Spin-Polarized Transport in Semiconductor Nanostructures Semion Saikin, Min Shen, Ming-Cheng Cheng and Vladimir Privman
NSF Center for Quantum Device Technology, Department of Physics, Department of Electrical and Computer Engineering Clarkson University email:
[email protected]
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NSF Center for Quantum Device Technology Modeling of Quantum Coherence for Evaluation of QC Designs and Measurement Schemes
Task: Model the environmental effects and approximate the density matrix Task: Identify measures of decoherence and establish their approximate “additivity” for several qubits Task: Apply to 2DEG and other QC designs; improve or discard QC designs and measurement schemes
Use perturbative Markovian schemes
Relaxation time scales: T1, T2, and additivity of rates
P in Si QC
Q-dot QC
New short-time approximations
“Deviation”
measures of decoherence and their additivity
P in Si QC
(De)coherence in Transport Measureme Coherent Coherent Measurement by charge spin spin by nt charge carriers transport carriers transport
Q-dot QC
Improve and finalize solid-state QC designs once the single-qubit measurement methodology is established
How to measure spin and charge qubits
Spin polarization relaxation in devices/ spintronics
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Goals • The general goal of the project is to establish device modeling tools at various levels of description, from microscopic transport equations to moment equations (e.g., hydrodynamic, energy-transport, drift-diffusion), with the latter description involving parameters extracted from more microscopic, numerically demanding simulations.
Problem • To model spin polarized transport in a 2DEG at a semiconductor heterointerface in a spintronic device structure. The study is motivated by proposals for SpinFETs (S. Datta, B. Das, Appl. Phys. Lett. 56, 665 (1990); J. Schliemann, J. C. Egues, D. Loss, Phys. Rev. Lett. 90, 146801 (2003)).
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Hierarchy of transport models for spintronic devices Boltzmann transport equation (quantum potential corrections, spin) Expansion of moments
•
Hydrodynamic model
•
Energy transport model
•
Drift-diffusion model
Parameters
Monte Carlo simulation model
Ballistic models Compact models 4
Spin dynamics in semiconductor quantum wells 2 p • Hamiltonian: H = * + V (r ) + H SO 2m
• Spin-orbit interaction: H SO = kAσ
(Linear in an electron momentum)
1. Effect of quantum well asymmetry (Rashba term): (Isotropic with respect to crystallographic axes.)
H R = η (k yσ x − k xσ y )
2. Effect of crystal inversion asymmetry (Dresselhaus term): (Anisotropic with respect to crystallographic axes.) H D = β k z2 (k yσ y − k xσ x )
(2D)
H D = σ x {βk y ((k z2 − k||2 )sin 2ξ + βk x2 sin 2ξ + η ) − βk x (k z2 − k||2 )cos 2ξ }+
σ y {βk x ((k z2 − k||2 )sin 2ξ + β 2 k y2 sin 4ξ − η ) + βk y (k z2 − k||2 )cos 2ξ }
(Quasi 2D)
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Drift-Diffusion Equations Expasion in moments of the Wigner function equation. • Wigner function for an electron with spin: Wss′ ( R, k , t ) = ∫ ρ ( R, ∆r, s, s ′, t )e − ik∆r d 2 ∆r
• Transport equation:
∂W 1 ⎧⎪ ∂W ⎫⎪ 1 ∂V ∂W + ⎨v j , + ik j [v j , W ] = StW ⎬− ∂t 2 ⎪⎩ ∂x j ⎪⎭ h ∂x j ∂k j
• Particle density and current density definitions: nn = ∫ Wn d 2 k ,
J nj = ∫ ( vnjWn + vσjα Wσ α )d 2 k ,
nσ α = ∫ Wσ α d 2 k ,
J σjα = ∫ ( vnjWσ α + vσjα Wn )d 2 k .
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Drift-Diffusion Equations ∂nn ∂J nj + = 0, ∂t ∂x j
•
J nj = −
∂nσ ∂J 2m + − [vσj × J σj ] = 0. ∂t ∂x j h j σ
*
Anisotropy of spin transport.
5 4 3 2 1
0
50
100
150
∂nn ∂V ⎞⎟ nn , + ∂x j ∂x j ⎟⎠
⎞ ∂nσ ∂V 2m *kT j J = − * kT nσ − [vσ × nσ ]⎟. + ⎟ ⎜ m ⎝ ∂x j ∂x j h ⎠
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0
m * ⎜⎝
kT
τ ⎛⎜
j σ
Spin dephasing length L⊥ (µm)
Spin precession length Lp (µm)
Case η = β k z2
•
τ ⎛⎜
200
Angle ξ (deg)
250
300
350
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E = -0.01 V/cm E = -100 V/cm E = -300 V/cm E = -1000 V/cm
5 4 3 2 1 0
0
50
100
150
200
250
Angle ξ (deg)
300
350
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Monte Carlo modeling •
• Charge transport (BTE + Poisson equation)
•
f (r, k, t )
• Spin dynamics (spin density matrix)
⎛ ρ ↑↑ (t ) ρ ↑↓ (t ) ⎞ ⎟⎟ ρ i (t ) = ⎜⎜ ρ ( ) ρ ( ) t t ↓↓ ⎝ ↓↑ ⎠
⎛∂ f ∂f 1 ∂V +v ⋅∇f − ⋅ ∇ k f = ⎜⎜ ∂t h ∂x ⎝ ∂t
∇V =− 2
•
ρ i (t + dt ) = e − iH
•
H SO = H R + H D
SO dt / h
e2
εs
⎞ ⎟⎟ ⎠C
(n(r) − N ) eq
ρ i (t )eiH
SO dt / h
• Statistics
Z Pz
φ X
•
|P|≤1
1 n Spin polarization: Pα = ∑ Tr (σ α ρ i ) n i =1
θ Px
Py
Y
• Spin current:
J αβ
1 n i = ∑ vβ Tr (σ α ρ i ) n i =1 8
Spin scattering • Spin dynamics ρ i (t ) = S ( k n , dtn ) S (k n −1 , dtn −1 )...S (k1 , dt1 ) ρ i (t0 ) S (k1 , dt1 )...S (k n −1 , dtn −1 ) S ( k n , dtn )
• Scattering matrix
α ⎞ ⎛ i sin(| α | dt ) ⎟ ⎜ cos(| α | dt ) α ⎟ S ( k , dt ) = ⎜ * ⎟ ⎜ α i α dt α dt sin(| | ) cos(| | ) ⎟ ⎜ α ⎠ ⎝ α = h −1{( k y (η + β k z2 sin 2ξ ) − k x β k z2 cos 2ξ ) − i ( k x ( β k z2 sin 2ξ − η ) + k y β k z2 cos 2ξ )}.
k
S(k1,dt1) S(k2,dt2)
S(k3,dt3)
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Model 1
Study of spin polarized transport properties n-doped In0.52Al0.48As
injection
absorption
In0.52Al0.48As In0.53Ga0.47As In0.52Al0.48As
Z L = 550 nm
Y X
100
T = 300 K
• Injected electrons are 100% spin polarized. • Injected electrons are thermalized. • Electron injection condition is to preserve charge neutrality in the structure. • Absorption boundary is not spin selective. VDS = 0.05 V VDS = 0.1 V VDS = 0.15 V VDS = 0.2 V VDS = 0.25 V
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Velocity (10 m/sec)
80 70
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Energy (meV)
90
60 50 40
4 3 2 1
30 0.0
0.1
0.2
0.3
X (µm)
0.4
0.5
0 0.0
0.1
0.2
0.3
X (µ m)
0.4
0.5
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Spin polarization (1 0 0) transport direction 1.0
1.0
Spin polarization, P
0.6 0.4 0.2 0.0 -0.2
Injection Py=1
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4
-0.4 -0.6 0.0
0.1
0.2
0.3
0.4
-0.6 0.0
0.5
0.1
0.2
0.3
0.5
1.0
Injection Px Py Pz
Injection Pz=1
Spin polarization, |P|
0.8
0.4
X (µm)
X (µm)
1.0
Spin polarization, P
Px Py Pz
Spin polarization, P
Injection Px=1
0.8
0.6 0.4 0.2 0.0 -0.2
0.8 0.6 0.4 0.2
-0.4 -0.6 0.0
0.1
0.2
0.3
X (µm)
0.4
0.5
0.0 0.0
0.1
0.2
0.3
X (µm)
0.4
0.5 11
Spin scattering length (µm)
Anisotropy of spin dynamics
P Drain
Source
z
y
ξ
1.0 0.8 0.6 0.4 0.2 0.0 0
x
linear SO high order
50
100
0.6
0.6
0.4
0.4
Polarization
Polarization
0.8
0.2 0.0 -0.2 -0.4
-0.2 -0.4
-0.8
-0.8
0.1
0.2
0.3 X (µ m)
0.4
0.5
350
0.0
-0.6
0.0
300
0.2
-0.6 -1.0
250
Cubic term
1.0
Linear
0.8
200
Angle ξ (deg)
(1 -1 0) transport direction ξ=-45° 1.0
150
-1.0 0.0
0.1
0.2
0.3
X (µm)
0.4
0.5
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Model 2
Spin injection from a Fe contact into a GaAs 2D layer Thermionic emission
Energy (eV)
qφB0.8
Injection rate (a.u.)
1.0 Tunneling
0.6 0.4
Ef
0.2 0.0 -0.2
Ef -0.2
n+ 0.0 x tp
0.2
0.4
0.6
0.8
Ec
0.3 1.0
0.4
0.5
0.6 0.7 0.8 Energy (eV)
0.9
1.0
X (µm)
• Spin polarization of electrons for a given energy E in the metal contact is defined by the relative density of states for spin-majority and spin-minority carriers. • Electrons in the metal contact are thermalized. • The probability of an electron injection is defined according to the WKB approximation.
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0.5 0.0 -0.5 -1.0 -1.5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-2
JySx JySy JySz JyS
1.0
-1
1.0
JxSx JxSy JxSz JxS
19
-1
1.5
Linear SO
Spin current density (10 sec m )
-2
2.0
19
Spin current density (10 sec m )
Spin current
0.5
0.0
-0.5
0.0
0.1
0.2
0.5
0.6
0.5
0.6
0.7
Spin current density (10 sec m )
-2
1.0
-1
2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0.0
0.4
X (µm)
19
19
-1
-2
Spin current density (10 sec m )
X (µm)
0.3
0.1
0.2
0.3
0.4
X (µm)
0.5
0.6
0.7
0.5
0.0
-0.5
0.0
0.1
0.2
0.3
0.4
X (µm)
0.7
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Spin current
0.5 0.0 -0.5 -1.0 -1.5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-2 -1
1.0
0.4
19
-1
1.5
JxSx JxSy JxSz JxS
Spin current density (10 sec m )
-2
2.0
19
Spin current density (10 sec m )
Cubic SO
0.2
JySx JySy JySz JyS
0.0 -0.2 -0.4 0.0
0.1
0.2
0.5
Current spin polarization
-1
-2
Spin current density (10 sec m )
1.5
19
1.0 0.5 0.0 -0.5 -1.0 0.1
0.2
0.3
0.4
X (µm)
0.4
0.5
0.6
0.7
X (µm)
X (µm)
0.0
0.3
0.5
0.6
0.7
Current spin depolarization Longitudinal component Transverse component
0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
X (µm)
0.5
0.6
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0.7
Conclusions ● The ensemble Monte-Carlo method has been developed for investigation of spin polarized transport in semiconductor heterostructures. ● The problems of spin transport in finite length structures and spin injection through the Schottky barrier from a ferromagnetic metal contact into a QW have been studied. ● The characteristic length of non-equilibrium spin polarization transport in a III-V semiconductor QW is in the order of one micron at room temperature. ● The characteristic length of coherent spin dynamics is in a length scale of deep submicrometer. ● Boundary conditions appreciably affects spin transport properties in a spintronic device structure.
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