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# Rashba spin orbit interaction in a quantum wire superlattice

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## Abstract

In this work we study the effects of a longitudinal periodic potential on a parabolic quantum wire defined in a two-dimensional electron gas with Rashba spin-orbit interaction. For an infinite wire superlattice we find, by direct diagonalization, that the energy gaps are shifted away from the usual Bragg planes due to the Rashba spin-orbit interaction. Interestingly, our results show that the location of the band gaps in energy can be controlled via the strength of the Rashba spin-orbit interaction. We have also calculated the charge conductance through a periodic potential of a finite length via the non-equilibrium Green's function method combined with the Landauer formalism. We find dips in the conductance that correspond well to the energy gaps of the infinite wire superlattice. From the infinite wire energy dispersion, we derive an equation relating the location of the conductance dips as a function of the (gate controllable) Fermi energy to the Rashba spin-orbit coupling strength. We propose that the strength of the Rashba spin-orbit interaction can be extracted via a charge conductance measurement.
arXiv:1111.1534v1 [cond-mat.mes-hall] 7 Nov 2011
Rashba spin orbit interaction in a quantum wire superlattice
Gunnar Thorgilsson,1J. Carlos Egues,2, 3 Daniel Loss,3and Sigurdur I. Erlingsson4
1Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
2Departamento de F´ısica e Inform´atica, Instituto de F´ısica de S˜ao Carlos,
Universidade de S˜ao Paulo, 13560-970 S˜ao Carlos, SP, Brazil
3Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
4Reykjavik University, School of Science and Engineering, Menntavegi 1, IS-101 Reykjavik, Iceland
In this work we study the eﬀects of a longitudinal periodic potential on a parabolic quantum
wire deﬁned in a two-dimensional electron gas with Rashba spin-orbit interaction. For an inﬁnite
wire superlattice we ﬁnd, by direct diagonalization, that the energy gaps are shifted away from
the usual Bragg planes due to the Rashba spin-orbit interaction. Interestingly, our results show
that the location of the band gaps in energy can be controlled via the strength of the Rashba
spin-orbit interaction. We have also calculated the charge conductance through a periodic potential
of a ﬁnite length via the non-equilibrium Green’s function method combined with the Landauer
formalism. We ﬁnd dips in the conductance that correspond well to the energy gaps of the inﬁnite
wire superlattice. From the inﬁnite wire energy dispersion, we derive an equation relating the
location of the conductance dips as a function of the (gate controllable) Fermi energy to the Rashba
spin-orbit coupling strength. We propose that the strength of the Rashba spin-orbit interaction can
be extracted via a charge conductance measurement.
PACS numbers: 71.70.Ej, 85.75.-d, 75.76.+j
I. INTRODUCTION
During the last two decades there has been much inter-
est in using the electron spin in electronic devices. This
research ﬁeld, often referred to as spintronics, has already
made great impact on metal-based information storage
systems. There are hopes that a similar success can also
be achieved in semiconductor based systems1,2. Manip-
ulating the spins of the electrons via external magnetic
ﬁelds over nanometer length scales is not considered fea-
sible. Another, more attractive, method is to use electric
ﬁelds to manipulate electron spins via spin-orbit inter-
action. The spin-orbit interaction arises from the fact
that an electron moving in an external electrical ﬁeld ex-
periences an eﬀective magnetic ﬁeld in its own reference
frame, that in turn couples to its spin via the Zeeman
eﬀect3.
In condensed matter systems, the spin-orbit interac-
tion is found in crystals with asymmetry in the underly-
ing structure4. In bulk this is seen in crystals without an
inversion center (e.g zincblende structures) and is termed
the Dresselhaus spin-orbit interaction5. On the other
hand the structural asymmetry of the conﬁning poten-
tial in heterostructures gives rise to the so called Rashba
term6. The Rashba interaction has practical advantages
in that it depends on the electronic environment of the
heterostructure which can be modiﬁed in sample fabrica-
tion and in-situ by gate voltages7,8. This results in the
possibility of varying the spin-orbit interaction on the
nanometer scale. Interestingly, even structurally sym-
metric heterostructures can present spin-orbit interaction
provided that coupling between subbands of distinct par-
ities is allowed9,10.
The spin-orbit strength can be measured in a variety of
diﬀerent experimental setups11: In a magnetoresistance
measurement via Shubnikov-de Haas oscillations7,8,12–14,
weak (anti-) localization13,15–17, or electron spin reso-
nances in semiconducting nanostructures18–21 or quan-
tum dots22, or optically via spin relaxation23, spin
precession18, spin-ﬂip Raman scattering24, or radiation-
induced magnetoresistance oscillations25.
Backgate
2DEG
Fingergates
Topgates
Parabolic quantum wire
FIG. 1. Schematic of the proposed experimental setup. The
top gates produce the parabolic conﬁnement of the quantum
wire while the ﬁnger gates produce the longitudinal periodic
potential. The Rashba spin-orbit interaction is controlled by
the backgate.
In this paper we propose a method for extracting the
strength of the Rashba spin-orbit interaction via a charge
conductance measurement. This method does not require
the use of external magnetic ﬁelds or radiation sources.
We consider a quantum wire modulated by an external
periodic potential, Fig. 1. Essentially, the method relies
on the fact that the Rashba-induced shifts of the band
gap positions in energy dramatically alter the charge con-
ductance of the superlattice.
Via direct diagonalization, we determine the band
2
structure of an inﬁnite parabolically conﬁned quantum
wire. The energy bands clearly show band gaps that
are renormalized by the Rashba interaction. Interest-
ingly, the band gaps shift in energy as the strength of the
Rashba interaction is varied. The location of the band
gaps are at the crossing of energy bands from adjacent
Brillouin zones. These energy bands can be calculated
via an analytical approximation scheme26.
Using Non-Equilibrium Green’s Functions (NEGF)
and the Landauer formalism we calculate the charge con-
ductance through a ﬁnite region containing both a peri-
odic potential and the Rashba interaction. In the con-
ductance we ﬁnd several dips appearing at diﬀerent lo-
cation in energy. Moreover, these conductance dips coin-
cide with renormalized band gaps of the superlattice. As
with the band gaps the positions of the conductance dips
are at the crossing points of the energy bands from next
neighbor Brillouin zones. For a wide range of Rashba
spin-orbit interaction strengths some of the conductance
dips shift linearly in energy as function of the strength of
the Rashba coupling. For this range we derive a relation,
Eq. (12), describing the location of linearly shifting dips
in energy as a function of Rashba interaction strength.
The Rashba coupling can therefore be extracted by ﬁt-
ting the shift of conductance dips via Eq. (12). Figure 1
shows a schematic of the proposed experimental setup.
This paper is organized as follows. In Sec. II we calcu-
late the energy bands of the inﬁnite wire superlattice via
direct diagonalization. By using an analytical approxi-
mation scheme26 we then show how the resulting band
gaps depend on the Rashba spin-orbit interaction. Via
the NEGF method and the Landauer formalism we in-
troduce in Sec. III a numerical scheme to calculate the
charge conductance through a ﬁnite region. This region
is connected to electron reservoirs to the left and right,
and contains both a periodic potential and a Rashba spin-
orbit coupling. In Sec. IV we then show that the band
picture of the inﬁnite wire superlattice, developed in Sec.
II, is applicable to the ﬁnite length periodic potential. We
close Sec. IV with a discussion about possible experimen-
tal procedures. Lastly, in Sec. V we show that our results
are robust against ﬂuctuations in the strength and width
of the periodic potential.
II. MODEL SYSTEM: THE INFINITE WIRE
SUPERLATTICE
We investigate an inﬁnite quasi-1D parabolic wire with
a uniform Rashba spin-orbit coupling in the presence of
a longitudinal modulation described by the potential
Vp(x) = Vp0 X
n
cnein 2π
λx,(1)
where λis the period of the superlattice. The Hamilto-
nian that describes this system is
H=1
2m(p2
x+p2
y) + 1
2mω2
0y2
+α
~(pyˆσxpxˆσy) + Vp(x),(2)
were mis the eﬀective mass, pxand pyare the momen-
tum operators in the longitudinal and transverse direc-
tion of the wire, αis the Rashba spin-orbit strength, and
ω0is the conﬁnement frequency of the parabolic poten-
tial.
To ﬁnd the eigenvalues of the Hamiltonian in Eq. (2) it
is convenient to introduce the standard ladder operator
ˆaof the parabolic conﬁnement and rotate the spin oper-
ators so that the pxpart in the Rashba interaction term
couples to the ˆσzoperator26
e
H=eˆσx/4He ˆσx/4
=1
2k2π
λn2
+1
2+ ˆaˆaqRk2π
λnˆσz
+iqR
8σ+aˆa)h.c.] + Vp(x)
=e
H0+e
H1,(3)
here ˆσ+= ˆσx+iˆσyis the spin ladder operator, qR=mα
~2
is the rescaled Rashba strength and ~k2π
λnare the
eigenvalues of the operator px. In Eq. (3) we scale all
lengths in oscillator length l=p~/mω0and all energies
in ~ω0. We separate the Hamiltonian e
Hinto a diagonal
part,
e
H0=1
2k2π
λn2
+1
2+ ˆaˆaqRk2π
λnˆσz
+c0Vp0,(4)
and a non-diagonal part
e
H1=iqR
8σ+aˆa)h.c.] + X
n6=0
cnein 2π
λ.(5)
A. Zeroth-order eigenstates and eigenvalues
The eigenstates of the e
H0Hamiltonian are represented
by the kets |k, m, si. Here mis the quantum number of
the harmonic transverse energy bands, i.e. the eigenvalue
of the ˆaˆaoperator, and sis the eigenvalue of the ˆσz
operator with s= +1 and s=1 denoting the spin up
and spin down states, respectively. The corresponding
eigenenergies are
En
m,s(k) = 1
2k2π
λn2
+1
2+msqRk2π
λn
+c0Vp0.(6)
3
0
0.5
1.5
2.5
3.5
4.5
5.5
6.5
0 0.5 1 1.5 2 π
λ
E1
2Vp0 [~ω0]
k[l1]
(a)
k
c
(b)
k
c
E
(c)
k
c
E+~ω0
k
c
E+ 2~ω0
(d)
FIG. 2. (a) Energy bands of the zeroth-order Hamiltonian
e
H0for l= 50 nm, λ= 62 nm= 1.2l,α= 8 meVnm (qR=
0.21l1), k
c= 2.3l1,c0= 1/2, and Vp0 = 0.5. (b) Crossing
of the energy bands corresponding to the |k , 0,1i,|k, 1,+1i,
and |k2π/λ, 0,1istates. (c) Crossing of the energy bands
corresponding to the |k , 1,1i,|k, 2,+1i,|k2π/λ, 1,1i,
and |k2π/λ, 0,+1istates. (d) Same as (c) but for higher
bands, i.e. mm+ 1.
A plot of the En
m,s(k) energy bands vs kin half of the
Brillouin zone, i.e., k= 0 . . . π/λ, can be seen in Fig. 2
(a).
The Hamiltonian e
H1introduces couplings i) between
the |k, m, siand the |k, m ±1,sistates, due to the
Rashba interaction, and ii) between the |k, m, siand
the |k±n2π/λ, m, sistates, due to the periodic poten-
tial. Note that the coupling is strongest where the energy
bands corresponding to these states cross each other. For
some particular Rashba strength αthe energy bands
associated with the states |k, 0,1iand |k, 1,+1iof e
H0
cross at k
c= 1/(2q
R), where q
R=αm/~2. If we
choose the period λof the superlattice potential, Eq. (1),
as
λ= 2πq
R
1 + 2(q
R)2,(7)
the |k2π/λ, 0,1ienergy band will also cross at k
c.
This crossing point occurs in energy at
E=1
8q
R
+ 1 + c0Vp0,(8)
see Fig. 2 (b). In the following we refer to this par-
ticular choice of parameter αas the reference spin-
orbit coupling strength. Similar crossing also occur at
k
cwith the energy bands associated with the states
|k, m, 1i,|k, m + 1,+1i, and |k2π/λ, m, 1ifor en-
ergies E+m~ω0with the energy band corresponding
to the state |k2π/λ, m 1,+1icrossing close by. The
crossings for m= 1 and m= 2 can be seen, respectively,
in Fig. 2 (c) and (d).
B. Coupling between eigenstates
We calculate the eigenenergies of the full Hamiltonian,
e
H, via direct diagonalization. The resulting energy bands
are plotted in Fig. 3 (a). The energy bands that cross at
k
cand the one crossing close by, see Sec. II A, corre-
sponds to the states coupled by e
H1. A blow up of the
resulting energy gaps can be seen in Fig. 3 (b). At Ethe
coupling results in a double energy gap, see Fig. 3 (c),
and a triple energy gap at the higher energy crossings,
see Fig. 3 (d) and (e). In Fig. 3 the spin-orbit strength
is at the reference value α.
Note that for a non-zero Rashba coupling, as in Fig.
3, the energy gaps have shifted from the Bragg plane at
k=π/λ. This is because the spin-orbit interaction shifts
the wave-number kof the electrons in the longitudinal
direction and thus renormalizes the locations of interfer-
ences.
0.5
1.5
2.5
3.5
4.5
5.5
6.5
k
c
-π
λ0π
λ
E1
2Vp0 [~ω0]
k[l1]
(a)
2.5
3.5
4.5
5.5
6.5
2k
cπ
λ
E1
2Vp0 [~ω0]
k[l1]
(b)
k
c
E
k
c
E+~ω0
k
c
E+ 2~ω0
(c)
(d)
(e)
FIG. 3. (a) Energy bands of the Hamiltonian e
H=e
H0+e
H1
for the same parameters as used in Fig. 2. The energy bands
are calculated via direct diagonalization of e
H. (b) A blow up
focusing on the band gaps of the ﬁrst three crossings at k
c.
(c) The band gaps formed at the energy band crossing shown
in Fig. 2 (b). In (d) and (e), the energy band crossings shown
in Fig. 2 (c) and (d), respectively.
When the strength of the Rashba interaction is
changed the crossing points kcof the energy-bands shift
and with them the energy gaps. These crossing points
4
can be worked out analytically by using the eigenener-
gies of the eﬀective Rashba Hamiltonian of the parabolic
wire26, which are
εn
m,(k, qR) = kn2π
λ2
2+mq2
R
2(1 + 2qRkn2π
λ)
+ ∆mkn2π
λ, qR+c0Vp0,(9)
and
εn
m,(k, qR) = kn2π
λ2
2+m+ 1 q2
R
2(1 + 2qRkn2π
λ)
m+1 kn2π
λ, qR+c0Vp0,(10)
for k2π/λ 0. For k2π/λ < 0εm,s (k, qR) =
εm,s(−|k|, qR). In Eq. (9) and (10)
m(k, qR) = 1
2"12qRkq2
Rm
1 + 2qRk2
+ 2q2
Rm1q2
Rm
4(1 + 2qRk)2#1/2
,(11)
and we have added the energy constant c0Vp0 result-
ing from the periodic potential and replaced kwith
kn2π/λ. Note that the energy bands described by Eq.
(9) and Eq. (10) are derived for wires without a longitudi-
nal periodic potential and therefore do not contain energy
gaps that result from the periodic potential, i.e. they are
the solution of the e
H, see Eq. (3), with c06= 0 and cn= 0
for n6= 0. Having determined the crossing points, we in-
sert them into either of the crossing energy bands, Eq.
(9) or Eq. (10), to obtain the location of the crossing
points in energy as a function of α. In the appendix we
present the equation for the crossing point between the
energy bands ε0
m,(k, qR) and ε1
m1,(k, qR) correspond-
ing to states |k, m + 1,+1iand |k2π/λ, m, 1i, see
Eq. (A3). To determine the location in energy of this
crossing point as a function of αwe insert kcfrom Eq.
(A3) into either ε0
m,(k, qR) or ε1
m1,(k, qR).
Now, as the crossing points shift in energy with αthey
can be thought of as trajectories. The trajectories of the
crossing points, calculated from Eq. (9) and (10), can
be seen in Fig. 4. For higher energies there are further
trajectories.
We index the trajectories via mand the letters “C”,
“L”, and “R”. The trajectories that we label with the
letter C only shift a little to the left for α > α(see
lowest horizontal line in Fig. 4). Relative to the C labeled
trajectories and again for α > α, we see trajectories that
make a large shift to the left and right. Those tra jectories
we, respectively, label with the letters “L” and “R”.
For spin-orbit strengths close to the reference strength,
α, the crossing points follow nonlinear trajectories. As
the spin-orbit coupling becomes larger or lower than the
0
5
10
15
20
25
30
2.5 3.5 4.5 5.5 6.5
α[meVnm]
EF1
2Vp0[~ω]
α
α1
α0
0L 0C 1L 1C 1R 2L 2C 2R 3L
FIG. 4. (Color online) Trajectories of the crossing points be-
tween the energy bands. The black dashed lines are the lin-
earization of the left moving trajectories around α0. We also
mark the values α= 8 meVnm and α1= 12 meVnm onto
the y-axis. These values are used for the conductance results
in Fig. 6.
reference strength the trajectories quickly become more
linear. The choice of the reference Rashba spin-orbit
strength αdetermines the period of the periodic po-
tential, see Eq. (7).
C. Extracting the Rashba coupling
We will show in Sec. III that the band gaps appear as
dips in the charge conductance through a ﬁnite periodic
potential. By ﬁtting the measured energy shift of the con-
ductance dips to the trajectories in Fig. 4 it is possible to
extract the value of α. The linear parts of the trajectories
are best suited for ﬁtting. It would therefore be conve-
nient that the range of αextracted via the ﬁtting is con-
tained within the linear region. This can be achieved by
choosing a suﬃciently low α-value (and thus λ, Eq. 7).
We present below a linearized equation for the α-value of
the crossing point between the energy bands correspond-
ing to the states |k, m + 1,+1iand |k2π/λ, m, 1ias
a function of Fermi energy. By Taylor expanding around
some point α0in the linear region of the trajectories and
making a linear approximation we obtain
α=~2
m
~
lqR0 EF1
2Vp0 F(m)
G(m),(12)
here F(m) and G(m) are known functions of the energy
band index m, see Eq. (A7) and Eq. (A8). The derivation
of Eq. (12) is shown in the appendix. In Fig. 4 we plot as
dashed lines linearized trajectories described by Eq. (12)
where we have chosen α0= 20 meVnm.
5
III. FINITE PERIODIC POTENTIAL
In this section we calculate in the linear response the
conductance through a ﬁnite periodic potential. This is
done via the Landauer formula together with the NEGF
method.
A. The system setup
We consider a hardwalled wire of width Lywith a
transverse parabolic potential. We divide the wire into
a ﬁnite central region of length Lxand semi-inﬁnite
left and right parts, see Fig. 5. The central region in-
cludes a Rashba spin-orbit interaction described by a
symmetrized Hamiltonian which is turned on smoothly,
at both the left and right ends. In the central region we
also assume a longitudinal periodic potential that repre-
sents the potential due to the ﬁngergates. We describe
the central region by the Hamiltonian
HC=1
2m(p2
x+p2
y) + 1
2mω2
0y2
+1
2~{α(x, y), pyˆσxpxˆσy}
+Vp0
1
21cos 2π
λx (13)
Here {,}denotes an anticommutator.
The left and right leads contain the same parabolic
potential as in the center region but neither a Rashba
spin-orbit interaction nor a periodic potential in the lon-
gitudinal direction , i.e., they are described by the Hamil-
tonian
HL/R=1
2m(p2
x+p2
y) + 1
2mω2
0y2.(14)
The total system, HT=HL+HC+HR, is discretized
via the ﬁnite diﬀerence method on a grid of Nx×Ny
points with a mesh size a. A schematic of the system can
be seen in Fig. 5.
B. The numerical formalism
Here we use the NEGF method27–29 to calculate the
charge conductance. The method requires us to ﬁnd the
retarded Green’s function of the central region Gr
C. To
do this we have to isolate Gr
Cfrom the inﬁnite matrix
equation describing the retarded Green’s function of the
total system
(EIHT)Gr
T=I.(15)
HC
α6= 0
Vp0 6= 0
HL
α= 0
Vp0 = 0
HR
α= 0
Vp0 = 0
Lx
Ly
a
Rashba coupling
smoothly turned on
FIG. 5. Schematic of the system used in our numerical simu-
lations. The system is divided into a central region of length
Lx, described by the Hamiltonian HCand two semi-inﬁnite
wires, described by the Hamiltonians HLand HR. The total
system is discretized on a grid with mesh size a. At the left
and right edges of the central region the Rashba spin-orbit
coupling is smoothly turned on.
By separating the total Green’s function into a left, right,
and central part we can write Eq. (15) as
EIHLHLC 0
HCL EIHCHCR
0HRC EIHR
Gr
LGr
LC Gr
LR
Gr
CL Gr
CGr
CR
Gr
RL Gr
RC Gr
R
=
I0 0
0I0
0 0 I
.(16)
The matrices HCj =HjC =tI, couple together the cen-
tral region to the left (j= L) and right leads (j= R).
Here t=~/2ma2is the tight-binding hopping parame-
ter that results from the discretization. Multiplying out
Eq. (16) gives nine matrix equations from which we can
isolate a ﬁnite matrix equation for Gr
C. This is done
by treating the contributions from the inﬁnite leads as
self-energies27–30. The matrix equation for Gr
Cis
EIHCX
j
ΣjGr
C=I,(17)
and the self-energy of lead j=L, R is Σj=HCjGjj HjC
where Gjj is the Green’s function of lead jand is deter-
mined analytically27. Here HCis the discretized Hamil-
tonian of Eq. (2) over the central region. All the ma-
trices are 2NxNy×2NxNymatrices. In order to save
computational power only the necessary Green’s func-
tion matrix elements are calculated with the recursive
Green’s function method28 . Here we are interested in
low biases and hence focus on the linear response regime.
From the Green’s function the charge conductance at the
Fermi Energy EFcan be calculated, via the Fisher-Lee
relation31, as
G(EF) = 2e2
hTr [ΓRGr
CΓLGa
C],(18)
where Γj=2Im (Σj) and Ga
C= (Gr
C). In what fol-
lows we use Eq. (18) to calculate the charge conductance
through our system.
6
C. Numerical values used in the calculation
In our simulations we consider a Ga1xInxAs alloy
with an eﬀective mass m= 0.041me, with mebeing
the bare electron mass. The width of the wire is set as
Ly= 500 nm and its length as Lx= 4.92 µm. The oscil-
lator length is l= 50 nm which corresponds to ~ω0= 0.74
meV. For the discretization we use Nx×Ny= 600 ×63
points with a= 8.1 nm being the distance between
nearest neighbors. This corresponds to a tight-binding
hopping parameter t= 29 meV= 39 ~ω0. We choose
the reference Rashba strength as α= 8 meV nm (i.e.
k
R= 0.21 l1) which results in an energy band cross-
ing point at k
c= 1/(2q
R) = 2.3l1and a period
λ=π/ (k
c+qR) = 1.23 l= 7.62 a= 61.5 nm. This cor-
responds to Lx=80 ﬁngergates. The strength of the
periodic potential is set as Vp0 = 0.5~ω0= 0.37 meV.
IV. RESULTS
In Fig. 6 we compare the calculated conductance, Fig.
6 (a), through the ﬁnite wire to the energy bands of the
inﬁnite wire, Fig. 6 (b) and (c), as a function of Fermi
energy. Results for two values of αare presented, α= 8
meVnm and α1= 12 meVnm. For comparison we also
plot in Fig. 6 (a) the conductance through a non-periodic
potential Vp(x) = 1/2Vp0 for the same values αmen-
tioned above, i.e. the corresponding gapless systems32 .
Each energy gap is associated to its corresponding con-
ductance dip via a labeled arrow. The labeling on the
arrows is the same labeling as those of the trajectories
in Fig. 4. We see that there is a good correspondence
between the dips in conductance and the gaps in the en-
ergy band for both αand α1. This indicates that the
behavior of the ﬁnite periodic potential can be properly
described with the band model from the inﬁnite periodic
potential. In Fig. 4 we can see how the band gaps shift
in energy as a function of α. We want see if the energy
shift of conductance dips correlates with the shift of the
band gaps. This is most easily seen by considering the
diﬀerential conductance
∂G
∂EFG(EFi+1 )G(EFi)
EFi+1 EFi
.(19)
In Fig. 7 we plot G
∂EFas a function of EFand α. There
we see the conductance steps of the parabolic wire as rel-
atively straight vertical trajectories appearing at the tic
marks on the horizontal axis. The rest of the trajectories
in Fig. 7 correspond to dips in the charge conductance.
By comparing the trajectories, that the dips in charge
conductance form, to the ones of the band gaps in Fig. 4
we see an excellent match. This ﬁrmly conﬁrms that the
conductance dips of the ﬁnite wire can be described via
the band model for the superlattice. We also plot in Fig.
7 as light-gray lines the linearized trajectories of crossing
points between the energy bands corresponding to the
2
k
c
π/λ
(b)
2
3
4
5
6
7
8
9
G[2e2
h]
(a)
2
k
c
π/λ
2.5 3.5 4.5 5.5 6.5
EF1
2Vp0 [~ω0]
(c)
k[l1]k[l1]
0L 0C 1L 1C 1R 2L 2C 2R 3L
α1
α
0L 0C 1L 1C 1R 2L 2C 2R 3L
FIG. 6. (Color online) Comparison between energy gaps of
the inﬁnite superlattice and conductance dips of the ﬁnite
periodic potential. In (a) the results from the numerical cal-
culation of the conductance is shown for α1= 12 meVnm,
and α= 8 meVnm. Note that the conductance curve for
α1has been shifted by 2×2e2/h for clarity. The results for a
wire with a non-periodic potential Vp(x) = 1/2Vp0 is plotted
with black solid lines. In (b) and (c) the energy bands of the
inﬁnite wire superlattice are plotted for the same αvalues as
in (a), respectively. Note that (c) is the same ﬁgure as Fig.
3 (b), but rotated clockwise by 90. The correspondence be-
tween the nine pairs of dips and band gaps are indicated by
the labeled arrows.
|k, m + 1,+1iand |k2π/λ, m, 1istates, see Eq. (12).
The trajectories are linearized around α=20 meVnm and
are identical to the straight dashed lines in Fig. 4.
A. Discussion of possible experimental procedures
From Fig. 7 we can extract the Rashba spin-orbit cou-
pling by ﬁtting the linear energy shift of the conductance
dips, via Eq. (12). The energy shift, i.e. the change
in Fermi energy, is controlled by the chemical potential
of the leads. The backgate that controls the spin-orbit
strength introduces an extra shift in the Fermi energy
(due to the electrostatic coupling of the backgate to the
2DEG). This shift can be compensated by changing the
chemical potential of the leads by an equal amount. Sim-
ilar methods have been used to probe the energy spec-
trum of quantum dots using transport methods33. An-
other way to compensate for this shift is to use a com-
bination of back and front gates. This has been experi-
mentally demonstrated to control the Rashba spin-orbit
interaction strength without introducing charging in the
2DEG34.
7
α
α1
α0
α
α1
α0
2.5 3.5 4.5 5.5 6.5
EF1
2Vp0 [~ω0]
0
5
10
15
20
25
30
α[meVnm]
-20 -10 0 10 20 ∂G
∂EF[a.u.]
FIG. 7. (Color online) Plot of the diﬀerential conductance
∂G/∂EFthrough the ﬁnite superlattice as a function of the
Fermi energy EFand spin-orbit coupling, α. The linearized
trajectories of the crosspoints between the energy bands cor-
responding to the |k , m + 1,+1iand |k2π/λ, m 1,+1i
states are shown with gray solid lines. The trajectories are
linearized around α0= 20 meVnm. The Rashba spin-orbit
values, α= 8 meVnm and α1= 12 meVnm, correspond-
ing to the curves in Fig. 6 are marked on the right border.
Note that qR=mα/~2land that the diﬀerential conduc-
tance ∂G/∂EFis in arbitrary units.
V. VARIATIONS OF PARAMETERS IN THE
PERIODIC POTENTIAL
The periodic potential plays a crucial role in in the
formation and behavior of the energy gaps. We there-
fore devote this section to study the eﬀect that variations
of parameters in the periodic potential has on the con-
ductance. In what follows we consider variations on the
length and strength of the periodic potential (Sec. V A)
as well as random ﬂuctuations on the period and strength
of each ﬁnger gate potential (Sec. V B).
A. Changing the periodic potential parameters
Here we examine the eﬀects of changing the strength
of the periodic potential, Vp0, and the length Lx. In Fig.
8 (a) the length of the wire is varied such that it contains
10, 40, 80, or 160 ﬁngergates with a ﬁxed period λ= 63
nm. There we see the dips easily for &40 ﬁngergates
and they become clearer for more ﬁngergates. There is
not much change between the conductance results for 80
and 160 ﬁngergates, i.e. 80 ﬁngergates is an adequate
number of ﬁngergates. Fig. 8 (b) shows results where
the strength of the periodic potential has been varied be-
tween values of 0.2, 0.5, 1.0, and 1.5 ~ω0. We see that the
width of the conductance dips grows with larger Vp0. But
2
4
6
8
10
12
G[2e2
h]
(a)
2
4
6
8
10
12
2.5 3.5 4.5 5.5
G[2e2
h]
EF1
2Vp0 [~ω0]
(b)
160 ﬁngergates
80 ﬁngergates
40 ﬁngergates
10 ﬁngergates
Vp0 = 1.5~ω
Vp0 = 1.0~ω
Vp0 = 0.5~ω
Vp0 = 0.2~ω
FIG. 8. (Color online) Charge conductance Gthrough the
ﬁnite periodic potential as a function of Fermi energy EF
for (a) diﬀerent values of Vp0 and (b) diﬀerent number of
ﬁngergates. All the conductances were calculated with α= 12
meVnm. Note that the conductances have been separated by
2×2e2/~.
as Vp0 increases the total conductance strength Gdeteri-
orates due to interferences. This makes the detection of
the conductance dips diﬃcult.
B. Eﬀects of ﬂuctuations in the periodic potential
The periodic potential used in the previous section can
be considered as being ideal. A more realistic case would
be if there were some ﬂuctuations in the potential. To
verify the robustness of the results in the previous sec-
tion we rerun the simulations with a non-ideal periodic
potential. This is done by introducing a 5% Gaussian
error in both the height and length of each potential hill
created by the ﬁngergates. Figure 9 (b) shows that extra
noise is added to our conductance result; still the tra-
jectories of the conductance dips are fairly visible. To
counter against the noise introduced by ﬂuctuations we
could add extra periods to the periodic potential. This
helps averaging out the noise, as can be seen in Fig. 9 (a)
where we have quadrupled the length of the wire and the
number of ﬁngergates. This (self)averaging is however
slow and requires a large number of ﬁngergates. Another
solution, as we are mainly interested in the slope of the
crosspoint trajectories, is actually to reduce the number
of ﬁngergates to an optimal number. As can be seen
in Fig. 9 (c) we obtain many extra trajectories resulting
from the imprecision of the period. But the slope of the
trajectories of the conductance dips can be easily ﬁtted
for α&10. As expected if we go down to as few as 10
ﬁngergates the eﬀects of the periodic potential is nearly
washed out.
8
(c) (40 ﬁngergates)
(a) (320 ﬁngergates) (b) (80 ﬁngergates)
(d) (10 ﬁngergates)
2.5 3.5 4.5 5.5
EF1
2Vp0 [~ω0]
0
5
10
15
20
25
30
α[meVnm]
0
5
10
15
20
25
30
α[meVnm]
2.5 3.5 4.5 5.5
EF1
2Vp0 [~ω0]
-20 -10 0 10 20 ∂G
∂EF[a.u.]
FIG. 9. Plots of the diﬀerential conductance ∂G/∂EFthrough
the ﬁnite periodic potential as a function of Fermi energy
EFand spin-orbit coupling, α. Four cases are considered: A
system with (a) 320 , (b) 80, (c) 40, and (d) 10 ﬁngergates. In
all the plots we introduce a 5% Gaussian error in the length
and height of the potential in each period.
VI. CONCLUSION
We have studied a parabolic quantum wire with
Rashba spin-orbit interaction and a longitudinal periodic
potential. We ﬁnd that the energy gaps resulting from
the periodic potential split up and shift from the Bragg
plane due to the Rashba spin-orbit interaction. Above a
certain spin-orbit strength some of the new energy gaps
shift linearly in energy as a function of the Rashba spin-
orbit strength. We propose that this eﬀect can be used to
measure the change in the Rashba spin-orbit strength, e.
g. resulting from a voltage gate8. The energy gaps result
in dips in the charge conductance. The energy shift of
these dips can be ﬁtted, by an analytical equation that
we derive, Eq. (12), to extract the strength of the Rashba
spin-orbit interaction. The advantage of the this method
is that it only requires conductance measurement and not
any external magnetic or radiation source.
Appendix A: Energy crossing points
In this appendix we calculate the crossing point kcof
the |k, m, 1iand |k2π/λ, m 1,+1ienergy bands.
We need to solve the equation
ε0
m,(k, qR)ε1
m1,(k, qR) = 2π
λk2π2
λ2+ 1
+A(k, qR)m+1(k , qR) + ∆m(k+ 2 π
λ, qR) = 0
(A1)
for k. Here we used Eq. (9) and (10) and have deﬁned
A(k, qR) = q2
R
21
12qRk+4π
λqR1
1 + 2qRk.(A2)
Equation A1 is a nonlinear equation which is hard to
solve directly. An easier way is to linearize the A(k)
and ∆m(k) functions around the point k=k
c. This
approximation introduces little errors and allows us to
write the crossing point as
kc=
2π2
λ21B(m, qR) + C(m, qR)k
c
2π
λ+C(m, qR).(A3)
where
B(m, qR) = A(k
c, qR)m+1(k
c, qR)
+ ∆m(k
c+ 2π/λ, qR),(A4)
and
C(m, qR) = A(k, qR)
∂k k=k
cm+1(k, qR)
∂k k=k
c
m(k, qR)
∂k k=k
c+2π/λ
.(A5)
The trajectories of the crossing points across the
energy-Rashba spin-orbit coupling surface are then
ε0
m,(kc, qR) or ε1
m1,(kc, qR). The crossing points and
their trajectories for the other energy bands, namely
the ones corresponding to the states |k, m + 1,+1iand
|k2π/λ, m, 1i, the |k, m, 1iand |k2π/λ, m, 1i,
and the |k, m + 1,+1iand |k2π/λ, m 1,+1i, can be
worked out in the same way.
The kvalue of the crossing point between the
ε0
m,(k, qR) and ε1
m1,(k, qR) bands remains almost con-
stant as a function of α. This is because these bands
move at similar rate in opposite directions in kspace as
αis changed. We take an advantage of this when ﬁnding
a linear equation around some point α0in the straight
segments of the trajectories. We know that k
cis a cross-
ing point for α=αso we can, instead of using Eq.
(A3), make the approximation that kck
cfor all α. We
then linearize ε0
m,(k
c, qR) by Taylor expanding around
α0. Solving for qRin ε0
m,(k
c, qR) and scaling back to α
we obtain
α=~2
m
~
l"qR0 ε0
m,F(m)
G(m)#,(A6)
9
where
F(m) = 1
2(k
c)2+m+ 1 q2
R0
2(1 + 2qR0k
c)m+1(k
c, qR0)
= 3.7 + m
p6.0×105m34.1×102m2+ 0.20m+ 0.79,
(A7)
and
G(m) = k
cq2
R0 +qR0
4(k
c)2q2
R0 + 4k
cqR0 + 1 +m+1(k
c, qR0)
∂qRqR=qR0
= 9.8×102
+5.1×104m32.7×102m2+ 0.82m+ 4.4
2.4×104m31.6×102m2+ 0.79m+ 3.1.
(A8)
In the last steps of the Eq. (A7) and Eq. (A8) we have
plugged in the values k
c= 2.35 and qR0 = 0.537, which
correspond to α= 8 meVnm and α0= 20 meVnm.
Note that all lengths are scaled in the oscillator length
l=p~/mω0and all energies in ~ω0.
ACKNOWLEDGMENTS
This work was supported by the Icelandic Science
and Technology Research Program for Postgenomic
Biomedicine, Nanoscience and Nanotechnology, the Ice-
landic Research Fund, the Research Fund of the Univer-
sity of Iceland, the Swiss NSF, the NCCRs Nanoscience
and QSIT, CNPq, and FAPESP
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We introduce an analytical approximation scheme to diagonalize parabolically confined two-dimensional (2D) electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts W and V, which couple crossing and noncrossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with W can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the V coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength kRℓ of V, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the nth Landau-level gn factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk g factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.
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The manipulation of electric charge in bulk semiconductors and their heterostructures forms the basis of virtually all contemporary electronic and opto-electronic devices. Recent studies of spin-dependent phenomena in semiconductors have now opened the door to technological possibilities that harness the spin of the electron in semiconductor devices. In addition to providing spin-dependent analogies that extend existing electronic devices into the realm of semiconductor "spintronics," the spin degree of freedom also offers prospects for fundamentally new functionality within the quantum domain, ranging from storage to computation. It is anticipated that the spin degree of freedom in semiconductors will play a crucial role in the development of information technologies in the 21st century. This book brings together a team of experts to provide an overview of emerging concepts in this rapidly developing field. The topics range from spin transport and injection in semiconductors and their heterostructures to coherent processes and computation in semiconductor quantum structures and microcavities.
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Introduction.- Band Structure of Semiconductors.- The Extended Kane Model.- Electron and Hole States in Quasi 2D Systems.- Origin of Spin-Orbit Coupling Effects.- Inversion Asymmetry Induced Spin Splitting.- Anisotropic Zeeman Splitting in Quasi 2D Systems.- Landau Levels and Cyclotron Resonance.- Anomalous Magneto-Oscillations.- Conclusions.- Notation and Symbols.- Quasi Degenerate Perturbation Theory.- The Extended Kane Model: Tables.- Band Structure Parameters.
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