We use semiclassical Monte Carlo approach along with spin density matrix calculations to model spin polarized electron transport. The model is applied to germanium nanowires and germanium two-dimensional channels to study and compare spin relaxation between them. Spin dephasing in germanium occurs because of Rashba Spin Orbit Interaction (structural inversion asymmetry) which gives rise to the D’yakonov-Perel (DP) relaxation. In germanium spin flip scattering due to the Elliot-Yafet (EY) mechanism also leads to spin relaxation. The spin relaxation tests for both 1D and 2D channels are carried out at different values of temperature and driving electric field, and the variation in spin relaxation length is recorded. Spin relaxation length in a nanowire is found to be much higher than that in a 2D channel due to suppression of DP relaxation in a nanowire. At lower temperatures the spin relaxation length increases. This suggests that spin relaxation in germanium occurs slowly in a 1D channel (nanowires) and at lower temperatures. The electric field dependence of spin relaxation length was found to be very weak.

Of late, intensive experimental and theoretical studies have been conducted on the physics of electron spins due to the enormous promise displayed by the spin-based devices [

The basic idea of the spintronic-based devices is to use the spin degree of freedom. At the source, information is encoded as spin state of individual electrons and is then injected into the material. During its motion in the material, the electrons undergo scattering and hence the electron spin states relax or depolarize as they move in the channel. This is the process of spin relaxation. Spin detection is done at the drain. Our paper here deals with the second process of spin relaxation in a material. Spin relaxation lengths or spin dephasing lengths represent the distance from the source in which the spin polarization of an ensemble of electrons gets randomized and thus loses the information. We do not want the electrons to lose encoded information before the operation is complete. Thus information regarding spin relaxation lengths is critical to realize any useful spintronic-based device.

Intensive theoretical and experimental research has been conducted to study spin relaxation in metals and semiconductors. Several III–V and II–VI materials have been studied to ascertain their spin properties [

Though silicon has been the workhorse of the semiconductor materials since long, germanium has some superior properties [

Spin relaxation in semiconductors can occur via different spin relaxation mechanisms, such as D’yakonov-Perel (DP) [

In addition to the above studies, charge transport in nanowires has been modeled using the nonequilibrium Green’s function technique [

The paper is organized as follows. In Section

A detailed account of the Monte Carlo method [

Germanium is an elemental semiconductor and possesses crystallographic inversion symmetry [

In the Monte Carlo method, transport is simulated by free flights occasionally disrupted by scattering events. The flow chart of the Monte Carlo method is shown in Figure

Flow chart of the Monte Carlo simulation [

During free flight, in which no scattering occurs, the temporal evolution of spin [

Using (

We consider here that since the L valleys are lower in energy than the other two valleys, majority of electrons are concentrated in these

Also we assume here that the electric field applied is in the

The scattering processes considered are intravalley and intervalley phonon scattering, surface roughness scattering, and ionized impurity scattering. While considering phonon scattering, both optical phonons and acoustic phonons have been taken into account. The electrons will not always remain in the lowest ground subband and will make transitions to the higher subbands. Therefore, subbands are included in the simulation, and intervalley and intravalley intersubband scatterings are thus accounted for.

Spin flip scattering [

The formula for scattering rates calculations in nanowire and 2D channels are taken from [

The 2D channel has 5 nm as the thickness and 125 nm as the width. The nanowire is taken to be of cross-section 5 nm × 5 nm. The doping density is taken to be 4 × 10^{25}/m^{3}. The effective field is taken to be 100 kV/cm which is a reasonable value for germanium channels. This effective field acts as the transverse symmetry breaking field and leads to the Rashba spin orbit coupling.

Accounting for the confinement, four subbands are taken for the sake of simulation in each valley for both the channels. The moderate values of driving electric field (100 V/cm to 5 kV/cm) used ensure that the majority of electrons are restricted to the first four subbands. Also due to very small (5 nm) transverse dimensions of the channels, the higher subbands will be too much higher up in energy to the effect that we can assume them to be depopulated. Similarly in [^{6} such time steps to ensure that steady state has been reached. Data is recorded for the last 50,000 steps only. The ensemble average is calculated for each component of the spin vector for the last 50,000 steps at each point of the wire.

A spin transport study was done at room temperature (300 K) for both 2D channels and 1D nanowires at a moderate driving electric field of 1 kV/cm. Since the initial polarization is along the thickness of the wire, that is, in the

Variation of magnitude of spin along the channel of a germanium nanowire.

Variation of magnitude of spin along a 2D germanium channel.

Spin-dephasing length in a nanowire is found to be around 210 nm compared to around 12 nm in a 2D channel. Thus the spin-dephasing lengths for a nanowire are about 18 times larger than two-dimensional channels. This result bears conformity with similar studies conducted earlier by researchers where improvements in spin relaxation lengths in 1D channels over 2D channels have been reported [

Explaining this difference in terms of difference in scattering rates in between nanowires and 2D channels meets with failure owing to the fact that mobility in nanowires has been found to be smaller than that in a 2D channel. The origin of this difference is due to the fact that the dominant spin-relaxing mechanism, DP relaxation, is suppressed by confinement [

Figures

Variation of magnitude of spin along the channel for germanium nanowire at different temperatures at a driving electric field of 1 kV/cm.

Variation of magnitude of spin along a 2D germanium channel at different temperatures at a driving electric field of 1 kV/cm.

On increasing the crystal temperature, the phonon scattering rates increase. These increased scattering rates cause the electron to undergo scattering after very short time intervals and thus very short distances. This in turn randomizes the

Figures

Variation of magnitude of spin along the channel for germanium nanowire at different driving electric fields at a temperature of 300 K.

Variation of magnitude of spin along 2D germanium channel at different driving electric fields at a temperature of 300 K.

Any dominance of drift velocity over the scattering rates makes the electron and hence the spin penetrates further into the channel leading to larger spin relaxation lengths. The reverse happens when scattering rates dominate over drift velocity and the increased scattering rates dephase the spin faster. The overall effect is decided by the dominant effect amongst the two.

In our four subband model, the intersubband scattering saturates after a point as we increase the driving electric field. At higher driving electric fields, the scattering rates remain fairly constant with only a slight variation. The drift velocity starts to dominate the scattering rates in this regime, and spin relaxation length starts to increase. This explains the increase in relaxation length at higher electric fields. Comparing with our work on spin transport in Si in [

In our work we show that confining the motion to only one direction can improve drastically upon the spin relaxation length (more than an order of magnitude to around 210 nm for a nanowire compared to 12 nm for a 2D channel at 300 K and driving electric field of 1 kV/cm). Thus the information stored in the spin of electrons remains polarized up to a larger length on using nanowires due to suppression of DP relaxation. This larger spin relaxation length leads us to believe that spintronic devices can be efficiently implemented with nanowires. Also we observe that the spin relaxation length in nanowires can be further increased by reducing the temperature, in which case it increases to 940 nm at 77 K and 2.19

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