Content uploaded by Daniel Loss

Author content

All content in this area was uploaded by Daniel Loss

Content may be subject to copyright.

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

2m∗ω2

0y2

+α

~(pyˆσx−pxˆσy) + Vp(x),(2)

were m∗is 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=e−iπˆσx/4Heiπ ˆσx/4

=1

2k−2π

λn2

+1

2+ ˆa†ˆa−qRk−2π

λnˆσz

+iqR

√8[ˆσ+(ˆa†−ˆa)−h.c.] + Vp(x)

=e

H0+e

H1,(3)

here ˆσ+= ˆσx+iˆσyis the spin ladder operator, qR=m∗α

~2

is the rescaled Rashba strength and ~k−2π

λnare the

eigenvalues of the operator px. In Eq. (3) we scale all

lengths in oscillator length l=p~/m∗ω0and all energies

in ~ω0. We separate the Hamiltonian e

Hinto a diagonal

part,

e

H0=1

2k−2π

λn2

+1

2+ ˆa†ˆa−qRk−2π

λnˆσz

+c0Vp0,(4)

and a non-diagonal part

e

H1=iqR

√8[ˆσ+(ˆa†−ˆa)−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 ˆa†ˆaoperator, 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

2k−2π

λn2

+1

2+m−sqRk−2π

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

E−1

2Vp0 [~ω0]

k[l−1]

(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=

0.21l−1), k∗

c= 2.3l−1,c0= 1/2, and Vp0 = 0.5. (b) Crossing

of the energy bands corresponding to the |k , 0,−1i,|k, 1,+1i,

and |k−2π/λ, 0,−1istates. (c) Crossing of the energy bands

corresponding to the |k , 1,−1i,|k, 2,+1i,|k−2π/λ, 1,−1i,

and |k−2π/λ, 0,+1istates. (d) Same as (c) but for higher

bands, i.e. m→m+ 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 α∗the energy bands

associated with the states |k, 0,−1iand |k, 1,+1iof e

H0

cross at k∗

c= 1/(2q∗

R), where q∗

R=α∗m∗/~2. If we

choose the period λof the superlattice potential, Eq. (1),

as

λ= 2πq∗

R

1 + 2(q∗

R)2,(7)

the |k−2π/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 α∗as 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 |k−2π/λ, m, −1ifor en-

ergies E∗+m~ω0with the energy band corresponding

to the state |k−2π/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 E∗the

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

E−1

2Vp0 [~ω0]

k[l−1]

(a)

2.5

3.5

4.5

5.5

6.5

2k∗

cπ

λ

E−1

2Vp0 [~ω0]

k[l−1]

(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) = k−n2π

λ2

2+m−q2

R

2(1 + 2qRk−n2π

λ)

+ ∆mk−n2π

λ, qR+c0Vp0,(9)

and

εn

m,↓(k, qR) = k−n2π

λ2

2+m+ 1 −q2

R

2(1 + 2qRk−n2π

λ)

−∆m+1 k−n2π

λ, qR+c0Vp0,(10)

for k−2π/λ ≥0. For k−2π/λ < 0εm,s (k, qR) =

εm,−s(−|k|, qR). In Eq. (9) and (10)

∆m(k, qR) = 1

2"1−2qRk−q2

Rm

1 + 2qRk2

+ 2q2

Rm1−q2

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

k−n2π/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

m−1,↑(k, qR) correspond-

ing to states |k, m + 1,+1iand |k−2π/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

m−1,↑(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]

EF−1

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 α∗determines 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 |k−2π/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 −EF−1

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

2m∗ω2

0y2

+1

2~{α(x, y), pyˆσx−pxˆσy}

+Vp0

1

21−cos 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

2m∗ω2

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

(EI−HT)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

EI−HL−HLC 0

−HCL EI−HC−HCR

0−HRC EI−HR

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=~/2m∗a2is 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

EI−HC−X

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 Ga1−xInxAs 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 l−1) which results in an energy band cross-

ing point at k∗

c= 1/(2q∗

R) = 2.3l−1and a period

λ=π/ (k∗

c+qR) = 1.23 l= 7.62 a= 61.5 nm. This cor-

responds to Lx/λ =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 α∗and 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

∂EF≈G(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

EF−1

2Vp0 [~ω0]

(c)

k[l−1]k[l−1]

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 |k−2π/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

EF−1

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 |k−2π/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=m∗α/~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]

EF−1

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

EF−1

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

EF−1

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 |k−2π/λ, m −1,+1ienergy bands.

We need to solve the equation

ε0

m,↓(k, qR)−ε1

m−1,↑(k, qR) = 2π

λk−2π2

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

1−2qRk+4π

λqR−1

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

λ2−1−B(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∗

c−∂∆m+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

m−1,↑(kc, qR). The crossing points and

their trajectories for the other energy bands, namely

the ones corresponding to the states |k, m + 1,+1iand

|k−2π/λ, m, −1i, the |k, m, −1iand |k−2π/λ, m, −1i,

and the |k, m + 1,+1iand |k−2π/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

m−1,↑(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=α∗so we can, instead of using Eq.

(A3), make the approximation that kc≈k∗

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

−p6.0×10−5m3−4.1×10−2m2+ 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

= 9.8×10−2

+5.1×10−4m3−2.7×10−2m2+ 0.82m+ 4.4

√2.4×10−4m3−1.6×10−2m2+ 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~/m∗ω0and 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

1S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).

2N. S. D.D. Awschalom, D. Loss, Semiconductor Spintron-

ics and Quantum Computation (Springer, 2002).

3W. Greiner, Relativistic quantum mechanics (Springer,

1987).

4R. Winkler, Spin-orbit Coupling Eﬀects in Two-

Dimensional Electron and Hole Systems (Springer Berlin

Heidelberg, 2010).

5G. Dresselhaus, Phys. Rev. 100, 580 (1955).

6Y. A. Bychkov and E. I. Rashba, Journal of Physics C:

Solid State Physics 17, 6039 (1984).

7G. Engels, J. Lange, T. Sch¨apers, and H. L¨uth, Phys. Rev.

B55, R1958 (1997).

8J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.

Rev. Lett. 78, 1335 (1997).

9E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and

D. Loss, Phys. Rev. Lett. 99, 076603 (2007).

10 R. S. Calsaverini, E. Bernardes, J. C. Egues, and D. Loss,

Phys. Rev. B 78, 155313 (2008).

11 W. Zawadzki and P. Pfeﬀer, Semiconductor Science and

Technology 19, R1 (2004).

12 T. Matsuyama, R. K¨ursten, C. Meißner, and U. Merkt,

Phys. Rev. B 61, 15588 (2000).

13 V. A. Guzenko, T. Sch¨apers, and H. Hardtdegen, Phys.

Rev. B 76, 165301 (2007).

14 P. J. Simmonds, S. N. Holmes, H. E. Beere, and

D. A. Ritchie, J. Appl. Phys. 103, 124506 (2008), ISSN

00218979.

15 T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys.

Rev. Lett. 89, 046801 (2002).

16 J. B. Miller, D. M. Zumb¨uhl, C. M. Marcus, Y. B. Lyanda-

Geller, D. Goldhaber-Gordon, K. Campman, and A. C.

Gossard, Phys. Rev. Lett. 90, 076807 (2003).

17 G. Yu, N. Dai, J. H. Chu, P. J. Poole, and S. A. Studenikin,

Phys. Rev. B 78, 035304 (2008).

18 Y. Kato, R. C. Myers, A. C. Gossard, and D. D.

Awschalom, Nature 427, 50 (2004).

19 M. Duckheim and D. Loss, Nature Physics 2, 195 (2006).

20 L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch¨on, and

K. Ensslin, Nature Physics 3, 650 (2007).

21 M. Duckheim, D. L. Maslov, and D. Loss, Phys. Rev. B

80, 235327 (2009).

22 V. N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B

74, 165319 (2006).

23 P. S. Eldridge, W. J. H. Leyland, P. G. Lagoudakis, O. Z.

Karimov, M. Henini, D. Taylor, R. T. Phillips, and R. T.

Harley, Phys. Rev. B 77, 125344 (2008).

24 B. Jusserand, D. Richards, H. Peric, and B. Etienne, Phys.

Rev. Lett. 69, 848 (1992).

25 R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayana-

murti, W. B. Johnson, and V. Umansky, Phys. Rev. B 69,

193304 (2004).

26 S. I. Erlingsson, J. C. Egues, and D. Loss, Phys. Rev. B

82, 155456 (2010).

27 S. Datta, Electronic Transport in Mesoscopic Systems

(Cambridge University Press, 1995).

28 D. K. Ferry and S. M. Goodnick, Transport in Nanostruc-

tures (Cambridge University Press, 1997).

29 H. M. Pastawski and E. Medina, Rev. Mex. Fis. 47, S1,

1-23 (2001).

30 M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and

J. Rubio, Journal of Physics F: Metal Physics 15, 851

(1985).

31 D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).

32 F. Mireles, and G. Kirczenow, Phys. Rev. B 64, 024426

(2001).

33 L. P. Kouwenhoven, and D. G. Austing, and S. Tarucha,

Rep. Prog. Phys. 64, 701 (2001).

34 D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).