Simulation of the Rashba Effect in a Multiband Quantum Structure

Simulation of the Rashba Effect in a Multiband Quantum Structure

O. Morandi ¹ and L. Demeio ²

1 Dipartimento di Elettronica e Telecomunicazioni, Universit` a degli Studi di Firenze, Via S. Marta 3, I-50139 Firenze, Italy

2 Dipartimento di Scienze Matematiche, Universit` a Politecnica delle Marche, Via Brecce Bianche 1, I-60131 Ancona, Italy

Introduction

Devices containing asymmetric quantum wells where quantized states are spin-split by the Rashba effect have been proposed. In this work we introduce a multiband model derived within the Bloch-Wannier formalism, which gives a full description of the coupling between the conduction and the valence band, including the effect of the degenerate bands and the spin-orbit coupling. In particular, we present the six-band version of our model and some numerical results related to an asymmetric resonant interband tunneling diode.

Multiband envelope function model

We consider an electron of mass m ₀ moving in a periodic potential V _L and subject to an additional external potential U which is treated as a perturbation. The Hamiltonian which governs the motion of the electron is given by

H = H ⁰ + U (r) − iζ (∇U(r) ∧ ∇) · σ H ⁰ = − ^~

2

2m ₀ ∇ ² + V _L (r) − iζ (∇V ^L (r) ∧ ∇) · σ,

where ζ = ^{~ 2} /(4m ² ₀ c ² ), and σ is a vector whose components are the Pauli spin matrices.

The evolution of the electron wave function Ψ(r, t) is determined by the Schr¨odinger equation with the perturbed Hamiltonian H. The Bloch basis ψ n ^α (k, σ, r) provides a set of functions for the expansion of the wave function

Ψ(r, t) = ^X

n,α,σ

Z

B ϕ ^α _n (k, σ) ψ _n ^α (k, σ, r, t) dk,

Here B indicates the first Brillouin zone, the index n denotes the bands and α runs over the possible n _α degenerate states related to the eigenvalue E _n . Finally, the index σ labels the spin of the electron.

The aim of the “kp” approach is to separate the fast oscillating contributions to the Hamiltonian, due to the periodic potential V _L , from the slower contributions, due to the external potential U . This is achieved by expanding the equation of motion with respect to |k| and by taking the Fourier transform of the resulting system. To the first order we obtain

i ^~ ∂ϕ ^α _n (r, σ)

∂t = E _n ^σ ( −i ^~ ∇) ϕ ^α _n (r, σ) + ^~ m ₀

X

n

⁰

6= n, α

⁰

, σ

⁰

Z

B ∇U(r) · Q ϕ ^α _n ^{0 0} (r, σ ⁰ ) dk,

Q = π ^α,α

0

n,n ⁰ (σ, σ ⁰ ) E _n ^σ − E _n ^{σ 0 0} +

*

u ^α,σ _n

    

σ

    

u ^α _n 0 ^{0 ,σ 0}

+

∧ ∇ − ζ

*

u ^α,σ _n

    

σ ∧ ∇

    

u ^α _n 0 ^{0 ,σ 0}

+

,

π =

*

u ^α,σ _n

    

~

4m ₀ c ² (σ ∧ ∇V ^L ) − i ^~ ∇

    

u ^α _n 0 ^{0 ,σ 0}

+

Six-band model

The symmetry properties of the crystal lattice considerably reduce the number of independent parameters in the previous expressions. In particular, we consider the six-band model, which includes the conduction and the light and heavy holes valence bands.

By taking the y coordinate along the growth axis, we have ϕ(r) = ϕ(y)e ^{ik ⊥ ·r} , where k _⊥ is the transverse momentum, which is conserved. The equations for ϕ(y) are

i ^~ ∂

∂t

ϕ _c ϕ _h

!

= H _cc H _cv H ^† _cv H _vv

!

ϕ _c ϕ _h

!

ϕ _h (r) = ^ ϕ ^3/2 _h , ϕ ^1/2 _h , ϕ ^−1/2 _h , ϕ ^−3/2 _h ^{ T} ϕ _c (r) = ^ ϕ ⁺ _c , ϕ ⁻ _c ^{ T}

where with I _n×n we indicate the n × n unit matrix, and H _cc = I _{2 ×2} E _c − E + U − ^~

2

2m _c

d ² dy ²

!

+ ζk _⊥ ∂U

∂y σ _z H _vv = I _{6 ×6} (E _v − E + U) + I ^{h ~}

2

2

d ²

dy ² + k _⊥ ζ 3

∂U

∂y J ^3/2 _z H _cv = λ ∂U

∂y T _y I _h = diag 1

m _hh , 1

m _lh , 1

m _lh , 1 m _hh

!

J ^3/2 _z = diag (3, 1, −1, −3) T _y = − i

3 √ 2

√ 3 0 1 0 0 1 0 √

3

!

The other parameters are defined in the Table Symbol Physical Meaning

m _c Effective mass in conduction band

m _lh , m _hh Light and heavy holes effective masses

u ^α,σ _n Periodic part of Bloch function related to k = 0 λ = ζ ^√ _~ ³ P + _m ^~

0 π ^K Interband coupling coefficient π ^K = ^{√ 3} ₂ e _z · π _c,h ^+,1/2 Kane momentum

P = −i _m ^{~ 2} ₀ ^D ψ _{S    ∂} ^∂

z

 

 ψ _Z ^E Kane momentum without spin

Numerical results

Band alignments of the

InAs/AlSb/GaSb/AlSb/InAs double barrier structure used in the simulation.

Transport through this system involves resonant tunneling of electrons from the InAs emitter, through unoccupied elec- tron states in the subbands of the GaSb well, and subsequently back into the con- duction band of the collector.

0 10 20 30 40 50

0 0.5 1 1.5 2

x (nm)

Potential (eV)

E

c

E

v

0.0118 0.0118 0.0118 0.0118 0.0118 0.0118 0

0.1 0.2 0.3 0.4 0.5 0.6

E

i

(eV)

Transmission coefficient

Calculated transmission coefficient for the resonant diode for the six band system.

The in-plane wave vector is k _⊥ = ^2π _a (0.03, 0, 0) where a is the lattice constant. The resonant peak is related only to the spin-up conduction electrons, and it disappears completely for the spin-down states. In this way, the device is able to select electrons both from the energy and form the spin direction.

By using a Runge-Kutta scheme, we calculate the envelope function solution for incre- mental values of the electron beam energy for both directions of the spin. The figures show the electron density in the valence bands n _v (y) = ^P j=±1/2,±3/2 |ϕ ^j _h | ² and in the conduction band n _v (y) = ^P _j=± |ϕ ^j c | ² for of spin up electron.

When the electron energy approaches the resonant level, charge cumulates in the va-

lence quantum well. On the other hand, no resonance effects are present for spin down

electron.