where T1 is the spin relaxation time, which is also referred to the longitudinal time or spin-lattice time [18].
1.3 D’yakonov-Perel’ mechanism
In the discussion of EY mechanism in Sec. 2.1, the combined effect of inversion symmetry of space and time reversal symmetry yields a twofold degeneracy of single-particle energies
E+(k) = E−(k), (1.10)
where for convenience (+) and (−) denote the two states (2.4) and (2.5). If the spatial inversion symmetry is lifted, the spin-orbit interaction shall lead to a spin splitting of the electron state even at zero magnetic field, B = 0. The spin splitting can be caused by the bulk inversion asymmetry (BIA) of the underlying crystal structure. Examples include the zinc blende structure of III-V (such as GaAs and InSb) and II-VI (such as ZnSe and HgCdTe) compounds without center of inversion. These materials are different from Si and Ge, which have a diamond structure. Furthermore, the spin splitting can be caused by the structure inversion asymmetry (SIA) of the confined potential V (r). This potential may contain a built-in or an external potential, as well as the effective potential from the position-dependent band edges. To the lowest order of the wave vector k, the BIA induced spin splitting is caused by the so-called Dresselhause term, whereas the SIA induced one is generated by the so-called Rashba term. The spin splitting of higher orders of k can be described by, for instance, the 8 × 8 or the 14 × 14 extended Kane model.
For BIA, examples can be found in the conduction bands Γ-point of [001]
grown GaAs/AlAs and alike (type-I) quasi-2D quantum wells or two dimensional electron gas (2DEG) systems. In these systems, the Hamiltonian matrix Hkk of
the Dresselhause term is y directions, m∗ is the Γ-point conduction-band effective electron mass, γ is the spin-splitting parameter, and ˜kz is the operator id/dz [19, 20]. This Hamiltonian can be reduced as
with a material-specific coefficient β, where σx and σy are two components of the Pauli matrices.
For SIA, examples can also be found in the conduction bands Γ-point of [001]
grown GaAs/AlAs and alike (type-I) quasi-2D quantum wells or two dimensional electron gas (2DEG) systems. In these systems, the Hamiltonian matrix Hkk of the Rashba term is
Hkk = α (σ × k) · ν, (1.13)
where α is a pre-factor which depends on the constituting materials and on the geometry of the quasi-2D or 2DEG systems, σ are the Pauli matrices, ~k is the electron momentum, and ν is a unit vector perpendicular to the 2D plane. If we assume that ν is in the z direction, then this Hamiltonian becomes
HSIA = α (σxky− σykx) . (1.14)
A comparison shows that the energy degeneracy of spin orbit interaction or the quasi-spin up and down states can be lifted in different ways in BIA and SIA. The former can be achieved by removing spatial inversion symmetry or time reversal symmetry, while the latter can be accomplished by applying an external magnetic field. The expressions (1.12) and (1.14) have a general form
HSOI = 1
2~σ · Ω(k), (1.15)
where SOI denotes the spin orbit interaction, σ are the Pauli matrices, B(k) is a k-dependent effective magnetic field around which electron spins precess
with the Larmar frequency Ω(k) = (e/m∗)B(k), with the effective electron mass m∗ [12,13,15]. This expression gives a clear picture why the effective magnetic field B(k) causes the electron spin relaxation. That is, a collision event will change the electron momentum ~k and subsequently the electron spin precession axis. There-fore, the randomized precession axis will help smearing the electron spin coherence.
If the momentum relaxation time is longer than the spin precession period, e.g., under dilute impurities, the electron spins shall precess freely and lose their coher-ence between two collision events. In contrast, if the momentum relaxation time is shorter than the spin precession period, e.g., under dense impurities, the electron spins will not precess much before the carrier changes its momentum or the carrier spin changes its precession axis. It shall lead to the dynamical narrowing which helps preserving spin coherence.
Owing to different underlying mechanisms, the EY and DP spin relaxation times have opposite impurity density dependence. Under the EY mechanism, spin flip is a temporally discrete event and can only happen at a electron-impurity collision. Thus, more frequent collision events will cause faster spin relaxation.
Under the DP mechanism, each single spin precesses between two collisions. Less collision events will lead to longer precession and faster spin decoherence. Accord-ingly, there exists a trend between the momentum relaxation time τp and the spin relaxation times τsEY and τsDP of EY and DP mechanisms, respectively
τsEY ∝ τp and τsDP ∝ 1
τp. (1.16)
1.4 Supplement: general definition of spin relaxation time
To study spin relaxations, let us consider a system consisting of N+ electrons with spin state |Sz+i and N− electrons with spin state |Sz−i. The total electron number is N = N++N−and the net magnetization at any instance can be defined as
D = N+− N−. (1.17)
If at equilibrium the net magnetization is Deq, one expects that the evolution of D will follow the equation
dD
dt = Deq− D
T , (1.18)
with the relaxation time T . Let N∓→± be the number of spins which flip per second from |Sz∓i to |Sz±i and W∓→± be the transition rate of electrons from
|Sz∓i to |Sz±i. The evolution of D is then given by dD
dt = Deq− D
T = 2 (N−→+ − N+→−)
1 = 2 (W−→+− W+→−) . (1.19) These transition rates W∓→± are proportional to the square of the matrix elements of an interaction Hint between the electron and the impurity or the lattice that causes a spin flip [17].
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Chapter 2
Methods for Carrier Transport, Carrier Scattering, and Spin Evolution
Using full quantum mechanical approach to study the carrier transport, carrier scattering, and spin evolution of many-body systems in solids is a formidable task.
To overcome this complexity, we introduce the Ensemble Monte Carlo (EMC) method and Semiclassical Path Integral to tackle these dynamical problems.
2.1 Carrier transport
When we consider the carrier transport, e.g., electron transport, in a semi-conductor crystal, it is essentially an extremely complicated many-body problem.
However, we can focus on the motion of an electron and approximate the effective influence of atomic nuclei and other electrons on the studied electron by a poten-tial V (r). Then the original many-body problem can be reduced to the problem of a single electron [1,2]. Under this reduction, V (r) is still periodic with the same periodicity as that of the crystal lattice. The electronic state under such V (r) can be obtained by solving the Schr¨odinger equation
· p2
2m + V (r)
¸
Ψ(r) = EΨ(r), (2.1)
where m is the free electron mass, Ψ(r) is the eigenfunction to be determined, and E is the energy eigenvalue. The Bloch theorem tells us that the solutions for a perfectly periodic potential have the form
Ψk(r) = uk,n(r)eik·r (2.2)
where uk,n(r) is periodic with the same periodicity of V (r), k is the wave vector of electron and n is the index of bands. Besides, the energy eigenvalue Ek,n is periodic with the periodicity of the reciprocal lattice.
The relation between Ek,n and k, that is the energy band structure, can be expressed in one period of the reciprocal lattice because of the periodicity of Ek,n. Conventionally, the first Brillouin zone, which is a period centered about at the origin of the k-space, is used to show the energy band structure. This structure is usually depicted along some significant crystallographic orientations, such as Λ,
∆, and Σ directions. The energy band structure reveals an energy region where electronic states can not be found. This forbidden energy region is termed the energy gap and electronic states are permitted above and below this gap. While the bands above the gap are the conduction bands, those below it are the valence bands. The energy separation between the minimum of the lowest conduction band and the maximum of the highest valence band is the band gap energy Eg. The band model offers the information about the energy levels of the band extremes and the relations between the electron energy Ek and the electron wave vector k, described by various band parameters.
The structures near the conduction band minima and the valence band maxima are important, because carriers located near the band edges are responsible for the transport property. The conduction band near the minimum is frequently approximated by a quadratic function of k. If the band minimum is located at
|k| = 0, Ek can be expressed as
Ek= ~2k2
2m∗, (2.3)
where k2 = (kx2+ k2y+ kz2) and 1 m∗ ≡ 1
~2
∂2Ek
∂k2 (2.4)
is the inverse of the effective mass. The Ek relation given by (2.3) shows that the electrons in a crystal behave just like electrons moving in a free space, except for a change in the mass. Here ~k plays the role of momentum, which is termed the crystal momentum. Ek represents the electron kinetic energy measured from the conduction band minimum. Such simple model is rather widely used for simplifying the calculation of carrier transport.
Since electrons in crystal behave just like electrons in free space, except for the change in the mass. This picture suggests that the motion of electrons in
a crystal may be described by the classical equations of motion. The idea is valid when the potential energy felt by the electrons varies slowly compared to the crystal potential so that quantum mechanical effects such as reflection, interference and tunneling can be ignored. Following this concept, the classical motion of an electron can be described by the equation of motion based on its total energy Hamiltonian
H = Ek+ U, (2.5)
where Ek is the kinetic energy and U is the potential energy. For an electron in a conduction band, one has
H = Ek+ Ec(r), (2.6)
where Ek represent the kinetic energy in terms of the crystal momentum and the effective mass and Ec(r) is the conduction band minimum. Then the equations of motion of the system are the Hamiltonian dynamics
dk
dt = −1
~∇rH (2.7)
dr dt = 1
~∇kH, (2.8)
where ∇r is del operator with respect to position vector r and ∇k is the del operator with respect to wave vector k. We can easily check that for the quadratic band, the group velocity v = dr/dt simply gives
v = ~k
m∗, (2.9)
which has the similar form of the free electron momentum divided by mass.
Due to the advances of modern semiconductor fabrication techniques, we can easily grow compositionally non-uniform heterostructure semiconductors. For in-stance, by placing two compositionally different materials next to each other, a heterojunction is established. A thin two-dimensional conducting layer, termed two dimensional electron gas (2DEG), is formed at the interface of the heterojunc-tion, for example, the interface between GaAs and AlGaAs. Besides, with the use of modern epitaxial growth techniques, the alloy composition can be varied on an atomic scale, so a very sophisticated layer structures consisting of several barriers and wells can be fabricated. For example, a triple alloys compound AlGaAs-GaAs-AlGaAs offers a quantum well in the layer GaAs which the thickness of the well may be about 100˚A or less. Because of the confined electron motion in the well,
the electrons behavior just like a quasi-two dimensional motion. The electrons running at the thin two dimensional conducting layer or the confined well are called the two dimensional electron gas (2DEG). Studying 2DEG is important, since quantum confinements frequently exist in modern heterostructure devices.
Let us assume the aforementioned electron motion is confined in the z di-rection and the electron can move freely in the xy plane. The corresponding three-dimensional Schr¨odinger equation is
− ~2
2m∗∇2Ψ(r) + Ec(r)Ψ(r) = EΨ(r). (2.10) The strategy to solve (2.10) is trying to separate the variables. We assume the plane wave solutions in the xy direction since the electrons are free to move on the xy plane. So the total wave function is represented as
Ψ(r) = Cψ(z)eikxxeikyy, (2.11) where C is the normalization coefficient. Substituting (2.11) into (2.10), we get an equation for ψ(z),
− ~2 2m∗
∂2ψ(z)
∂z2 + Ec(z)ψ(z) = Enψ(z), (2.12) where
En= E − Ek (2.13)
is the energy associated with confinement in the z direction and Ek = 2m~2∗(kx2+ky2) is the kinetic energy associated with the motion parallel to xy plane.
2.2 Carrier scattering
Carrier motion in semiconductor crystals is mainly made up of the scattering and drift processes. Here we briefly introduce the theory for scattering process, which plays a role in our study. The scattering theory is based on Fermi’s golden rule, which is derived from the first-order time-dependent perturbation theory.
It gives the transition probability per unit time between two eigenstates of the unperturbed Hamiltonian H0caused by the perturbation potential H0(r, t). While H0can be the free electron Hamiltonian in general, it is the effective mass electron
Hamiltonian in solid crystals. At first let us write down the Schr¨odinger equation
[H0+ λH0(r, t)] Ψ(r, t) = i~∂Ψ(r, t)
∂t , (2.14)
where λ is a real dimensionless parameter. We assume the equation for the un-perturbed Hamiltonian H0 has been solved as
H0ψk= Ekψk, (2.15)
where Ek is the energy eigenvalue and ψk is the corresponding eigenfunction. The time evolution of the eigenfunction can be represented as
Ψ0k(r, t) = ψk(r)e−iEkt~ . (2.16) Since the eigenfunctions Ψ0k(r, t) form a complete and orthonormal set, the solution of the perturbed problem can be constructed by the linear combinations of Ψ0k(r, t),
Ψ(r, t) =X
k
ck(t)Ψ0k(r, t), (2.17)
where the coefficient ck(t) describes how the perturbation makes the component at Ψ0k(r, t) vary with time. Substituting (2.17) into (2.14) and multiplying both sides of the arranged equation by ψk∗0e−iEk0 t~ , integrating with respect to r, and using the orthogonality of ψk, we obtain the following differential equation for ck(t)
i~∂ck0(t)
∂t = λX
k
hk0|H0|kick(t)ei(Ek0 −Ek)t
~ , (2.18)
where hk0|H0|ki is the expectation value defined as
hk0|H0|ki = Z
Ω
ψk∗0(r)H0ψk(r)dr, (2.19) with Ω the volume of the crystal.
The expression (2.18) indicates that ck(t) depends on time if λ is not zero.
Since ck(t) is expected to vary slowly with time if the perturbation is weak, it can be expanded as a power series of λ,
ck(t) = c(0)k + λc(1)k (t) + λ2c(2)k (t) + · · · . (2.20) Substituting (2.20) into (2.18) and equating terms of like powers of λ on the two
sides, we have
The first equation of (2.21) shows that the zero-order coefficients c(0)k0 are time independent. The first-order approximation c(1)k0 (t) of ck0(t) can be evaluated from the second equation of (2.21). The first-order approximation will be sufficient precise, provided that the interaction is very weak.
For an initial state in a definite unperturbed eigenstate ki, the above results give
As an application of (2.23), we consider a constant perturbation turned on at t = 0
H0(t) =
( 0, for t<0
H0, for t≥0. (2.24)
Substituting (2.24) into (2.23) and carrying out some integrations, we obtain
c(1)k0 (t) = 1
i~hk0|H0|kiick(t)eωt2 sin(ωt2 )
¡ωt
2
¢ t, (2.25)
where ω = (Ek0−E~ ki). The probability of finding an electron with the wave vector k0 at time t is then given by |c(1)k0 (t)|2. Thus, the transition rate S(ki, k0) from the
By using the relation limt→∞ 1 πsin2αx
αx2 = δ(x), the transition rate becomes S(ki, k0) = 2π
~ |hk0|H0|kii|2δ(Ek0 − Eki). (2.27)
Integrating S(ki, k0) given by (2.27), with respect to all accessible final states k0, we obtain the scattering rate,
W (k) = Ω (2π)3
Z
S(ki, k0)dk. (2.28)
This formula is independent of the dimension of the systems. For quasi 2DEG, we need only to put the wave function given by (2.11) to obtain the corresponding scattering formula.
2.3 Ensemble Monte Carlo method
The Monte Carlo transport calculation is usually referred to the single particle Monte Carlo method or the ensemble Monte Carlo (EMC) method. As discussed above, carrier transport in a semiconductor crystal is a many-body problem with a huge number of mutually interacting carriers. However, if in some parameter regimes the carriers can be approximately treated as an ensemble of independent carriers, the macroscopic behaviors of the system might be approached by the long time behavior of a single particle. It is the principal idea of the single particle Monte Carlo method. This method is a useful for calculating carrier transport, especially in the case of steady-state carrier transport under a static and uniform electric field. However, if the problems of interest are not steady, the long time average has to be replaced by ensemble average, which gives rise to the ensemble Monte Carlo method. This method can be used more widely for many other purposes, such as carrier diffusion, the carrier transport in an inhomogeneous field, the non-stationary behavior of carriers, etc. In the study of spin evolution, we need to use the ensemble Monte Carlo method to monitor the transient process of electron spin. But each member, i.e., single particle, in the ensemble follows the same calculation process as that in the single particle Monte Carlo method.
The ensemble Monte Carlo method is based on the successive and simultaneous calculations of the motions of many carriers during a small time increment ∆t.
The method is essentially dynamic and thus is suitable for the analysis of transient carrier motion. A key step to execute the ensemble Monte Carlo calculation is deciding the free flight time, that is the duration between two successive scattering events. This duration depends on the total scattering rate which is the sum of various scattering rates of individual scattering mechanisms. The probability density, P (τ ), of finding an electron traveling for a time τ without being scattered
is expected to follow the relation
where the total scattering rate
WT(Ek) = XN
j=1
Wj(Ek) (2.30)
is the sum of the scattering rates of N different scattering mechanisms. Since the scattering rate of each scattering mechanism is a function of electron energy Ek, the total scattering rate is also a function of Ek.
The solution of (2.29) is
P (τ ) = WT(Ek) exp
To determine the free flight time by P (τ ), we have to evaluate the integral in (2.31). Unfortunately, there is no analytical form for that integral because of the complicated form of general Wj(Ek). A simple strategy to get rid of this problem is adding a virtual scattering process, called self-scattering, with the scattering rate W0(Ek) to the original total scattering process, so that the new total scattering rate Γ becomes a constant [3],
W0(Ek) = Γ −
The inclusion of the self-scattering makes no change to the k wave vector of the particle and has an advantage that (2.31) can be recast simply as
P (τ ) = Γ exp−Γτ . (2.33)
The free flight time τ for a carrier scattering process is a random variable following the distribution P (τ ). This distribution gives the mean free flight time τm = R∞
0 τ P (τ ) dτ = 1/Γ. To see which value Γ should is taken, let us consider a carrier with Fermi velocity vF, which has the free flight length l = vFτ and the mean free path lmfp=R∞
0 lP (τ ) dτ = vF/Γ. That is, once vF and lmfp of a system are known, Γ can be decided. In numerical simulations, τ can be generated by substituting a uniformly distributed random number x ∈ [0, 1] into the following
the formula
τ = −ln(x)
Γ . (2.34)
Equivalently, one can calculate its mean free path by
l = −lmfpln(x). (2.35)
This classical picture is valid, when the sample size is larger than the de Broglie wave length of the carriers.
2.4 Semiclassical Path Integral formalism
With the knowledge of carrier transport and scattering, now we proceed to the evolution of carrier spin. Since the spin dynamics of the semiconductors discussed below is related to the electron dynamics by the spin-orbit coupling, how the spin evolves is decided by the carrier evolution discussed in the last chapter. While the former is stochastic due to impurity collisions, the latter is deterministic and fully decided by the former. In the following, in combination with the ensemble Monte Carlo method, we apply the Semiclassical Path Integral (SPI) method to the EY and DP spin relaxation mechanisms in a quasi-2DEG system to reveal the intriguing behaviors of carrier spin evolution.
The original semiclassical path integral method was formulated for Rashba systems [4, 5, 6], which has the Hamiltonian
H = H0+ HSOI, (2.36)
where H0 consists of the kinetic and potential energies of an electron in the system and HSOIrepresents its spin orbit interaction (SOI). Since the energy of spin orbital coupling in the interested material is usually much smaller than the kinetic and potential energies, the electron trajectory γ can be determined purely by H0. The spin dynamics of this electron will be described by an evolution operator in the path integral formalism,
Uγ = exp
·
−i
~ Z
γ
HSOI(t) dt
¸
. (2.37)
If the electron collides with the impurities or boundaries nγ times at points ξ1, ξ2,
· · · , ξnγ, its trajectory γ will comprise nγ straight segments,
γ = γnγ + · · · + γ2+ γ1. (2.38) The corresponding spin evolution operator Uγ becomes a product
Uγ = Uγnγ × · · · × Uγ2 × Uγ1 (2.39) of the individual operators Uγj = exp£
−~iHSOItγj¤
−~iHSOItγj¤