fig. 1~b!, in which Y3and Y4ha e
al eady mo ed om hei ini ial posi ions O(
e
ln
e
)a he
beginning o his s age#. I may be seen ha he injec ing
laye becomes uns able and sheds a new wa e when i s wid h
eaches a c i ical size Dy5O(
e
ln
e
)@18#.
I
is la ge enough, we ha e a s age whe e he old wa e
on loca ed a Y1,1 coexis s wi h he newly o med pulse
bounded by he wo wa e on s loca ed a Y3and Y4:
5u1~J!1@u3~J!2u1~J!#~12Y11Y42Y3!1O~
e
!.
~14!
Di e en ia ing his equa ion and using ha Y1and Y3mo e
wi h speed c1whe eas Y4mo es wi h speed c2, we ob ain
dJ/ds5A(2c12c2). S a ing om Jc,Jdec eases u he
o J†@ he ze o o (2c12c2)#i 2c1(Jc),c2(Jc)~ he
s able case wi h Jc.J†in Fig. 2!. A e he old wa e eaches
y51, we again ob ain Eqs. ~6!–~10!and eco e he ini ial
si ua ion. Thus a ull pe iod o he Gunn oscilla ion is de-
sc ibed; see Fig. 1. On he o he hand, i 2c1(Jc).c2(Jc)
(J*,Jc,J†), Jinc eases a e he o ma ion o he new
pulse, and i is possible o he injec ing laye o shed mo e
wa es in o he bulk, as shown by he nume ical simula ions
o Fig. 3. How many wa es a e shed depends bo h on he
alue o Jc~and he e o e on he injec ing esis i i y
!and
on he leng h L. A ough es ima ion would gi e
(n11)c1(Jc)5nc2(Jc)as
e
→0 o he numbe no shed
wa es. This shedding mechanism seems o ha e he e ec o
b eaking he spa ial cohe ence o he sample, which may
lead o complex spa io empo al phenomena ~in e mi encies
wi h a a ying numbe o pulses p esen in he sample a
di e en imes!. The uns able case will be u he analyzed in
he nea u u e.
FIG. 2. The unc ions c6(J) o K52, and g(u) as in Fig. 1.
We ma ked he alue J5J† o which 2c1(J)5c2(J).
3630 56
L. L. BONILLA AND I. R. CANTALAPIEDRA
In conclusion, we ha e in es iga ed wha a e he main
ea u es ha a gi en model should ha e in o de o p esen
he Gunn e ec . These ea u es a e demons a ed by s udying
a simple model by means o a gene al asymp o ic analysis
co obo a ed by di ec nume ical simula ions. As a esul he
Gunn e ec is educed o sol ing a sequence o e y simple
p oblems ~one equa ion o Jeach ime!plus a canonical
p oblem o shedding new pulses. Ou asymp o ic analysis
explains quali a i ely and quan i a i ely he o ma ion, mo-
ion and annihila ion o pulses in he Gunn e ec . This wo k
sheds ligh on se e al puzzling aspec s o he Gunn oscilla-
ions ~see he chap e on open p oblems in Re . @19#!:~i!
Why do pulses mo e wi h he well-known equal-a ea- ule
eloci y a cons an Jwhen hey a e a om he con ac s
@ he co esponding cu en is a s able equilib ium o Eq.
~10!#?~ii!How does he wa e speed change when i a i es
o he ecei ing con ac ? ~iii!How a e new wa es c ea ed a
he injec ing con ac ? In addi ion, we ha e desc ibed an in-
s abili y mechanism consis ing o mul iple pulse shedding
du ing each oscilla ion o J, which appea s o app op ia e
alues o he bounda y pa ame e s a he injec ing con ac .
Simila wo k has been pe o med in di e se semiconduc o
models: Gunn oscilla ions in ul apu e closely compensa ed
p- ype Ge @18#, K oeme ’s model o Gunn oscilla ions in
bulk n- ype GaAs @20#, and slow oscilla ions in semi-
insula ing GaAs @21#. A modi ica ion o he asymp o ic
me hod p esen ed he e desc ibes he cha ge monopole ecy-
cling esponsible o he sel -oscilla ions in n-doped weakly
coupled supe la ices @22#. I espec i e o he physical
mechanism esponsible o he exis ence o he wa e on
and pulses, ou asymp o ic me hod desc ibes he Gunn oscil-
la ions in hese models. The model p esen ed he e pe haps
illus a es in he simples way wha he me hod consis s o :
~i! ind he equa ions and bounda y condi ions which cha ac-
e ize he shape o he wa e on s and hei speed as unc-
ions o he cu en densi y J.~ii!De i e he equa ions which
de e mine Jas a unc ion o he slow ime scale depending
on he numbe o wa e on s p esen in he sample. The
ield p o ile ollows adiaba ically he e olu ion o J.~iii!
Add he semiin ini e p oblems esponsible o wa e shedding
a he con ac s. The solu ion and ma ching o hese p oblems
yields an app oxima ion o he Gunn e ec in he gi en
model. O cou se, sol ing some o hese s eps may be in
i sel a a he complica ed echnical p oblem o pa icula
models equi ing special asymp o ics @21#.
This wo k was suppo ed by he DGICYT G an No.
PB94-0375, and by he EC Human Capi al and Mobili y P o-
g am Con ac No. ERBCHRXCT930413. We hank M. J.
Be gmann, P. J. He nando, M. A. He e o, F. J. Higue a, M.
Kindelan, M. Moscoso, S. W. Tei swo h, J. J. L. Vela
´zquez,
and S. Venakides o ui ul discussions.
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¨ll, and H. L. G ubin, The
Physics o Ins abili ies in Solid S a e Elec on De ices ~Ple-
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@2#H. K oeme , in Topics in Solid S a e and Quan um Elec onics,
edi ed by W. D. He shbe ge ~Wiley, New Yo k, 1972!,p.20.
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in Semiconduc o s and De ices ~Re . @5#!,p.1.
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~1992!.
@16#While he Cahn-Hillia d equa ion is he simples g adien low
which conse es mass and is local, he cons ained Cahn-Allen
equa ion is a simple nonlocal g adien low ha conse es
mass. See a discussion and compa ison o he wo models in
Re . @15#. A nice analysis o he on mo ion in he one-
dimensional Cahn-Allen model can be ound in he pape : L.
G. Reyna and M. J. Wa d, Eu . J. Appl. Ma h. 5, 495 ~1994!.
@17#J. D. Mu ay, Ma hema ical Biology, Lec u e No es in Biom-
FIG. 3. ~a!Densi y plo o u(x, ) wi h
54,
50.3 and
g(u) as in Fig. 1 ~ligh e colo means la ge u). He e mul iple
shedding o pulses occu s a he injec ing laye : wo pulses a e
o med du ing each pe iod. The second shed eaches and o e akes
he i s one. ~b!The co esponding p o ile o J( ).
56 3631UNIVERSALITY OF THE GUNN EFFECT: SELF-...
a hema ics Vol. 19 ~Sp inge -Ve lag, New Yo k, 1990!, Chap.
11; P. C. Fi e, Ma hema ical Aspec s o Reac ing and Di using
Sys ems, Lec u e No es in Bioma hema ics Vol. 28 ~Sp inge -
Ve lag, New Yo k, 1979!.
@18#L. L. Bonilla, P. J. He nando, M. A. He e o, M. Kindelan, and
J. J. L. Vela
´zquez, Physica D ~ o be published!.
@19#V. L. Bonch-B ue ich, I. P. Z yagin, and A. G. Mi ono , Do-
main Elec ical Ins abili ies in Semiconduc o s ~Consul an s
Bu eau, New Yo k, 1975!.
@20#L. L. Bonilla, I. R. Can alapied a, G. Gomila, and J. M. Rubı
´,
Phys. Re . E 56, 1500 ~1997!. See also G. Gomila, J. M. Rubı
´,
I. R. Can alapied a, and L. L. Bonilla, ibid. 56, 1490 ~1997!.
@21#L. L. Bonilla, P. J. He nando, and M. Kindelan ~unpublished!.
@22#L. L. Bonilla, M. Kindelan, M. Moscoso, and S. Venakides,
SIAM ~Soc. Ind. Appl. Ma h.!J. Appl. Ma h. ~ o be pub-
lished!.
3632 56
L. L. BONILLA AND I. R. CANTALAPIEDRA