Relaxation of uniform spin modes

The spin relaxation times obtained in the experiments of Ref. [7] were measured in a 2D n-InGaAs channel of the length L = 200 µm and the width w = 0.42 ∼ 20 µm. The SOI in the sample is dominated by the Rashba coupling. In the notation of Ref. [7] it corresponds to the characteristic length lSP ' 1 µm, which is related to the above defined spin rotation length L_so as L_so = 2 l_SP.

sample (star) versus channel width w are taken from the experiments in Ref.

[7]. The τ_s calculated by the EMC and SPI methods are extracted from the polarization curve Pz(t), fitted by Eq. (4.1) with free parameters A and c (red solid curve) and with fixed parameters A = 1 and c = 0 (black dash-dotted curve). The experimental τ_s saturates at 11.5 µm (pink dashed straight line) for large w, the same as the analytically estimated value for Lso = 2.19 µm. The inset demonstrates three examples of P_z(t) for channel width w = 1.6 µm, 2.8 µm, and 4.4 µm.

The sample is characterized by the electron mean free path l = 0.28 µm, the

momentum scattering time τ_M = 0.76 ps, and its Fermi velocity can accordingly be estimated as v_F = 0.28 µm÷0.76 ps ≈ 0.37 µm/ps. For the carrier concen-trations ns = 5.4 ∼ 7.0 × 10¹¹ cm⁻² used in Ref. [7], the de Broglie wavelength λf = p

2π/ns of electrons in the 2DEG is around 30 ∼ 34 nm. The sample was patterned along various crystallographic directions and electron spins have been optically oriented parallel to the growth direction [0, 0, 1]. The relaxation times τ_s measured in Ref. [7] are replotted by the circles and the stars connected by the blue and the green curves in Fig. 4.1.

Since the width range 0.4 µm ≤ w ≤ 20 µm used in our calculations is much larger than the de Broglie wavelength λ_f, the quantum effects are negligible and the validity of the EMC and SPI approaches is justified. With the above exper-imental parameters, the EMC and SPI calculations are represented in Fig. 4.1, where the inset shows the relaxation curves P_z(t) for three channels of different widths. All electron spins were initially aligned in z direction. The relaxation time τ_s can be determined by a fitting of these P_z(t) curves with the exponential function

Pz(t) = A exp(−t/τs) + c. (4.1) For example, the (red) solid curve in Fig. 4.1 represents the relaxation time of 1.2 × 10⁷ electrons in channels of different widths w’s. A comparison with the experimental data (circles and stars) leads to following conclusions:

(i) At large widths (w > 15 µm), the electron spin can be regarded as relaxing in bulk systems. In the experiments in Ref. l_SP was estimated to be 1.0 ± 0.1 µm, corresponding to L_so = 2.0 ± 0.2 µm. This experimental uncertainty results in τ_s = 9.7 ± 2.1 ps, when calculated by the EMC and SPI methods.

However, each τs obtained from the EMC and SPI methods agrees very well with that determined by the analytical expression of DP relaxation τs = L²_so/(4vF l) for boundless systems. Thus, if the experimental samples are governed by pure Rashba Hamiltonian, as in our calculation, these samples most likely have L_so = 2.19 µm. This value is used in our EMC and SPI simulations to obtain the red and black curves in Fig. 4.1.

(ii) For intermediate widths (1.4 µm< w < 15 µm), there is no an analytical expression for τ_s to compare with. The EMC and SPI result deviates slightly from the experimentally measured τ_s. The maximum deviation is around 3 ps for [1, 1, 0] sample and 4 ps for [1, 0, 0] sample at w = 5 µm. The calculated τs is closer to the τs of the [1, 1, 0] sample.

(iii) For small widths (w < 1.4 µm), the experimentally measured τ_s saturates at 28 ps for [0, 0, 1] sample and 20 ps for [0, 1, 1] sample. It is assumed in Ref. [7] that this saturation might be related to other mechanisms, like the bulk inversion asymmetry. However, the calculated τs in Fig. 4.1 is also bounded by a maximum value around 24 ps, although in our calculation only the Rashba Hamiltonian was considered, without any additional mechanisms involved.

Figure 4.2: In the inset, a spin configuration in a channel of the length 8π µm relaxes to zero, depicted at three different times. These configurations are spa-tially uniform up to the ripples at two ends caused by boundary effect. Record-ing the polarization Pz(t) at the middle point of the channel gives the relax-ation curve (green solid thick) in the main plot. This curve can be fitted by the exponential function in Eq. (4.1) with [τ_s, A, c] = [7.127, 1.274, −0.055] (blue dashed curve) and [τs, A, c] = [4.323, 1, 0] (red dotted curve). For t close to zero, most electrons have not been reflected by impurities or boundaries. In this range P_z(t) does not behave as an exponential function. Later, after most of the electrons and their spins have been randomized by impurities or bound-aries, P_z(t) became more exponential-like. The physical parameters used are [w, L_so, v_F, l] = [0.1 µm, 2 µm, 0.37 µm/ps, 0.3 µm] with 6 × 10⁴ electrons.

An important factor affecting the interpretation of the experimental data is how to determine the relaxation time τ_s from the function P_z(t). The solid curve in Fig. 4.2 is an example of Pz(t) in a channel with w = 0.1 µm and l = 0.3 µm.

At first sight it looks like an exponential function to be fitted with a relaxation

time τ_s in Eq. (4.1). But a closer look shows that it is not a pure exponential function. Indeed, if we gradually increase l by reducing the number of impurities in the channel, the monotonically decreasing Pz(t) in Fig. 4.2 will transform to an oscillatory function. In the extreme case of an infinitely thin impurity-free channel, we shall prove in the next section that the evolution of P_z(t) will follow the Bessel function

P_z(t) = J₀

µ2v_F t Lso

¶

. (4.2)

This analytical formula is depicted by the smooth (red) dashed curve in Fig. 4.3.

A corresponding result of a numerical EMC and SPI simulations is plotted as a (green) rugged solid curve in the same figure.

0 10 20 30 40 50

−1

−0.5 0 0.5 1

Observation time t (ps) Spin polarization P z(t)

Figure 4.3: In a 1D channel without impurities, a spin polarization P_z(t) behaves like a sinusoidal function Eq. (4.3) (blue dash-dotted curve). In an infinitely thin channel without impurities, P_z(t) behaves like a Bessel function Eq. (4.2) (smooth red dashed curve), which agrees with the numerically obtained P_z(t) simulated by 5 × 10⁴ electrons (rugged green solid curve). The physical parameters are [w, L_so, v_F, l] = [0.1 µm, 2 µm, 0.37 µm/ps, 10⁴µm].

During transition from the diffusive to the ballistic regime, P_z(t) will undergo a crossover from an exponential to a Bessel function. In principle, it is meaningless to use an exponential function to extract τ_sfrom such a crossover function, especially when it is far from an exponential behavior. But if one would like to carry out this procedure, the so obtained τ_s will depend on the choice of the parameters A and c in Eq. (4.1):

(A) If A = 1 is chosen, Eq. (4.1) can precisely fit the real initial polarization

P_z(t) = 1 at t = 0 (red dotted curve in Fig. 4.2).

If A 6= 1, Eq. (4.1) can provide a better fitting to P_z(t) in a wider range of times at t > 0 (blue dashed curve in Fig. 4.2). On this reason, such a choice of A seems to be more appropriate.

(B) Further, if c 6= 0 the fitted values of c and τ_s will be strongly dependent on the observation time cutoff. The reason is that usually the tail of P_z(t) is oscillating, if the system within the considered range of times is not in the diffusive regime. The closer the system to the ballistic regime, the larger is the oscillation amplitude. The Bessel function in Eq. (4.2) for ‘pure’ ballistic regime has the largest amplitude. If the non-oscillating Eq. (4.1) is used to fit an oscillating Eq. (4.2) truncated at some cutoff, the fitted τ_s and c will depend on the cutoff. The corresponding uncertainty of τ_s will decrease with an increasing observation time.

In the experiments [7], the width of the channel varies between w ≈ 1.5 l and 70 l. Since at the smallest w the system is not far from the ballistic regime, the difference between P_z(t) and the exponential function should be observable. In-deed, the value of τs fitted by Eq. (4.1) with A = 1 and c = 0 (black dash-dotted curve in Fig. 4.1) is somewhat distinct from τs at A 6= 1 and c 6= 0 (red solid curve in Fig. 4.1). Since our observation time is sufficiently long, the fitted value of c is close to zero. A disagreement produced by different fitting procedures will become more remarkable when the system approaches the ballistic regime with strongly oscillating P_z(t). Hence, when comparing τ_s’s obtained by different re-search groups, it is important to know the whole set of the fitting parameters (A, c, and observation time). Even when the same P_z(t) curve is considered, the re-ported τ_s’s could be different. One more problem with the fitting procedure is that even in the diffusive regime the evolution of the spin polarization not necessarily follows the exponential behavior with a single relaxation time. For example, a homogeneous P_z distribution is not an eigenstate of the diffusion equation in a 2D channel. Therefore, as shown in Ref. [6], edge states can contribute to the P_z(t) evolution with the relaxation time different from that of the bulk eigenstate. The weight of edge states increases with decreasing w.

In regime (ii), the experimental data deviate slightly from the EMC and SPI calculations with a maximum difference τ_s ≈ 3 ∼ 4 ps at w ≈ 5 µm. This discrepancy is too large to be attributed to different fitting procedures. One of the explanations for such a behavior might be a specific role of long-lived edge states. The lifetime of such modes depends on the boundary conditions [6]. Our

EMC and SPI calculations assumed a specular reflection of electrons from hard wall boundaries of the wire. Probably, the experimental situation in Ref. [7]

corresponds to other boundary conditions which give rise to the edge states with larger τs. This problem requires a more thorough analysis.

In regime (iii), the relaxation time goes to a finite value at w → 0 in both experimental and EMC and SPI calculated plots in Fig. 4.1. For a homogeneous spin distribution along the channel, the diffusion theory [4] also predicts a sat-uration of τ_s at w → 0. The saturated value should be twice of the bulk DP spin relaxation time. With the experimental bulk value τ_s=11.5 ps, one expects τ_s=22.8 ps at w = 0. Experimental and EMC and SPI curves at Fig. 4.1 are not far from this value, although the diffusion approximation fails at w ' l. At the same time, one should not forget that in a narrow channel the time evolution of the spin polarization strongly deviates from the exponential function. On this reason, in regime (iii) τ_s can not be a representative parameter to describe the spin relaxation.