Spin evolution under DP mechanism

For systems with Rashba SOI, the Hamiltonian consists of two parts

H = H₀+ H_R, (2.48)

where H₀ represents the sum of the kinetic and potential energies of an electron with its effective mass in a quasi-2DEG. The second part H_R = α (σ × k) · z represents the Rashba SOI, where α is the spin orbit coupling constant, σ stands for Pauli matrices, ~k denotes the electron momentum, and z is the unit vec-tor perpendicular to the quasi-2D sample. The Hamiltonian H_R will cause spin precession, when the carrier of the spin moves along a classical trajectory. A char-acteristic length determining this precession is the spin rotation length Lso = _αm^~2∗, where m^∗ is the effective mass of the particle. In real semiconductor materials, the energy ratio H_R/H₀ can be as large as 1/10, such as that in the InSb sample [9].

But even for this ratio, H_R is still small compared with H₀. In such systems, the electron dynamics is not affected strongly by its spin dynamics, so that in the leading approximation classical trajectories are determined by H₀. Thus, we can apply the ensemble Monte Carlo method to determine the trajectory and scattering process for each electron in quasi-2DEG system.

For a free electron moving along a straight trajectory γ of length l, the dynam-ics of its spin state is governed by the evolution operator U in the path integral formalism [6]

where b = z × k/|k|. This operator represents simply the spin rotation. If an electron collides with impurities or boundaries nγtimes, its trajectory γ will consist of nγ straight segments γ = γnγ+ · · · + γ2+ γ1, The corresponding spin evolution operator U_γ becomes a product

Uγ = Uγ_nγ × · · · × Uγ2 × Uγ1, (2.50)

where the individual operators

U_γj = exp

·

−i l_j L_sob_j· σ

¸

= 1 cos µ l_j

L_so

¶

− i(b_j · σ) sin µ l_j

L_so

¶

(2.51)

along different straight segments do not commute with each other. In analogy to the case in of EY mechanism, the microscopic information on each individual spin from (2.50) allows us to calculate any macroscopic average of a crowd of spins, as discussed in (2.46), in analogy to (2.47).

Bibliography

[1] C. Jacobono and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation (Springer-Verlag Wien New York, 1989).

[2] K. Tomizawa, Numerical Simulation of Submicron Semiconductor Devices (Artech House, Boston, London, 1993).

[3] K. Kurosawa, Journal of the Physical Society of Japan 21, Supplement, 424 (1966).

[4] A. G. Mal’shukov, V. V. Shlyapin and K. A. Chao, Phys. Rev. B 66, 081311(R) (2002).

[5] C.-H. Chang, A. G. Mal’shukov and K. A. Chao, Phys. Lett. A 326, 436 (2004).

[6] C.-H. Chang, A. G. Mal’shukov and K. A. Chao, Phys. Rev. B 70, 245309 (2004).

[7] S. Datta, Quantum Phenomena (Addison-Wesley, 1989), 185.

[8] J. Tsai and C.-H. Chang, J. Phys.: Condens. Matter 24, 075801 (2012).

[9] H. Chen, J. J. Heremans, J. A. Peters, A. O. Govorov, N. Goel, S. J. Chung, and M. B. Santos, Appl. Phys. Lett. 86, 032113 (2005).

Chapter 3

EY Spin Relaxation in Quantum Wells and Narrow Wires

In the following, we apply the ensemble Monte Carlo method and the semiclas-sical path integral method to study the spin relaxation of the Elliott-Yafet mecha-nism in low-dimensional systems. In quantum wells, the spin properties calculated by these methods confirmed the experimental results. In two dimensional narrow wires, size and impurity effects on the Elliott-Yafet relaxation were predicted, in-cluding the wire-width-dependent relaxation time, the polarization evolution on the sample boundaries, and the relaxation behavior during the diffusive-ballistic transition. For ballistic narrow wires, we derived an exact relation between the Elliott-Yafet relaxation time and the wire width, which confirmed the above sim-ulations.

3.1 Introduction

Spin relaxation is one of the central issues in the study of spintronics [1, 2, 3].

This phenomenon is ubiquitous in materials with spin polarization and has a long research history dating back to the Elliott-Yafet (EY) relaxation in simple met-als (see [3] and recent papers citing this review). The study in this context is largely motivated by a fundamental interest in material properties. However, pur-suing efficient spin manipulation in devices might further boost the progress in this field. Today, several types of mechanisms responsible for different spin relax-ations have been found [4, 5, 6, 7], and among these, the D’yakonov-Perel’ (DP) and EY mechanisms play an essential role. The former is due to spin precession between the momentum scattering events, while the latter happens ”during” the

momentum scattering events. These mechanisms affect the spin dynamics in var-ious materials. For instance, in zinc-blende semiconductors at low temperatures, the spin relaxation is dominated by the DP mechanism [8, 9, 10, 11, 12, 13]. In In-GaAs/InP multiple-quantum wells at room temperature [14,15] and the Te-doped InSb/Al_0.15In_0.85Sb at low temperatures [16], the spin lifetime depends mainly on the EY mechanism [17, 18, 19, 20]. In the past, a large number of experimental and theoretical studies have been devoted to the DP mechanism, either in the 3D bulk or in low-dimensional systems like quantum wells (QWs) and 2D narrow wires [13, 21, 22, 23, 24, 25, 26, 27, 28]. However, comparatively less effort has been put into studying the EY relaxation, especially in low-dimensional systems [29].

In this work, we apply the ensemble Monte Carlo method and the semiclassi-cal path integral method to investigate the EY relaxation in QWs and 2D narrow wires in both diffusive and ballistic regimes. The study gave results in accor-dance with the experimentally measured values in real samples [16]. Based on this consistency, we used these methods to study the impurity and sample size effects on the EY relaxation under broad sample conditions. The main issues were how the relaxation time changed with sample width, how the polarization evolved on the boundary, and how the impurity density variation from diffusive to ballistic regimes affected the EY relaxation. Furthermore, the DP relaxation was calculated under the same sample conditions in order to compare it with the EY results. Finally, an analytical formula was derived for ballistic narrow wires, which confirmed our simulations and revealed exactly how the EY relaxation time varied with the wire width.

This chapter is organized as follows. In Sec. 2, the validity and precision of using the EMC and the SPI methods on the experimental samples are examined and compared with the theoretical results. In Sec. 3 and Sec. 4, the effects of size and impurity, respectively, on the EY relaxation are studied and compared with the DP relaxation. Finally, a summary and discussion are given in Sec. 5, and a supplementary material for spin relaxation process is represented in Sec. 6.

3.2 The EMC and SPI methods on experimental samples

The spin relaxation caused by the EY mechanism has been explored by some experimental groups [14,15,16]. In [16], a sample is InSb/Al_0.15In_0.85Sb single QW grown by MBE on the GaAs substrates. The QW has a well width of 20 nm (corresponding to the height in Figure 3.1) and was uniformly Te-doped (sample number me1831F). The electron density in this sample is 5.7×10¹¹ cm⁻² at 77K

W ~200( m)

Figure 3.1: A quasi-2D sample and an observation window which is a stripe of area 1 × 200 µm². For a quantum well with large L and W , the stripe is long and the average spin behavior therein is almost the same as that in the whole well (for cases in Figures 3.2 and 3.3). For a narrow wire with large L but small W , the stripe is short and the average spin behavior inside it is a local spin dynamics along the wire (for the case in Figure 3.7).

and 7.3×10¹¹ cm⁻² at 300K. Since the carrier concentration of semiconductor is proportional to T [30], the concentration for other T in between can be linearly interpolated, as n_e(T ) ≈ (0.0072T + 5.15) × 10¹⁰ cm⁻². The mobility of this sample was measured by means of the Hall effect and behaves as log₁₀µ(T ) ≈ 0.28 × log₁₀T − 0.55 m²V⁻¹s⁻¹ within T = 50 ∼ 300K. For more temperature dependent factors in spin relaxations, it is referred to [23].

Figure 3.2 shows the product of the spin relaxation time with the temperature, τ_sT , versus the carrier mobility µ. Its inset depicts the spin relaxation time versus the temperature of the sample. In both plots, the triangles are the experimental data measured from the sample me1831F, which is mainly governed by the EY mechanism. The black dots are calculated from the formula [16]

1

Therein, m is the free electron mass, m^∗ denotes the effective mass in the con-duction band, E_g represents the band gap, E_1e stands for the confinement energy of the lowest electron subband, τ_s is the EY mechanism induced spin relaxation time which is equal to T₁ in Eq. (1.9), and η = ∆/(E_g + ∆) with the spin orbit splitting energy ∆. The momentum relaxation time τ_p is related to the mobility µ by τ_p = µm^∗/e and the dimensionless constant C_EY is believed to be of the order of unity. The black dots in Figure 3.2 are calculated from (3.1) by using the following parameters of me1831F: ∆ ≈ 0.81 eV, Eg ≈ 0.24 eV, m^∗/m ≈ 0.014,

Table 3.1: The simulation protocols. T₁ is calculated by (3.1), τ_p = µm^∗/e, v_F = (~/m^∗)√

2πn_e, l_mfp= v_Fτ_p, and φ is calculated by (2.45).

T(K) 50 70 100 120 150 170 200 250 300

T1(ps) 2.54900 1.99790 1.54320 1.35230 1.15060 1.05090 0.93424 0.79486 0.69656 τp(ps) 0.06619 0.07263 0.08014 0.08428 0.08963 0.09278 0.09703 0.10320 0.10852 vF(µm/ps) 1.5381 1.5580 1.5874 1.6067 1.6353 1.6540 1.6817 1.7270 1.7710 lmfp(µm) 0.10180 0.11316 0.12722 0.13541 0.14656 0.15345 0.16318 0.17821 0.19219 φ 0.01282 0.01785 0.02531 0.03022 0.03749 0.04228 0.04937 0.06096 0.07227

E_1e ≈ 0.08 eV and C_EY ≈ 7.5 [16]. Recall that τ_s can be affected by various scattering potentials mentioned in Sec. 1.2. Among others, phonons will become more significant at high T .

Figure 3.2 shows that both the experimental and theoretical studies give the relation τ_sT ∝ µ for most µ. But two experimental points have an opposite trend τ_sT ∝ µ⁻¹ at high µ, which corresponds to the high T regime in the sample me1831F, as known from the empirical µ(T ) relation mentioned at the beginning of this section. One believes that this opposite trend is because the DP mechanism overrides the EY mechanism in the high µ regime, according to the current under-standing that τsT ∝ µ for EY mechanism and τsT ∝ µ⁻¹ for DP mechanism [16].

The latter is supported by the observation on the sample me1833 (remotely n-doped with Te 20 nm above the well) in [16], which follows the DP mechanism and has the property τ_sT ∝ µ⁻¹.

Next, the relaxation properties will be calculated by the ensemble Monte Carlo method and the semiclassical path integral method. To compare with above ex-perimental results, the simulations needs to insert the following exex-perimental pa-rameters. First, the spin flip probability φ will be calculated by (2.45), where how τ_p = µm^∗/e and T₁ vary with T is based on the above empirical relation µ(T ) and the black dots in the inset of Figure 3.2, which are calculated from (3.1). Second, vF can be derived from vF = ~/m^∗√

2πne with the above empirical electron den-sity n_e(T ). Notice that since n_e lies between 5.5 × 10¹¹ and 7.3 × 10¹¹ cm⁻², the corresponding de Broglie wavelength λ_F =p

2π/n_e ranging from 34 to 30 nm is larger than the sample hight 20 nm, as shown in Figure 1. Thus, the electrons are confined in the z direction of the sample. Third, the size of the experimental sample was not explicitly mentioned in [16]. However, (3.1) therein is referred to [14, 15], where the sample sizes are about 2 inches (approximately 5 × 10⁴ µm) in length. Our simulation is performed on a smaller square of 2 × 10² µm in length for less computational consumption. Both the experimental and simulation sam-ples belong to bulk systems. Since their scales are much larger than the de Broglie

wavelength λ_F (30 ∼ 34 nm), the electron motion on the xy plan is more particle-like and the validity of ensemble Monte Carlo method and the semiclassical path integral method are justified. We put 4 × 10⁶ electrons into our 2D sample, which are initially in the standard initial condition and follow the simulation protocols at 50 ∼ 300K in Table 1. The time course of the polarization P_z(t) is recorded in the middle of the sample (Figure 3.1).

0.8 1 1.2 1.4

Figure 3.2: A comparison between the analytical, numerical and experimental relations between τ_sT and mobility µ, as well as between τ_s and temperature T (inset). The large triangle size indicates the experimental error bar.

The observed P_z(t) is an exponential function with a relaxation time τ_s. During the temperature variation in Table 3.1, the relations (τ_sT, µ) and (τ_s, T ) can be calculated, which are plotted as red squares in the main plot and the inset of Figure 3.2, respectively. Note that our recent theoretical study and simulation reveal that the Pz(t) of the DP relaxation in a narrow wire will transit from an exponential function to a Bessel function during the impurity density decline [27]. Such P_z(t) deviation from an exponential function will not occur in the EY mechanism, as we shall prove in Sec. 4. Thus, here we can characterize P_z(t) properly by the parameter τ_s without worrying its deformation.

The red squares in Figure 3.2 calculated by our methods show very close values to the theoretical and experimental results for most µ, with the same relation τ_sT ∝ µ. The opposite experimental trend τ_sT ∝ µ⁻¹ in the high µ regime is not to see in our simulation. It indirectly supports the previous hypothesis that τsT ∝ µ⁻¹ arises from other mechanisms, because the pure EY mechanism in

our ensemble Monte Carlo method and the semiclassical path integral method simulation cannot produce this trend. The main plot of Figure 3.2 does not explicitly tell us how τsvaries with T . In fact, T can influence the sample me1831F through two ways. First, a large T will increase the electron mobility µ and subsequently τ_p = µm^∗/e, which in turn will reduce the electron scattering pro unit traveling distance. However, a large T also will enhance the spin flip probability φ (see Table 3.1) and make spin flip more frequently. When two effects blend together, it is hard to predict how τ_s will change with T . Nevertheless, an obvious τ_s decay is readily seen, when we transform the (τ_sT, µ) data into the (τ_s, T ) plot in the inset.

3.3 The size effect on the EY relaxation

The size effect on the spin relaxation is another interesting issue in spintronics.

For instance, the group of Awschalom has carried out some measurements on the DP relaxation in narrow wires of different widths [26]. However, to the best of our knowledge, very few experiments have investigated the size effect on the EY relaxation. A study close to this topic was the EY relaxation in the granular systems [29], but the sample size there was fixed. In this section, we will study how the EY spin relaxation changes with the width of a wire. Our sample has 200 µm in length, while its width varies between 0.1 µm (narrow wire) and 200 µm (2D quantum well). We take 8 × 10⁵ electrons in the standard initial condition and use the parameter values from Table 3.1 for simulations as before.

Figure 3.3 depicts the relaxation time τsversus the sample width W at various temperatures T . Three conclusions can be drawn from this plot:

(1) τ_s decreases with T .

(2) τ_s is nearly a constant for W > 1 µm at all T . (3) τ_s drops abruptly to zero, when W < 1 µm.

Phenomenon (1) has the same reasoning as that at the end of Sec. 3.2. To account for phenomena (2) and (3), remember that for W > 1 µm the sample is like a bulk system. The electron spin in this system are flipped mainly by the impurities in the bulk and less by the sample boundaries. Therefore, the relaxation time τ_s is almost fully determined by the impurity density and is thus a constant of W . However, for W < 1 µm, the boundary induced spin flip becomes more significant. The smaller the sample width, the higher the collision frequency will

0 50 100 150 200

Figure 3.3: The EY spin relaxation time versus the wire width at five different temperatures, observed on a stripe of area 1 × W µm². The inset is magnified from the main plot.

be, and the faster the spin will flip. When W approaches zero, τ_s tends to zero, because almost all electrons collide with the boundaries infinitely often, except the minority electrons moving exactly along the wire axis.

It is well known that the DP relaxation near the sample boundary behaves differently from that far from the boundary [27, 31, 32]. An interesting question is whether the EY relaxation will behave similarly on the boundary. To answer this question, we shrink the observation window to a small square of area 1 × 1 µm² and use this window to scan the local τs at different places along the width direction of a wire. Figure 3.4 depicts the EY relaxation time τ_s, which remains close to a constant inside the sample, up to the drops near two boundaries. How close to the boundaries τ_s will begin to drop is an open question requiring further study. Moreover, the inset of Figure 3.4 shows that P_z(t) is almost flat up to the slight drops on the boundaries. These drops will become apparent, if we magnify the individual P_z(t) curves.

For a comparison, we calculate the τ_s of the DP relaxations along a wire of width 6 ∼ 50 µm, as shown in Figure 3.5. The initial electron and spin states in the simulation are the same as above EY cases, while its temperature is as low as 5K to mimic the real experimental environment. The corresponding Fermi velocity and mean free path are vF = 0.37 µm/ps and lmfp= 0.28 µm and its spin

0 5 10 15 20

Figure 3.4: The EY spin relaxation time versus four wire widths at 50K, observed on a square of area 1 × 1 µm² scanning along the wire width in the middle of the sample. The inset is the evolution of spin polarization along the width W = 20 µm.

Figure 3.5: The DP spin relaxation time versus eight wire widths at T = 5K and L_so = 2 µm, observed on a square of area 1 × 1 µm² scanning along the wire width in the middle of the simulation sample. The inset is the evolution of spin polarization along the width W = 50 µm.

rotation length (as defined in [24,25,27]) is L_so = 2 µm. The simulation method is referred to [27]. In contrast to the EY relaxation, the DP relaxation time near the boundary is larger than that elsewhere (Figure 3.5). Moreover, the Pz(t) of the DP relaxation on the boundary exhibits a hump structure (inset of Figure 3.5), which is opposite to the EY relaxation and is a main difference between these two relaxations. The τ_s increase on the boundary in the DP relaxation is ascribed to the reversal rotation of spins [27, 32], whereas the τ_s decrease on the boundary in the EY relaxation is due to more frequent boundary collisions.

3.4 The impurity effect on the EY relaxation

When the mean free path of the electrons exceeds the wire width, the system will enter the ballistic regime. The P_z(t) of the DP relaxation undergoes a drastic change from an exponential function to a Bessel function during the diffusive-ballistic transition [27]. In this section, we will examine how the P_z(t) of the EY relaxation behaves in the ballistic regime.

Suppose an ensemble of electrons in the standard initial condition are put in a narrow wire as in Figure 3.6. The spin polarization Pz(t) observed at the origin p₀ at time t is averaged from the spin states of all electrons which arrive at p₀ at time t. These electrons can arrive through a straight trajectory or various zigzag ones, like p₁p₀ and gp₂p₀ in Figure 3.6, all of which have the same length l = v_Ft.

Depending on the trajectory types, these electrons will launch at different x at time 0. Suppose ˜s_z(x) is the z component of the spin state of an electron at p₀ at time t when it starts at x at time 0. Then P_z(t) is an average over all these spin states [27],

P_z(t) = R_l

−l˜sz(x)ρw(x)W dx R_l

−lρw(x)W dx , (3.2)

where w(x) is a weight proportional to the number of electrons starting at x and contributing to ˜s_z(x) and ρ denotes the constant surface density of the electrons in the wire. Notice that since the spin flip in the EY mechanism is a stochastic process, two electrons, even when running along the same trajectory, may have different final spin states. Thus, the spin state ˜s_z(x) should be understood as an ensemble average taken from all electrons running along the same trajectory. The fluctuation around this average is extremely small for real materials having the

typical electron density 10¹¹ cm⁻². If an electron starting between x and x + ε (for the case x ≥ 0) has to arrive at p₀ at time t, its initial outgoing angle must lie between θ(x) and θ(x + ε), with ε ¿ 1 (see the example for x = ξ in Figure 3.6). Thus, the weight w(x) becomes the fraction of the electrons at x running within these two angles over those within the whole 2π angle. If the electrons are uniformly distributed in the wire with isotropic outgoing angles, the fraction of electron number equals the fraction of orientation range [27],

w(x) = 1

2π · 2 · [θ(x) − θ(x + ε)]

= 1

π

·

arccos³x l

´

− arccos

µx + ε l

¶¸

, (3.3)

where the factor 2 arises from the two mirror-symmetric orientations ±θ.

Figure 3.6: In a ballistic narrow wire, a straight trajectory p₁p₀ and a zigzag trajectory gp₂p₀ have the same length l = v_Ft. The length of gp₂p₀ is equal to that of p₂p₃, since the former can be regarded as a multiple mirror reflection of the latter with respect to the horizontal dashed lines. The values θ(ξ) and θ(ξ + ²) are the outgoing angles of the electrons at ξ and ξ + ², respectively, along which the electrons can reach p₀ after running through the same distance l. The inset is a magnification of gp₂p₀.

While the fraction w(x) for EY relaxation is similar to that for DP relaxation,

˜

sz(x) is completely different for both. To obtain ˜sz(x) of the EY mechanism, we need to know how many scatterings an electron running from x to p0 will encounter

and how its spin state will be changed by these scatterings. If an electron is initially polarized in z direction and encounters n times of scattering, its z component of

and how its spin state will be changed by these scatterings. If an electron is initially polarized in z direction and encounters n times of scattering, its z component of

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