This journal is © the Owner Societies 2024 Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 | 13995
Cite this: Phys. Chem. Chem. Phys.,
2024, 26, 13995
Field-dependent THz transport nonlinearities in
semiconductor nano structures†
Quentin Wach,
a
Michael T. Quick,
a
Sabrine Ayari
b
and
Alexander W. Achtstein *
ac
Charge transport nonlinearities in semiconductor quantum dots and nanorods are studied. Using a
density matrix formalism, we retrieve the field-dependent nonlinear mobility and show the possibility of
intra-pulse gain. We further demonstrate that the dynamics of master equations can be captured in an
analytical formula for the field-dependent charge carrier mobility, e.g. for two-level systems. This
equation extends the linear response theory based Kubo–Greenwood result to nonlinear processes at
elevated field strength, easily reached in THz transport spectroscopy. With these tools we analyze the
field strength, chirp, temperature and dephasing dependence of the charge carrier mobility in the model
system of CdSe quantum dots and wires. Stark broadening and Rabi splitting result in strong alterations
of the mobility spectra, pronounced at low temperatures. The mobility spectra are strongly temperature
and pulse shape dependent in the nonlinear regime. The findings are of immediate interest e.g. for non-
linear THz generation, conversion and amplification in 6G technology and nano electronics. Our results
further enable experimentalists to fit and understand measured charge transport nonlinearities with ana-
lytical expressions and to design nanosystems with engineered material properties.
1 Introduction
Charge transport in nanostructures and materials is a highly
active research field. These systems promise to optimize and
facilitate processes e.g. for hydrogen generation, solar energy
harvesting or electronics at a nano scale, nano photonics as
well as high frequency THz generation and detection by nano-
materials in emerging 6G technology.
1–6
THz time-domain and
pump–probe spectroscopy have proven to be an efficient tool to
investigate charge mobility and exciton polarizability in low
dimensional systems like semiconductor quantum wells, nano-
wires and dots.
7–22
Nonlinear transport phenomena in bulk
and nano semiconductors occur at elevated field strengths,
giving rise to high harmonics generation (e.g. in graphene
23
)
and Bloch oscillations
24
(at very high fields) as well as THz
generation by rectification
25
for example. These phenomena
show great promise for upcoming technologies like 6G tele-
communication in the THz range as they allow efficient
frequency generation and conversion. Although there is in-
depth understanding of linear transport in bulk and nano
semiconductors based on Drude, Drude-Smith and Kubo–
Greenwood based models,
26–32
understanding of nonlinear
transport and charge carrier mobility remains scarce.
33–36
Recent density matrix calculations show strong THz transport
nonlinearity in nanowires as well as a yet unexplored equili-
bration current, suppressing the low-frequency charge (or
exciton) mobility.
32,34,37,38
The latter can be understood in the
frame of a generalized Kubo–Greenwood theory including
charge carrier relaxation and dephasing.
In this paper, we study the field-dependent nonlinear mobi-
lity based on master equations and demonstrate that it can be
understood with an analytical equation, e.g. for two-level sys-
tems, extending the linear response theory based Kubo–Green-
wood result to nonlinear processes, which occur at elevated
field strength, e.g. in THz spectroscopy of transport phenom-
ena. Using master equations for the density matrix, we analyze
the full field strength, chirp, temperature dependence of the
charge carrier mobility as well as dephasing by phonon scatter-
ing, exemplified with CdSe wires and quantum dots. The
obtained frequency-dependent mobility spectra are shown to
be strongly temperature and THz pulse shape dependent,
especially in the nonlinear regime. Our results are of direct
interest e.g. for nano electronics or nonlinear THz generation
and conversion (e.g. from the high, electronically accessible
GHz range to THz frequencies) as well as amplification via gain
a
Institute of Optics and Atomic Physics, Technische Universita
¨t Berlin, 10623
Berlin, Germany
b
Laboratoire de Physique de l’E
´cole normale supe
´rieure, ENS, Universite
´PSL,
CNRS, Sorbonne Universite
´, Universite
´Paris-Diderot, Sorbonne Paris Cite
´, Paris,
France
c
Fakulta
¨tfu
¨r Physik, Universita
¨t Bielefeld, Universita
¨tsstr. 25, 33615 Bielefeld,
†Electronic supplementary information (ESI) available. See DOI: https://doi.org/
10.1039/d4cp00952e
Received 4th March 2024,
Accepted 22nd April 2024
DOI: 10.1039/d4cp00952e
rsc.li/pccp
PCCP
PAPER
13996 | Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 This journal is © the Owner Societies 2024
in 6G technology. There nonlinear currents are the basis for
efficient frequency mixing, harmonics generation and amplifica-
tion, seen as re-radiation in the spectrum emitted by the
nanostructure. We remark that our paper is focused on introdu-
cing the methodology of nonlinear charge transport simulation
on a quantum mechanical basis and identifying the relevant
parameters that govern the resultant system properties, while
detailed analysis of experimental findings is not the focus.
2 Theoretical foundations
2.1 Density matrix formalism
We describe the mobility of a charge carrier on a semiconductor
quantum dot or rod (represented by cuboids) within the frame of
density matrix theory. Starting point of our discussion is a
photogenerated electron in the conduction band (CB) or hole
left in the valence band (VB). The time-dependent dynamics
of this charged particle is given by action of the Hamiltonian H
on the density matrix rthrough the Liouville–von Neumann
equation.
34,39
Subjecting the system to a time-varying perturba-
tion, e.g. a THz-pulse, we write
_
r¼
_
I
hH;r½¼
_
I
hH0þH0;r
¼
_
I
hH0MEðtÞ;r
;(1)
where we divided the system into an unperturbed system
Hamiltonian H
0
and a perturbation H0=ME(t), with E(t)a
THz field and Mthe transition dipole moment. The energy
transferred between THz field and nanosystem relates to a
relative charge displacement, so that we write in the 1D case
M=q
e
(xL/2). We assume parallel orientation of THz field and
rod axis. Eqn (1) is the short-hand notation for a set of coupled
differential equations – the density matrix master equations.
40
The corresponding elements of rrelated to the quantized
electron (hole) states kcan be selected by r
ij
=hi|r|ji,sothat
we expand eqn (1) above
_
rij ¼
_
I
hX
N
n
H0
inrnj rinH0
nj
Minrnj rinMnj
EðtÞ
hi
þRij:
(2)
We introduced a decay contribution R
ij
reflecting population i=j
and polarization iajdecay. The way these phenomena are
addressed differs from many modeling approaches. We consider
the nanorods and dots we are dealing with as 4-level systems.
Our intention is to model coupling between energetic states
within the relaxation-time approximation. We write
Rii ¼X
N
kai
Gik rkk req
kk
Gki rii req
ii
ðÞ
;(3)
where G
ik
is the population decay rate coefficient from |ki-|ii.
On the other hand, population shifting among the energetic
states can only destroy phase coherence, prior imposed onto the
system by coherent coupling to a (THz) electric field. For that
reason, every single population distribution process is translating
into a loss of phase relation information, the decoherence. For the
off-diagonal elements, we write
Rij ¼gij rij with
gij ¼1
2X
N
kai
Gik þGki
ðÞþ
X
N
laj
Gjl þGlj
"#
þgpure
(4)
and, referring to polarization among states |iiand | ji,sumoverall
terms that either populate or deplete i(first sum) as well as j(second
sum). The last term g
pure
istheso-calledpuredephasing,aterm
corresponding to random resonance frequency fluctuations
41,42
or
inelastic scattering.
43
Due to the pronounced presence of (acoustic)
phonons in nanoparticles, we assume the dephasing to be governed
by phonon mediated relaxation. The corresponding population
redistribution and polarization decay processes are indicated in
Fig. 1a.
2.2 Relaxation rate coefficients
After intraband excitation of charge carriers, e.g. by THz
photons of a pump beam, a hot, non-thermalized charge carrier
distribution arises, which cools predominantly by the emission
of phonons.
44–47
The charge carrier dynamics can then be
probed by THz photons. The rate of electron or hole scattering
from initial state i to the final state f under assistance of a
phonon with wave vector ~
q¼qx;qy;qz
is given by Fermi’s
golden rule:
Gi!f
n;l¼2p
hX
q
Ci
nHl
nph
Cf
n
DE
2
dEf
nEi
nElðqÞ
(5)
Here n= e, h stands for electron or hole respectively. C
i(f)
n
is the
initial (final) state, lis the phonon mode. The dfunction
represents energy conservation, DE
n
=E
f
n
E
i
n
is the energy
Fig. 1 (a) 1D Quantum well with relaxation and dephasing channels for
the electron populations and polarization exemplified for the |2iand |3i
state pair. (b) Illustration of an electron being driven and excited by an
electric field E
-
inside a 1D nanorod including electron–phonon interac-
tions. Population (c) and polarization (r
12
(t)) (d) dynamics for a 20 nm
length CdSe rod at E
0
=20kVcm
1
and 10 K.
Paper PCCP
This journal is © the Owner Societies 2024 Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 | 13997
separation between the initial and final state, E
l
(q) is the
phonon energy. Hl
nph is the electron/hole phonon interaction
Hamiltonian. The upper sign corresponds to emission and the
lower sign to the absorption of a phonon. In non-centrosymmetric
materials, the coupling to acoustic phonons arises from both the
deformation potential and the piezoelectric coupling
mechanisms.
48,49
However, different studies on CdSe nanomater-
ials have shown that the acoustic phonon scattering mechanisms
are dominated by deformation potential interaction rather than
piezoelectric coupling.
48–52
As we focus on dot-like and short rod-
like CdSe quantum structures treated as quantum boxes charac-
terized by discrete densities of state, we will omit the considera-
tion of optical phonons in our investigation. In fact, in contrast to
acoustic phonons, optical phonons have no continuous-energy
spectrum in our dispersionless approach. For a OD system, the
discrete electron and LO phonon energies don’t have any first-
order (Frohlich) interaction, except for the special cases E
f
E
i
=
E
LO
, so that the scattering rates are low and we do not consider
them.Hence,wewillconsideronly the electron-longitudinal
acoustic (LA) phonon scattering. We remark that the intention
of this paper is to study the basics of nonlinear charge transport
physics and not a refined model for all kinds of phonon related
processes in the considered nanostructures. Within our acoustic
deformation potential approach eqn (5) turns into
Gi!f
n¼2p
hX
q
Nqþ1
21
2
aðqÞ
jj
2Y
Z¼x;y;z
Fni
Z;nf
ZqZ
2
dDEnhoq
;
(6)
where N
q
stands for the Bose distribution function, Fni
Z;nf
ZqZ
is
the overlap integral in Z=x,y,zdirection, n
i(f)
Z
is the initial(final)
state quantum number in Zdirection. We will consider in our
work an infinitely deep, rectangular quantum box. a(q)
2
=D
n2
ho
q
/
2r
d
c
s2
V,whereV=L
x
L
y
L
z
is the principal volume of the nanocrys-
tal, D
n
is the acoustic deformation potential, r
d
is the mass
density, o
q
=c
s
qis the long wave approximation of the acoustic
phonon dispersion, and c
s
is the sound velocity in the material.
See (ESI,†Section S1) for further details. Tables S1–S6 (ESI†)
display the calculated electron (hole) acoustic phonon scattering
rates for three different CdSe nanoparticle sizes (6 66nm
3
,
20 66nm
3
and 40 66nm
3
) for several transitions and
three different temperatures T= 10, 70 and 300 K. We approx-
imate our nanosystems as four-level systems in the THz range,
since the relevant transitions fall in the typically observed range in
(broadband) THz probe experiments. Tables S1–S6 (ESI†)show
the size and transition energy dependence of scattering rates. The
coupling to acoustic phonons decreases with the nanoparticles
size and increases with decreasing transition energy for fixed size.
The calculated scattering rates are then implemented in our
master equation description. Fig. 1c and d shows the results for
the dynamics of populations r
ii
and polarizations r
ij
as obtained
for20nmCdSerodsat10K.Withtheseresultsthecharge
carrier mobility on semiconductor nanoparticles is obtained in
the following.
2.3 Charge carrier mobility
With the Liouville–von Neumann equation fully set up, we can
solve numerically for all population and polarization elements.
34
In the next step, we can calculate observables, which represent the
microscopic properties of our system. Aiming to determine the
frequency-dependent mobility m(o) of charge carriers, we write
53–56
mðoÞ¼ ~
vðoÞ
~
EðoÞ¼FT vðtÞfg
FT EðtÞ
fg
:(7)
Making use of the fundamental property of Fourier transformation
~
vðoÞ¼FT d
dtxðtÞ
¼
_
IoFT xðtÞ
fg (8)
we have reduced the calculation of mobility to evaluation of charge
carrier position xon a 1D nanorod
xðtÞ
hi
¼Tr rðtÞx
fg
¼X
N
ij
rij ðtÞxji;(9)
so that the final measured (complex) mobility spectrum is given by
mðoÞ¼
_
Io
FT P
N
ij
rij ðtÞxji
()
FT EðtÞ
fg
:(10)
Inspecting eqn (10) as well as revisiting eqn (2), we see that
the numerical modeling is going to depend on the choice of the
electric field E(t). In accordance to experiment, we describe the
field of our THz (probe) pulse as
EðtÞ¼1
2E0eðt=tÞ2e_
Io0taðt=tÞ2þf
½
þc:c:(11)
with E
0
the electric field strength, o
0
the carrier frequency, athe
chirp parameter (negative aequals up-chirp)
53
and tthe 1/ewidth
of the Gaussian envelope, which relates to the full width at half
maximum by t¼tfwhmffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4lnð2Þ
p.
Before delving into the discussion of results, we should
differentiate between possible size and temperature effects on
the electronic structure of the charge carrier itself vs. its
surrounding, e.g. phonons in the semiconductor. Charge carrier
states, described in the infinite-well approximation, populated
according to Fermi–Dirac statistics,
37,57
follow a 1/L
2
energy
scaling. For that reason, smaller systems (with larger energy
differences) accumulate more population close to ground state,
from where transitions are going to dominate. At the same time,
transition matrix elements hi|M|jishrink with increasing j,but
grow, when looking at elements hj|M|j+1icorresponding to
absorption between neighboring states. Similarly, for a fixed size
but lowered temperature, the population accumulates close to
ground state, as well, implying the same effect.
31
At room
temperature (RT), however, the Fermi energy is elevated and
the Fermi-edge smeared, so that states in energetic proximity are
gradually populated.
31
Inspecting eqn (5), size effects on electronic energies
and wavefunctions, impact the coupling rate to phonons
as well. The phonon coupling Hamiltonian matrix elements
jhijHLA
ephjjij2Nbjhije_
I~
q~
rjjij2=Vcan be expanded for small
PCCP Paper
13998 | Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 This journal is © the Owner Societies 2024
wave vectors -
q, so that we see for a transition iaj
ie_
I~
q~
r
j
DE
_
I~
qij~
rjj
hi
_
IqxLxþqyLyþqzLz
:(12)
An indirect length scaling is the result, which becomes obvious
when L
x
4L
y
,L
z
. Hence, the L
2
influence from the squared
matrix element is countered by the 1/V= 1/(L
x
L
y
L
z
) dependence,
so that roughly a proportional scaling with increasing nanorod
length is observed (see ESI†).
Temperature, on the other hand, affects the phonon-
coupling rate solely through the LA phonon occupation number
N
b
. Given a certain energy (or wave vector, respectively) of LA
phonons, this number grows strictly with rise of temperature –
linearly for very small LA phonon energies. The dephasing rate
coefficient is translating into the FWHM w
L
by w
L
=2gof the
transition between states hi| and | ji. Using eqn (4) omitting
g
pure
as discussed, we find
wL
ij ¼X
N
kai
Gik þGki
ðÞþ
X
N
laj
Gjl þGlj
:(13)
Each of the pairs in parentheses refers to an up- and down-
scattering transition rate constant proportional to either N
b
or
N
b
+ 1, respectively. Only at higher temperatures we can expect
a clear linear growth dependence of linewidth (where N
b
EN
b
+
1pk
B
T). Although its precise behavior might not be immedi-
ately clear, we reasonably expect the number of phonons and
thus the linewidth through eqn (13) to shrink towards cryo-
genic temperatures.
3 Results and discussion
Fig. 2 shows the comprised results for temperature dependent
complex linear mobility spectra of three different sizes of CdSe
nanoparticles (6 66nm
3
Fig. 2(a) and (d), 20 66nm
3
Fig. 2(b) and (e) and 40 66nm
3
Fig. 2(c) and (f)). If not
specified otherwise, for modeling purposes of linear mobility
spectra our standard E-field parameters are E
0
= 0.1 kV cm
1
,
o
0
=2p1 THz, a=3, f= 0 and t
fwhm
= 0.5 ps.
Starting with the special case of the 6 66 nanocrystal
structure (Fig. 2a and d), it is striking that regardless of
temperature seemingly no real part is present (compare axis scale),
while the imaginary contribution is small, but non-negligible and
decreasing linearly. Commonly, such feature is attributed to
excitons, whose polarizability is claimed to merely cause a THz
field phase lag.
58,59
While this is true at low temperature, where
the exciton ground state is solely populated,
17,60
it has been shown
recently to render an incomplete picture at room temperature.
38
Here, however, the behavior of an electron can be fully explained
given the eigenenergy scaling by B1/L
2
.Ataneffectivemassof
m
e¼0:18m0;thefirstresonanceoccursat42THzoutsideofthe
THz window any common experimental setup can provide. With
that in mind, even at RT only the ground state is populated, so that
the single line of width w
L
=2.37ps
1
leaves no detectable real part
in the 0–8 THz window, while the initial slope of the pole function
in the imaginary part appears to be linear.
This scenario is educational for the case of larger nanorod
species, as it renders the contribution of short orthogonal
extensions to total THz mobility negligible. An electron on
the 20 nm rod being more resonant within the 0–8 THz
window, shows pronounced peak and pole functions Fig. 2(b)
and (e) up to 4000 cm
2
V
1
s
1
(10 K) decreasing with tempera-
ture down to still 450 cm
2
V
1
s
1
(inset Fig. 2(b)) for the larger
spectral range at room temperature. However, this amplitude
decrease is not directly proportional to temperature, because
(i) at the same time total population is distributing among the
4-level system and (ii) the width of (multiple somewhat over-
laying) transition lines is growing non-linearly with tempera-
ture between 10–300 K. This effect is amplified, when the
nanorod is lengthened even further to 40 nm Fig. 2(c) and (f).
Here, two additional effects become apparent, best seen at
Fig. 2 Size dependence of real (a)–(c) and imaginary (d)–(f) electron mobility for 6 66nm
3
,2066nm
3
and 40 66nm
3
CdSe quantum dots
and wires at 10 K, 70 K and 300 K and a weak THz field with E
0
= 0.1 kV cm
1
,o
0
=2p1 THz, a=3, f= 0 and t
fwhm
= 0.5 ps. Results are obtained with
eqn (10) from 4-level master equation approach. Insets in (b) and (c) display the room temperature case in a zoomed version.
Paper PCCP
This journal is © the Owner Societies 2024 Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 | 13999
cryogenic level: (i) the total dephasing rate is increasing with the
growth of respective up- and down-scattering rates as discussed
with eqn (12) and (ii) the 1/L
2
dependence of the energy levels is
driving the resonance frequencies close to the DC limit. The sharp
mobilitydroptowards0THzisreflecting a purely quantum
mechanic equilibration current contribution,
34
coming from
non-unitarian system dynamics (i.e. dephasing and relaxation).
32
For such nanowire like structures, the familiar Drude-Smith
response shape emerges for long wires, however for completely
different reasons than assumed backscattering at domain
boundaries.
16,30
Approaching room temperature, we are con-
fronted with another exception: the particularly small mobility
(Fig. 2(c) inset) is an unrealistic byproduct of our 4-level modeling.
Due to the Fermi-distribution reaching out to many higher states
for large L, modeling of more than 10 levels deems necessary. As
the computational effort is growing (N
2
N2)/2 with the
number of states N,
34
we choose the 20 nm nanorod for our
nonlinear discussion in the following. Fig. 3 displays the field
strength dependent real and imaginary parts of mobility spectra
from a 20 nm CdSe nanorod in case of 3 different temperatures.
For reference the linear case (E
0
=0.1kVcm
1
)isalwaysplottedas
startoftheseries.TheESI,†Section S4 contains our results for the
hole transport in the CdSe nanoparticles, in analogy to Fig. 2 as
well as the total mobility.
In order to understand the underlying physics, we expand
the time-dependent velocity v(t) into a linear and third order
nonlinear contribution (omitting even order terms for symme-
try reasons)
33,61,62
v(t)=v
(1)
(t)+v
(3)
(t) (14)
We realize that in combination to eqn (7) two distinct contribu-
tions to the measured spectrum arise
D~
EðoÞ
~
EðoÞ/mexpðE;oÞ¼ ~
vðoÞ
~
EðoÞ¼mð1ÞðoÞþmð3Þ
eff ðE;oÞ(15)
We intentionally write m
eff
for the third order related contribu-
tion, as it will become clear that an experimental measurement,
as performed by eqn (15), renders the term m
(3)
eff
itself field
dependent. Expanding the quantum mechanical expression for
the mobility from eqn (10) by means of a perturbation series
and defining r(t)=r
(0)
(t)+r
(1)
(t)+r
(3)
(t), we find
mexpðE;oÞ¼
_
Io
~
EðoÞFT X
N
ij
rð1Þ
ij ðtÞxji
()
þFT X
N
ij
rð3Þ
ij ðtÞxji
()"#
:
(16)
Since we ask for the relative displacement, only off-diagonal
elements are going to add to the total mobility. Hence, all
graphs in Fig. 3 will derive from the corresponding linear
spectra given in Fig. 2b and e and contain dynamic information
according to r
(3)
ij
in the second term of eqn (16) above. For that
reason, we revisit the generalized Liouville–von Neumann
eqn (1) and (2) to apply the common perturbation assumptions
r
ij
=r
(0)
ij
+lr
(1)
ij
+l
2
r
(2)
ij
+l
3
r
(3)
ij
and H¼H0þlH0¼H0lM
EðtÞfor the perturbative regime of interaction.
63
Further, we
expand the decay contribution in the same manner R
ij
=R
(0)
ij
+
lR
(1)
ij
+l
2
R
(2)
ij
+l
3
R
(3)
ij
. Sorting the respective terms by orders of l,
we find the q-th order off-diagonal elements to evolve
according to
_
rðqÞ
ij ¼ _
Ioij þgij
rðqÞ
ij þ
_
I
hrðq1Þ
jj rðq1Þ
ii
MijEðtÞ
þ
_
I
hX
N
naj
Minrðq1Þ
nj X
N
nai
rðq1Þ
in Mnj
"#
EðtÞ
(17)
This approach might seem similar to conventional perturba-
tion theory, but bears significantly different implications due to
the way relaxation among population elements is implemented.
Often the assumption of an overarching relaxation rate coeffi-
cient by means of G
i
(r
ii
r
eq
ii
) is applied, which is identical to
the way dephasing is treated here. Within such an approxi-
mation, we could easily derive nonlinear evolution of popula-
tion according to eqn (17) above (by setting i=j). Instead,
Fig. 3 Field strength dependence (a–f) of the electron mobility on a 20 66nm
3
CdSe nanorod for different temperatures using o
0
=2p1 THz, a=
3, f= 0 and t
fwhm
= 0.5 ps, obtained from 4-level master equation approach together with eqn (10).
PCCP Paper
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however, we need to define another evolution equation
_
rðqÞ
ii ¼
_
I
hX
N
n
rðq1Þ
ni rðq1Þ
ni
MinEðtÞ
þX
N
nai
GinrðqÞ
nn rðqÞ
ii X
N
nai
Gni (18)
As we can see, the first order polarization evolution (eqn (17),
q= 1) can easily be integrated, while this is not possible for the
first order population differential equation (DEQ) ((eqn (18),
q= 1)). Still, the relaxation terms enforce coupled DEQs we can
not disentangle. The shape of the perturbation equations,
however, teaches us that a polarization element of third order
q= 3 depends on population and polarization accumulated by
preceding order 2. In contrast, a population element is depend-
ing on even earlier installed polarization by order 1 only.
Nonlinear interaction is governed by the principle of polariza-
tion generation out of population and in turn, generating a
nonlinear population distribution etc.
Inspecting Fig. 3, generally speaking, we can see stronger
spectral alterations for lower temperatures. This observation is
rooted in the more pronounced accumulation of population (in
the ground state) at low temperatures. By iteratively inserting
eqn (17) and (18) into themselves, we can trace r
eq
jj
r
eq
ii
=
f
j
(T)f
i
(T)upto _
r
(3)
ij
. The less distributed a population is
among states prior to perturbation, the more scattered it can
become among all relevant levels during heavy excitation. This
constitutes a new situation, which is then immediately probed
by the same THz field. At 10 K (Fig. 3a and d) we can approach
best, what is seen. Ignoring the lineshape for a moment, we
crudely recognize the light curves (constituting the lone |1i-
|2iresonance at 3.8 THz) decrease, while another is emerging
at B6.3 THz – the |2i-|3itransition. The same phenomenon,
however in a weakened form, can be observed for higher
temperatures: low-frequency absorption and dispersion bands
are weakened and resonances occur at higher energies (Fig. 3b
and e and c and f). Elevated temperatures also include higher
levels, which brings the system closer and closer to the state of
equal occupation of individual levels. The graphs in Fig. 3(c)
and (f) describe the same finding, but over a frequency range
outside 0–8 THz.
In order to understand the lineshapes themselves, we are
going to look at several limiting cases of strong field coupling.
We bring the discussion down to a two-level system, where we
can easiest recognize phenomena known from optical spectro-
scopy. We adapt eqn (2) to write the closed two level DEQs as
_
r21 ¼ _
Io21 þg21
r21 þ_
I12r22
ðÞO
EðtÞ(19)
_
r22 ¼_
Ir
21 r21
O
EðtÞ r22 req
22
G21 þG12
ðÞ;(20)
where we introduced the Rabi-frequency O=M
21
E
0
/h, leaving the
normalized field E
¯(t)=E(t)/E
0
behind and made use of the
closure relation r
11
+r
22
= 1. Typically, from here one introduces
multiple constraints: (i) the electric field is monochromatic, (ii)
the excitation is quasi-resonant (rotating wave approximation) at
o
k
, hence (iii) polarization is separable into envelope and carrier
r21 ¼s21e_
Ioktand (iv) up-scattering is non-existent G
21
(due to
r
11
being the ground state), while (v) G
12
is slow. Many of these
approximations can not be applied rigorously in the THz regime,
hence the matrix master equations remain coupled at all times.
However, at low temperature, where relaxation and dephasing
become increasingly lower we can ask for the steady-state
solution of the system (eqn (19) and (20) above) under quasi-
resonant excitation E(t)BE
0
cos(ot). This allows to deduce
general properties expected upon growing coupling strength.
Making use of ref. 43, we find
m2LS ¼
_
Io
hqe
req
11 req
22
M21
jj
2oo21
ðÞþ
_
Ig21
oo21
ðÞ
2þg212þ4GO2(21)
for the of the 2-level mobility, containing resonant and anti-
resonant contributions already. We want to highlight the term
G=g
21
/(G
21
+G
12
), representing the ratio of dephasing to
relaxation, which is scaling the impact of the Rabi-frequency O
and thus the field-dependence itself. The result from eqn (21) is
quite similar to what is known from optical spectroscopy for the
susceptibility.
41
We recognize two emerging effects upon larger
Rabi-frequency affecting the real part: (i) the decrease of peak
maximum (decline of mobility amplitude due to reduced energy
transfer form the THz field to the charge carrier analogous to
absorption saturation in two-level systems). Setting dm2LS=do¼
!0,
we find
Re m2LS omax
ðÞ
fg
¼M21
jj
2
2hqe
req
11 req
22
g21
o21 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þg21
o21
2
þG2O
o21
2
s
0
@1
A
;
(22)
and (ii) linewidth growths due to power broadening. It is worth
mentioning that the equilibration current
32,34
causing the DC
mobility to vanish in nanosystems is the origin of rather compli-
cated amplitude and especially linewidth functions, see ESI,†
Section S2. This is, because the Lorentzian line is highly asym-
metric when the transition frequency o
21
is low (due to the
antiresonant contribution). Likewise, for the linewidth (full width
at half maximum) we obtain:
wL
Re m2LS
fg
¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3g212þ12GO2þ4o2121ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þg21
o21
2
þG2O
o21
2
s
0
@1
A
v
u
u
u
t;
(23)
While the change in amplitude is undoubtedly present in all
spectra from Fig. 3a and f, the broadening of linewidth is
especially difficult to discern in superposition with another very
prominent spectral feature: the energy shifting. Before entering
this discussion, though, we want to find a generalized solution to
the nonlinear two-level polarization (eqn (19) and (20)) valid for
any field form and especially room-temperature. Applying the
rules of perturbation expansion discussed prior, we write for third
Paper PCCP
This journal is © the Owner Societies 2024 Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 | 14001
order of two-level polarization ~
r
(3)
21
in the frequency domain
~
rð3Þ
21 ¼4O3req
11 req
22
oo21
ðÞþ
_
Ig21
ðð g21 _
Io00
~
Eo00
ðÞ
~
Eo0o00
ðÞ
~
Eðoo0Þ
G21 þG12
ðÞ
_
Io0
hi
o212þg21 _
Io00
2
do00 do0
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
~
RðoÞ
:
(24)
According to eqn (16), we can understand the complicated total
spectral lineshape, see ESI,†Section S3 for further details. For
simplification we introduce the third order lineshape distortion
function ~
RðoÞ¼ÐÐby the double integral term above. Further,
the evaluation of the velocity expectation value is carried out by
reinterpreting the relative displacement x
ij
from eqn (16) through
x
ij
=M
ij
/q
e
.Again,beingremindedthatFT{E(t)} = E
0
FT{E
¯(t)} =
E
0
E
˜(o), we finally write
mexpðoÞ¼_
Io
hqe
M21
jj
2req
11 req
22
oo21
ðÞþ
_
Ig21
14O2~
RðoÞ
~
EðoÞ
þa:r:
"#
;
(25)
where a.r. denotes the antiresonant part, i.e. m
12
(o)andm
21ðoÞ
denote the resonant and antiresonant contribution to m
exp
(o).
Closely inspecting the resulting complex mobility function reveals
several properties that could easily be overlooked. Firstly, as we
would expect, the overall spectrum is build from a linear spectrum
that is successively diminished by a growing nonlinear contribu-
tion scaling with the squared Rabi frequency O
2
. Usually, in the
optical regime large electric field strengths deem necessary to
introduce nonlinearities to the measurement. In the THz regime,
especially for nanostructures, the large transition dipole moments
(often corresponding to several nm of charge displacement)
impact the measurements already at low E
0
, as seen in Fig. 3.
With eqn (25) we have a new equation for the frequency-
dependent mobility in nanosystems including nonlinear third
order interaction being present at elevated field strength. Our
finding reproduces the linear extended Kubo–Greenwood equa-
tion recently discovered,
32
in the weak field limit of linear
response theory (O-0)
mlinear
exp ðoÞ¼
_
Io
hqe
req
11 req
22
M21
jj
2
1
oo21
ðÞþ
_
Ig21
þ1
oþo21
ðÞþ
_
Ig21
()
;
(26)
containing both resonant and antiresonant contributions. As
expected, reducing the interaction strength leads to the weak-
field (Lorentzian and pole like) lineshapes we observed in e.g.
Fig. 2b and e for the low temperature limit (where the pure two-
level approximation is expected to hold). Extending the rod length
from 20 to 40 nm Fig. 2(c) and (f) results in an assymetric
lineshape due to the antiresonant contribution (at negative fre-
quency) in eqn (26), having increased relevance once the reso-
nance shifts towards zero frequency, as well as an increasing
contribution of thermally populated higher states. Finally, to
fully set the stage for field-dependency discussions of the
electron mobility in nanorods, we shortly want to address the
energy level shifting and Rabi splitting. From diagonalization of
the total Hamiltonian H¼H0þH0¼E0
11
ji1
hj
þE0
22
ji2
hj
þ
hO
EðtÞ2
ji1
hj
þ1
ji2
hj
ðÞ(in the 2-level approximation), we can
determine the eigenvalues, reading
E1=2¼E0
1=21
2ho21 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ2O
o21
2
IðtÞ
2nce0
s1
8
<
:9
=
;
:(27)
This immediately shows that the strength of the electric field is
having an influence on the energy level positioning of the
eigenstates, as the perturbation field strength is entering the
eigenenergy expressions thorugh O. Starting from here, three
different cases can be distinguished: (i) either in case of vanish-
ingly low fields (OE0) or large detuning (o
21
cO)the
eigenenergies of H
0
are recovered. If the field intensity is increas-
ing, first (ii) the energy levels start to split further as the coupling
strength grows with O. The levels then oscillate according to the
temporal profile of I
¯(t)=2nc
0
e
0
|E
¯(t)|
2
,withn,therefractiveindex
and c
0
the speed of light. There is a regime in which the increase
of energy level distance becomes noticeable in absorption, the
states, however, still evolve almost unperturbed.
When the coupling strength becomes very large (as in the
THz regime) (iii), Rabi splitting occurs, as the evolution of
states is now coupled to the strong Rabi oscillation frequency as
well. This phenomenon is described for semiclassical light
fields, too.
41,64,65
The latter is especially relevant for the occur-
rence of gain in the THz regime as seen very prominently in
Fig. 3a around 4.4 THz. At strong fields, the system is no longer
resonating solely at the system eigenfrequency o
21
(Rayleigh
resonance), but at the Rabi sidebands o
21
O, which corre-
spond to resonances among the so called dressed states. The
latter constitute the AC Stark effect shifted resonances. It is a
core finding of dressed state theory that these strong light–
matter interaction related states contain their own population
and dynamics. Depending on the excitation field profile,
41,43
and especially the detuning from resonance absorption and
amplification can be obtained. It is extremely important to
realize that we are dealing with a polychromatic excitation, so
that some frequency components will be more resonant than
others and – at the same time – interact with the system at
different points in real time.
Rabi splitting is apparent predominantly at cryogenic tem-
peratures, where line broadening is smaller (see Fig. 3a, d, b
and e). While we clearly observe large splitting and gain at 10 K,
the superposed broadening and population effects at 70 K are
resulting in a strong lineshape asymmetrization only. Hence,
we find the possibility of THz gain in eqn (25), where the strong
driving via Ocan make the third order contribution overcome
the linear absorption leading to an amplification, as seen in the
negative real mobility around 4.4 THz in Fig. 3a. We remark
that this gain is related to intra-pulse frequency conversion via
third-order nonlinearity of our nanosystem and note that with
the absorption coefficient a= Re(s)/(nce
0
) for the intrinsic
PCCP Paper
14002 | Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 This journal is © the Owner Societies 2024
conductivity spm, with nthe refractive index, speed of light c
and vacuum permittivity, e
0
the amount of stimulated emission
and gain can be calculated. abecomes negative in this case. The
occurrence of nonlinear intra-pulse frequency conversion paves
the way for efficient frequency conversion and amplification in
the THz regime, useful for many applications, e.g. in 6G
technology. Our master equation model is suitable to investi-
gate these processes and make predictions, while eqn (25) and
(21) allow an interpretation of the results or are prospective for
fits to experimental results.
In order to shed more light on the role of the perturbing
field we investigate in Fig. 4 the chirp and carrier-envelope
dependence of the electron mobility calculated using the 4-level
master equation approach together with eqn (10). We remark
that depending on the chirp and envelope phase the pulses
have a different degree of unipolarity.
66
Unipolar pulses can
also propagate in the far field once a lens or mirror system is
used,
67
so that e.g. the waveform is recovered with inverted
amplitude in the image focus. We consider the nonlinear
transport regime, where the mobility becomes field strength
and pulse shape dependent, leaving linear response theory. In
panel (a) we see a pronounced dependence of the induced
charge velocity on the pulse chirp. Interestingly the electron
acquires less velocity amplitude in the case of an unchirped
pulse (with zero carrier-envelope phase). The reason is that the
unchirped pulse is strongly sub-resonant to the first electronic
transition at about 3.8 THz in 20 nm rods and has nearly no
field strength at this frequency, see Fig. 4e. The chirped pulses
have a much larger spectral amplitude resonant to the first
transition and hence induce a higher charge velocity, see
Fig. 4f. Altering the THz pulse carrier-envelope phase in
Fig. 4b leads to strong changes in the waveform while there is
a less pronounced effect on the acquired electron velocity. Only
at early times there are considerable differences, which result
in line width (and amplitude) changes of the frequency-
dependent velocity (Fig. 4f) around the lowest 3.8 THz system
resonance in 20 nm CdSe rods. Apart from that, the sub
resonant regime is strongly altered.
These findings translate in strong alterations of the
frequency-dependent (nonlinear) mobility spectra in Fig. 4c,
d, g and h. Near the resonances, strong alterations occur,
indicative for the discussed AC Stark effect causing non-
Lorentzian broadening and level splitting Fig. 4(c) and (g).
However, due to the broad pulse spectra, cf. Fig. 4(e), the
alterations of the resonances are complex. Interesting to see
is that an alteration of the pulse chirp from a=3toa=3
results in considerable optical gain near 5.5 and 6.5 THz
(Fig. 4c), an effect of the occurring strong nonlinearities in
the CdSe rods. This means that in the latter (down chirped)
pulse the parametric interaction of the THz field with popula-
tion and polarization (captured in the Liouville–von Neumann
evolution of the density matrix of eqn (2)) creates a transient
inversion, causing intra pulse gain and frequency conversion at
the expense of other frequency components in the pulse.
According to dressed state theory for strong coupling
41,43
gain
occurs above the electronic resonance at about o=o
21
+Ofor
strong fields, while there is induced absorption in the opposite
case. However, for spectrally broad pulses with respect to o
21
the situation gets more complicated and results in the seen
complex spectral shapes.
In Fig. 4g and h we visualize the effect of different carrier-
envelope phases from Fig. 4(b). We see that the phase changes
the nonlinear mobility spectrum especially by amplitude altera-
tion at the second transition. Hence it is important to notice,
that in contrast to linear mobility measurements at low field
strength, pulse carrier-envelope phase (CEP) is important. This
impacts e.g. the necessity of CEP-stabilization of pulsed THz
sources. E.g. in THz generation by Four Wave Mixing
68
CEP is
imprinted from a IR-pulse, while for optical rectification and
difference frequency generation
69
this is not the case.
Fig. 4 THz time-domain waveforms and resultant electron velocity (a) and (b) as well as THz field and charge velocity in the frequency domain (e) and (f)
for different chirp and carrier-envelope phase values, as obtained from 4-level master equation approach together with eqn (10) at 10 K. (First column for
f= 0, second column for a=3.) For all curves E
0
=10kVcm
1
,o
0
=2p1 THz and t
fwhm
= 0.5 ps. The resultant real and imaginary electron mobility
spectra for 20 nm CdSe nanorods at 10 K are displayed for different chirp (with f= 0) (c) and (g) and carrier envelope phase (d) and (h) (with a=3).
The a= 0 case delivers values in a 0–3 THz window only (overlaid by the other curves) due to finite pulse spectral bandwidth and center frequency.
Paper PCCP
This journal is © the Owner Societies 2024 Phys. Chem. Chem. Phys., 2024, 26, 13995–14005 | 14003
4 Conclusions
Transport nonlinearities in semiconductor quantum dots and
nanorods were investigated using a density matrix formalism for
the field-dependent nonlinear mobility. The charge dynamics
represented by the resultant master equations can be cast in an
analytical formula for the field-dependent charge carrier mobi-
lity, e.g. for two-level systems. Our equation extends the Kubo–
Greenwood result of linear response theory to nonlinear pro-
cesses at elevated field strength. We have demonstrated strong
field strength, chirp, temperature and dephasing dependence of
the charge carrier mobility on CdSe quantum dots and wires,
especially in the nonlinear regime. Stark broadening and Rabi
splitting result in pronounced alterations of the mobility spectra.
We have found the possibility of THz gain in the semiconductor
nanostructures based on intra pulse frequency mixing by third
order nonlinearities. The nonlinear changes are strongest for
strongly chirped pulses at low temperatures and depend weaker
on the THz pulse carrier-envelope phase. The findings are of
immediate interest e.g. for nonlinear THz generation and con-
version in 6G technology and nano electronics. Tunable and
intense THz sources e.g. from laser difference frequency
mixing,
70,71
plasms,
72
on chip nonlinear mixing
73
or quantum
cascade lasers
74
are interesting for implementation into future
6G communication systems with carrier frequencies operating in
frequency ranges of B300 GHz to 10 THz.
75,76
Nonlinear fre-
quency conversion and gain in semiconductor nanostructures,
like demonstrated in this paper may provide the needed func-
tionalities for signal mixing and amplification. As we have
shown, selective gain regions within the pulse spectrum can
occur so that intra-pulse amplification of THz signals at moder-
ate THz field strength in a finite spectral region and potentially
lasing may be feasible, an aspect potentially usable for amplifiers
or signal recovery in data links. Ultrafast, nonlinear THz imaging
techniques
77
may benefit from particles with high THz nonlina-
rities, like the presented dots and wires, and the gained under-
standing of how to model and tune their properties.
Furthermore, the presented formalism and simple analytical
expressions offer experimentalists the ability to analyze non-
linear transport phenomena
35
and interpret the frequency-
dependent conductivity of nano systems in the THz regime to
obtain microscopic system parameters as well as design nano-
systems with desired material properties.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
A. A. acknowledges support by DFG Grant AC290-2/2.
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