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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

We study spin relaxation in n-type bulk GaAs, due to the Dyakonov–Perel mechanism, using ensemble Monte Carlo methods. Our results confirm that spin relaxation time increases with the electronic density in the regime of moderate electronic concentrations and high temperature. We show that the electron-electron scattering in the non-degenerate regime significantly slows down spin relaxation. This result supports predictions by Glazov and Ivchenko. Most importantly, our findings highlight the importance of many-body interactions for spin dynamics: we show that only by properly taking into account electron-electron interactions within the simulations, results for the spin relaxation time—with respect to both electron density and temperature—will reach good

Recently, spin coherence in semiconductors has been the focus of both theoretically [

In n-type bulk GaAs, the main sources of spin relaxation are Elliot–Yafet and Dyakonov–Perel (DP) mechanisms [

Very recently, the ensemble Monte Carlo (EMC) method has been equipped for dealing with spin transport [_{i}^{16} to 2.5 ^{17} cm^{−3}).

Our results display good to very good agreement with the experimental results by Oertel ^{3}, suggested in the experimental paper. To our knowledge, this is the first time that EMC simulations can quantitatively reproduce spin-relaxation experimental results, and we will discuss the importance, to this aim, of properly taking into account electron-electron interactions.

Our results also confirm that, in the non-degenerate regime, SRT increases with the electron density, both including or excluding electron-electron scattering [

Finally, our findings suggest that the prediction made for two-dimensional systems by Glazov and Ivchenko [

We study carrier and spin dynamics in n-type bulk GaAs considering a single parabolic energy band (the central Γ valley), which determines the effective isotropic electron mass
_{e}

The scattering rates of electrons from an initial state,

where _{f}_{i}

By neglecting the exchange and correlation effects, Coulomb interaction between two charges in a homogeneous electron gas is usually estimated using the random phase approximation (RPA) [_{sc}

Here, _{q}^{2}^{2} the Fourier components with wavevector _{0}, for GaAs. We approximate _{TF}

Here, _{e}_{B}

with _{T}_{TF}

For

where υ is the magnitude of the electron (group) velocity and

Within the RPA, Bohm and Pines [

where r_{1}, r_{2} are the spatial coordinates of the colliding electrons. Only binary electron-electron collisions are considered here, as they are the most likely and effective scattering events. The quantum states of mobile electrons should be localized wave packets, but from the perspective of scattering theory [

where _{cr}_{k} is the electron occupation probability (or distribution function), in general, unknown, k_{0} is the wave-vector of the colliding electron and the sum runs over all the other electrons in the ensemble.

Within the EMC method, for any given scattering event, once the electron partner of wavevector k, involved in the collision is chosen, the final states, k^{0}_{0}, k^{0}

Here, _{0}, _{0} and

The expression for the scattering rate in _{TF}

First of all, we note that the Fermi golden rule entails first order Born approximation (BA) (usually simply referred to as “Born approximation”). From

All of this considered, assuming a parabolic band and a Yukawa screened potential, the formula for e-e scattering we use in our simulation is the following [

where _{lab}_{lab}_{βTF}

In the following, we shall use _{ee}_{ee}_{ee}

There are some important consequences about using the BA that we wish to recall.

The BA is well satisfied for sufficiently fast carriers assuming a weak scattering potential. It is indeed a high-energy approximation. At low energy (_{0} ≪ 1, where _{0} is the range of the scattering potential), a sufficient condition for the validity of the BA for a central potential (square well) is [

where _{0} is the typical strength of a short-range central scattering potential,

In the case of electron-electron scattering, assuming a screened Coulomb potential, the

with

The inequality in

There are indications that, when BA is not valid, it tends to overestimate the electron-electron total cross-section and, hence, the e-e scattering. Kukkonen and Smith [_{s}_{s}_{s}

To model electronic and spin dynamics in GaAs and to estimate the spin relaxation time, we employ the ensemble Monte Carlo method [

EMC consists of particles’ classical “free-flights”, during which the particles may be accelerated by classical forces, interrupted by scattering events, which alter the particles’ momentum. The probability of such scatterings and the momentum for each particle following a collision is determined computationally using stochastically generated random numbers.

Among the scattering mechanisms we consider (see Section 2), the scattering of carriers with the longitudinal polar optical phonons is the only source of thermal contact with the lattice. For convenience, we also introduce a fictitious scattering, known in the literature as “self-scattering”, which does not affect the particle, but simply ensures that the total scattering rate remains constant [

The free flight time,

with _{1} a random number generated from a flat distribution between (0, 1)
_{2} _{i}_{3}, if _{3} _{i}

The electron-electron scattering has to be handled somewhat differently, as it involves two particles. Traditionally, a number of approaches have been used, including treating the electron as scattering with a fictitious partner chosen from a Boltzmann distribution or being allowed to scatter with an actual simulated particle, whose momentum, though, was not updated. This second particle has been usually chosen irrespectively of its distance from the first particle.

In this work, we improve over previous EMC schemes and allow e-e scattering only between electrons that are within one screening length of each other. In our scheme, both electrons scatter, and their momenta are both updated. This approach prevents the unphysical accumulation of energy or momentum prevalent in other methods, as well as the scattering of electrons at opposite ends of the device.

To implement this, we effectively discretized the space into a grid of cubes of one screening length side. We keep track of the number of potential scattering events, which include each particle scattering off any in the same cube or in any of the neighboring cubes. Each time an electron- electron event is required, we choose randomly from each of these potential pairings and check that they are within one screening length of each other, and if they are, we carry out the scattering; if they are not within one screening length of each other, we choose a different electron as the second particle in the scattering.

In order to start collecting data, we need to wait for the electronic system to thermalize to the chosen lattice temperature. To do so, we initially allow the system to evolve for an appropriate time (thermalization run), during which the only source of thermal contact with the environment is provided by polar optical phonons. We note that the thermalization and the corresponding data collection runs always include the same type of scatterings; in particular, they will both include (or not include) electron-electron interactions.

The initial particle configuration (positions and momenta) for the thermalization run is chosen in the following way. The electron positions are generated randomly inside the bulk semiconductor using uniform pseudorandom numbers. Their velocity distribution is generated by choosing the

In order to ascertain that thermalization is reached, we have checked when the energy distribution of the carriers becomes a Boltzmann distribution function corresponding to the lattice temperature. Our simulations show that for the range of parameters of interest in this work and when electron-electron interactions are included, discarding the first 30 picoseconds from the simulation is sufficient to ensure thermalization: in particular, close to room temperature, the thermalization for the runs, including electron-electron interactions, appears to be completed after less than five picoseconds. This confirms the crucial role of electron-electron interactions in the thermalization process [

For the results shown in this work, after the thermalization run, we have reset the electronic spins to be fully polarized along one direction, namely, the

In bulk n-GaAs at room temperature and for the range of doping densities here considered, the main source of spin relaxation is the Dyakonov–Perel mechanism, a type of spin-orbit interaction. It is due to bulk inversion asymmetry, and it acts as an effective, momentum-dependent magnetic field, via the so-called Dresselhaus Hamiltonian [

where

Here, _{i}_{so}_{so}^{3} [

In the following, we neglect dipole-dipole interaction between spins. In this way, during free-flight, the spin of each electron undergoes an individual coherent evolution according to the Schrödinger equation.

Initially, each electron spin is assumed to be polarized in the _{D}

The time-evolution operator,

so that the spinor wavefunction, Ψ (

In order to integrate numerically

where Ψ^{n}

which is correct up to ^{4}.

This method is particularly convenient, as the inverse of the spin Hamiltonian can be written analytically, allowing for a significant improvement in computational efficiency compared to the exact solution, with insignificant loss of precision. The C-N method is particularly good for the problem of spin evolution, as it gives a unitary evolution of the spinor wavefunction in time; hence, it conserves its norm. In contrast to the commonly used Heun scheme, the C-N method has the advantage that we do not need to renormalise the spinor wavefunction after each time step. The explicit numerical scheme is:

where:

_{i}

and

At any given time, we can extract the expectation values of the _{x}_{y}_{z}

where _{z}

Using the above methodology, we are capable of simulating the time evolution of the total electronic spin and of its components in the sample. The quantity of interest to us is the characteristic spin relaxation time of the material. This can be extracted from the time evolution of

We assume that, after a transient period, the spin relaxation behavior in the bulk semiconductor takes the form:

It is then possible to fit the data from the simulation of the spin time evolution to such a curve (an example is plotted in ^{−}^{1} and is identified as the characteristic spin relaxation time of the sample, _{s}

In this section, we present and discuss our results for the spin relaxation time and compare them to experimental data.

Apart from assuming an exponential decay of the total spin polarization in the _{so}^{3}.

In _{s}_{s}_{e}^{16} to 2.5 ^{17} cm^{−}^{3}, we see that the inclusion of electron-electron scattering causes a net increases of _{s}

Additionally, we notice that the percentage increase of _{s}_{e}^{17} cm^{−}^{3}. Its absolute increment

We observe that, when including e-e interaction, our results for densities

However, at higher densities, our results for
_{s}_{e}^{17} cm^{−}^{3}.

We suggest that the overestimate of _{s}

We focus now on the effect of temperature on the spin relaxation time.

In order to compare our calculations with other experimental data from [_{e}^{16} cm^{−}^{3} and 3.8 ^{16} cm^{−}^{3}. In both cases, we will consider interacting carriers.

In _{e}^{16} cm^{−}^{3} and _{e}^{16} cm^{−}^{3} alongside the corresponding experimental data (empty square symbols). We find good agreement over the entire temperature range between

As noted before, the values of the spin orbit coupling for GaAs found in the literature vary greatly [_{so}

In order to let the reader appreciate how valuable this is, and in this respect, how relevant is the good agreement between our data and the experimental ones, in this section, we wish to show how sensible our simulations are with respect to the value of _{so}

In _{so}_{so}

We think that this is a convincing proof that the very good agreement between our results and the experiments is not accidental, but derives from the improvements we have devised in treating the e-e interaction within the EMC method. These improvements allow us to account properly for the electron-electron interaction within the simulations.

We wish to understand better the results relative to the e-e curve in _{s}^{17} ^{−}^{3}, while, as BA worsens at lower densities, we would expect that the SRT curve we obtained from our calculations lies above the experimental curve for the entire range of densities.

To explain this good agreement in the low density limit, we make some general considerations about Coulomb scattering, RPA and screening. Going towards low densities, the RPA starts to break down, which means that in our model, we are no longer allowed to split the e-e interaction into two parts. This can be also understood by looking at _{s}

From

The breakdown of RPA in low electronic densities may affect also the screening length, whose calculation strongly relies on this approximation in our model and, consequently, making the scattering probability less reliable.

The RPA breaking down means that e-e scattering should, in the real system, be more effective. However we still use a Yukawa potential in our calculations; so, as the density decreases, we should be underestimating the e-e scattering and, so, should obtain a
_{s}_{s}

In the opposite limit (_{s}_{s}_{e}^{16} ^{−}^{3}, which we do not simulate and which are not realistic, because the system becomes an insulator.

The third regime is intermediate and corresponds to _{s}_{s}_{s}

Another way of looking at the previous considerations is that, for low electronic densities, the system differential cross-section, as described by our simulations, is in some way mimicking a bare Coulomb potential one. Because the later is the exact differential cross-section of the system [

We will now demonstrate that indeed in our simulations and for the low density range, ^{bare}^{Y}^{bare}^{Y}^{ratio}^{Y}^{bare}^{ratio}

The Yukawa differential cross-section in BA is given by [

where _{TF}^{ratio}

where we have defined the dimensionless quantities _{βTF}_{lab}_{βTF}^{2} (^{ratio}^{ratio}_{θ}_{*}^{*}^{ratio}_{θ}_{*}^{*}^{ratio}

This integral can be solved analytically, and we get:

Because the system is at equilibrium, we can use the Boltzmann distribution, _{B}_{θ}^{ratio}

where
^{ratio}^{ratio}

The integral, ^{∗}_{e}^{17} cm^{−}^{3}, only 8% of the total number of carriers would scatter with a differential cross-section, such that 0.7 ^{ratio}

We have improved the treatment of e-e scattering in ensemble Monte Carlo and shown that our method allows one to reproduce, with no fitting parameters, the experimental results for spin relaxation by Oertel

For the highest electron densities considered, the Born approximation slightly overestimates the e-e scattering rate and, hence, the corresponding scattering cross-section. This implies a higher probability of having a third electron within the scattering cross-section. As future work, we wish to study the importance of this spurious “third-body” effect on spin dynamics in semiconductors and evaluate if an appropriate treatment of it can further improve the agreement with the experimental results.

We acknowledge support from EPSRC Grant No. EP/F016719/1 and from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement 281043 “FemtoSpin”. We thank Michael Oestreich for the experimental curves.

Gionni Marchetti and Irene D’Amico analyzed the data and wrote the manuscript. Matthew Hodgson and Gionni Marchetti together developed the code for the spin transport. James McHugh carried out part of the numerical simulations. Irene D’Amico and Roy Chantrell supervised the study. All authors contributed to the scientific discussion.

The authors declare no conflict of interest.

(_{e}_{i}

Histogram of the number of the electron-impurity scattering events against the scattering angle, _{e}^{16} cm^{−}^{3} (_{e}^{17} cm^{−3} (

The behaviour of _{s}_{s}

_{e}^{16} cm^{−}^{3} at

Results for _{s}^{3}, _{so}^{3}. The experimental data from [

The spin relaxation time, _{s}_{e}^{16} cm^{−}^{3} from simulations, including electron-electron interaction, and from the experimental results, as obtained in [^{3} and _{so}^{3} and include e-e scattering. Following [

The spin relaxation time, _{s}_{e}^{16} cm^{−}^{3} from simulations, including electron-electron interaction, and from the experimental results, as obtained in [^{3} and _{so}^{3} and include e-e scattering. Following [

Spin relaxation time _{s}_{so}^{3}. Other parameters: ^{3} and

The integral, ^{∗}