Monte-Carlo simulation studies of the effect of temperature and diameter variation on spin transport in II—VI semiconductor nanowires

  • 1. Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India
  • 2. Microelectronics Research Center, 10100 Burnet Road, University of Texas at Austin, Austin, TX, 78758, USA

Key words: spintronicsspin transportII—VI nanowirespin relaxation lengthsMonte Carlo method

Abstract: We have analyzed the spin transport behaviour of four II—VI semiconductor nanowires by simulating spin polarized transport using a semi-classical Monte-Carlo approach. The different scattering mechanisms considered are acoustic phonon scattering, surface roughness scattering, polar optical phonon scattering, and spin flip scattering. The II—VI materials used in our study are CdS, CdSe, ZnO and ZnS. The spin transport behaviour is first studied by varying the temperature (4—500 K) at a fixed diameter of 10 nm and also by varying the diameter (8—12 nm) at a fixed temperature of 300 K. For II—VI compounds, the dominant mechanism is for spin relaxation; D'yakonovPerel and Elliot Yafet have been actively employed in the first order model to simulate the spin transport. The dependence of the spin relaxation length (SRL) on the diameter and temperature has been analyzed.

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1.   Introduction
  • Spintronics devices harness spin and the associated magnetic moment of an electron to store and transmit information in terms spin polarized currents. By using a semi-classical Monte Carlo method[1, 2], spin transport has recently been successfully modelled. The feasibility of novel devices like spin-FET[3, 4], spin valves[5], etc. necessitates the study of spin transport in detail. Efficient data transfer in a spin device not only depends upon the injection and extraction of spin but also on the length and duration of the retention of injected spin. The critical length over which the injected spin retains 1/e times its initial injection value is referred to as the spin relaxation length, which eventually depends upon opto-electronic, magnetic and certain other parameters specific to the material of concern. Compounds that offer the provision to manipulate parameters to suit the conditions of smooth spin transport are required. II-VI compounds form a special class of materials for spintronics applications as they offer a wide range of combinations of various optical, magnetic and electronic properties. These direct band gap semiconductors have found usage in display technology and solar cell technology due to their excellent optical properties and electroluminescence[6, 7]. Recently, their magnetic properties are being studied and suitable fabrication processes are being developed and fine-tuned for drawing out better nanostructures and devices composed of these materials[8]. Band engineering in these semiconductors is possible allowing one to study the behaviour in detail with a wide range of band gaps ranging from 1.5 to 3.7 eV. Spin transport for Si and GaAs[9, 10, 11], the dependence of the spin relaxation length (SRL) on the field, temperature[12, 13], etc., are some of the recent results in this area. Quite recently, the dependence of spin behaviour on the electric field in the case of II-VI compounds was demonstrated[14]. The spin relaxation mechanisms used in II-VI compound semiconductors are D'yakonov-Perel (DP) relaxation and Elliott-Yafet (EY) relaxation, and the associated scatterings with these mechanisms are acoustic phonon scattering, polar optical phonon absorption and emission, surface roughness scattering and spin flip due to the EY relaxation mechanism. In Section 3, we present our results along with the discussion on them. Section 4 offers a conclusion and the scope of the results.

2.   Model
  • A Monte-Carlo based model is developed to simulate the spin based transport. The transport between scattering events is treated classically while the scattering events are treated quantum mechanically. The various scattering rates used are calculated using Fermi's Golden rule. The free flight time of the carriers is determined stochastically. At the end of the free flight time, scattering events are selected based on the relative probabilities of the scattering events. This process is repeated for a fixed number of iterations at the end of which desired information is extracted by taking an ensemble average of the spin for all the carriers at each position. The carriers inside the active region of the nanowire are treated quantum-mechanically in the direction of confinement, i.e., the electron motion is constrained along the $y$- and $z$-axes. The initial injection is along the $x$-axis with spin polarized along the $z$-direction. While the potential difference along the $x$-axis drives the current, a transverse field of 100 kV/cm applied along the $y$-axis introduces the effect of Rashba spin orbit coupling[4].

    The dimensions of the nanowire used for the simulations are 10000 $\times$ 10 $\times$ 10 nm$^3$ along $x, y$ and $z$ respectively, as shown in Figure 1. The simulations are run for 600000 iterations to ensure that the system reaches a stable state before the values are recorded and the statistics obtained are recorded and processed for the last 100000 iterations. The iterations corresponded to a step duration of d$t$ $=$ 0.2 fs, which has been chosen to ensure that no scattering takes place for a duration smaller than the mean free time.

    At the very outset, for calculating the different wave function ($\Psi )$ and the associated energy levels ($E$), the mass dependent Schrdinger equation was employed:

    Using $\Psi$, the probability of locating the electron position either in the core or the shell was calculated using

    Ensemble spin vector, calculated component-wise under conditions of steady state, is given by the expression
    $<S_{i}>(x, t)=\frac{\sum^{t=T}_{t=t_1}\sum^{n_x(x, t)}_{n=1}s_{n, i}(t)}{\sum^{t=T}_{t=t_1}n_x(x, t)}, $(3)
    where $n_x(x, t)$ is the number of electrons at time $t$ and position $x$ within an accuracy of $\Delta x$, $s_{n, i}(t)$ is the $n$th electron's spin at time `$t$', while $t_{1}$ and $T$ are the start and finish times, respectively. The Hamiltonians for Dresselhaus and Rashba spin-orbit interactions are given as follows:
    $H_{\rm R}=\eta(\boldsymbol{\sigma} \times \boldsymbol{p}), $(4)
    $H_{\rm D}=-\beta(\langle k_y \rangle^2-\langle k_z \rangle^2)k_x\sigma_x, $(5)
    where $\boldsymbol{\sigma}$ is the Pauli matrix, $\boldsymbol{p}$ is the momentum vector, $k$ is the wave vector, while $\eta $ and $\beta $ are Rashba and Dresselhaus components, which are material dependent. The spin orbit Hamiltonian is calculated from the expression:
    $H_{\rm SO}=\frac{ge \hbar}{8(m^*)^2c^2}\boldsymbol{\nabla}V(\boldsymbol{\sigma} \times \boldsymbol{p}), $(6)
    where $g$ is the Lande-$g$-factor of the material, $m^*$ is the effective mass, $c$ is the speed of light, and $\boldsymbol{\nabla}V$ is the potential gradient. These equations are used for updating spin along with the Larmor Precession equation given by
    $\frac{{\rm d}S}{{\rm d}t}=\Omega_{\rm eff}S, $(7)
    where $\boldsymbol{S}$ is the spin vector, and $\Omega_{\rm eff}$ is the angular frequency consisting of two components, one from the Rashba interaction and other from the Dresselhaus interaction, which is given by[12]
    $\Omega_{\rm D}=-\frac{2\beta_{\rm eff}k_xi}{\hbar}, $(8)
    $\Omega_{\rm R}=-\frac{2\eta k_xj}{\hbar}, $(9)
    where $\beta_{\rm eff}=-\beta(\langle k_y \rangle^2-\langle k_z \rangle^2) $.

    We have also used the Elliott-Yafet[15] relaxation mechanism, which is the cause of spin flip scattering given by[16]

    $\tau^{\rm EY}_{\rm s}$ here represents the spin relaxation time due to spin flip scattering, $\tau_{\rm p}$ represents the total momentum relaxation time, $E_{\rm g}$ is the band gap, and $\alpha $ is given by the following expression:
    $\alpha=\frac{\Delta}{E_{\rm g}+\Delta}, $(11)
    where $\Delta$ is the spin orbit splitting parameter. In our simulation, the scattering rates considered are acoustic phonon scattering[16, 17], polar optical phonon scattering[17, 18] and surface roughness scattering[16, 19] and spin-flip scattering[15].

    Parameters that have been used to calculate different scattering rates (mentioned above) are shown in Table 1.

3.   Results and discussion

    3.1.   Temperature variation

  • This section highlights the dependence of the spin relaxation length on the temperature in II-VI nanowires. The temperature is varied from 4 to 500 K in steps of 100 K (approximate) for both the materials, CdS and ZnS.

    Figures 2(a) and 2(b) show the dependence of the spin relaxation length on the temperature. It can be deduced from the figures that as the temperature increases, the spin relaxation length (SRL) decreases for both the II-VI materials. This is because both acoustic phonon scattering and polar optical phonon scattering that we have integrated in our model are temperature dependent; the rates of which increase with the increase in temperature, which reduces the ensemble spin thus reducing the spin relaxation length. Temperature dependence is clearly visible from the scattering expressions shown below.

    Acoustic phonon scattering[16, 17] is given by
    $\Gamma^{\rm ac}_{\rm n, m}(k_x)=\frac{\Xi^2_{\rm ac}k_{\rm b}T\sqrt{2m^*}}{\hbar^2\rho v^2}D_{\rm nm}\frac{(1+2\alpha \varepsilon_{\rm f})}{\sqrt{\varepsilon_{\rm f}(1+\alpha \varepsilon_{\rm f})}}\odot(\varepsilon_{\rm f}), $(12)
    where $\Xi_{\rm ac}$ is the acoustic deformation potential, $v$ and $\odot(\varepsilon_{\rm f})$ is the heavyside step function. $D_{\rm nm}$ is the overlap integral pertaining to the electron phonon interaction.

    The polar optical phonon scattering rate[17, 18] is given by
    \begin{split} {}& [W^{\rm e/a}_{\rm f, b}]_{{\rm PO}(E, \Gamma)}= \\[2mm]& \frac{e^2\omega\sqrt{2m^*}}{\hbar L_yL_z}\left(\frac{1}{\varepsilon^{\infty}}-\frac{1}{\varepsilon^{\rm s}} \right) \left(n+\frac{1}{2}\pm \frac{1}{2} \right) D(E\pm \hbar \omega, \Gamma)X \\[2mm]& \sum^{\infty}_{p=1} \sum^{\infty}_{r=1} \frac{\int^{L_y}_{\rm 0}\int^{L_z}_{\rm 0}|[\Psi_0(y, z)]^2 \sin \left(\frac{p \pi y}{L_y} \right)\sin \left(\frac{r \pi z}{L_z} \right){\rm d}y{\rm d}z|^2} {(q^{\rm e/a}_{\rm f, b})^2+\left(\frac{p \pi}{L_y} \right)^2+\left(\frac{r \pi}{L_z} \right)^2}, \\[2mm]& q^{\rm e/a}_{\rm f, b}=\frac{\sqrt{2m^*E}}{\hbar}-\frac{\sqrt{2m^*(E\pm \hbar \omega)}}{\hbar}, \quad n= \frac{1}{{\rm e}^{\frac{\hbar \omega}{k_{\rm b}T}}-1}, \end{split} (13)
    where $\omega$ is the angular frequency of a polar optical phonon, $\varepsilon^{\infty}$ and $\varepsilon^{\rm s}$ are the optical dielectric constant and static permittivity of the compound semiconductor, and $n$ is the phonon occupation number. The superscripts e/a stand for the emission and absorption of a phonon and subscript f(b) refers to forward (backward) scattering. It is clear from the above expression that with an increase in temperature, the acoustic phonon and polar optical phonon scattering rate increases and hence the spin relaxation length is reduced. Physically, it can be interpreted that when the temperature increases, the lattice vibrations would increase and there would be more chance of collision along the path of the electron and consequently more scattering would be observed.

  • 3.2.   Diameter variation

  • The dependence of the spin relaxation length on the variation of the dimensions of nanowire is shown for two other II-VI materials, ZnO and CdSe. The nanowire diameter is varied in steps of 2 nm from 8 to 12 nm.

    Figures 3(a) and 3(b) show the dependence of the spin relaxation length on the cross-sectional area of the nanowire. We can observe from the figures that as the nanowire cross-sectional area is reduced, the spin relaxation length (SRL) decreases for both the II-VI materials. Though the rate of reduction is different in two materials, the trend is evident. This can be explained on the basis of surface roughness scattering. In our model, we have assumed an exponentially correlated surface roughness. The intravalley surface roughness is given by[16, 19]
    \begin{split} \Gamma^{\rm sr}_{\rm n, m}(k_x, \pm)= {}& \frac{2e^2\sqrt{m^*}}{\hbar} \frac{2\Lambda \Delta^2\Lambda \sqrt{m^*}} {2+(q^{\pm}_x)^2\Lambda^2}|F_{\rm nm}|^2 \\[2mm]& \times \frac{(1+2\alpha \varepsilon_{\rm f})}{\varepsilon_{\rm f}(1+\alpha \varepsilon_{\rm f})}\odot(\varepsilon_{\rm f}), \end{split}(14)
    where $\Lambda$ and $\Delta $ are the correlation length and r.m.s. height of the fluctuations at the surface, respectively. $q^{\pm}_x=k_x \pm k'_x$ is the difference between the wave vectors of the initial and final states of electrons.

    $F_{\rm nm}$ is the surface roughness electron overlap integral at the top surface, and it is given by
    \begin{split} F_{\rm nm}= {}& \int\int {\rm d}y{\rm d}z \left[ -\frac{\hbar^2}{et_ym_y}\Psi_{\rm m}(y, z) \frac{\partial^2\Psi_{\rm n}(y, z)}{\partial y^2}+\Psi_{\rm n}(y, z) \right.\\[2mm]& \times \varepsilon_y(y, z)\left(1-\frac{y}{t_y} \right)\Psi_{\rm m}(y, z) +\Psi_{\rm n}(y, z)(\varepsilon_{\rm m}-\varepsilon_{\rm n}) \\[2mm]& \times \left.\left(1-\frac{y}{t_y} \right) \frac{\partial \Psi_{\rm m}(y, z)}{\partial y}\right].\end{split}(15)
    With reduced wire dimension, there is increased SRS due to the increase in the SRS overlap integral $F_{\rm nm}$. Overall, there is a net increase in scattering at lower reduced dimensions of nanowire resulting in the decrease of the spin relaxation length at lower widths. Similar results have been demonstrated in Reference [20].

4.   Conclusion
  • We have presented a first order calculation of the spin relaxation length (SRL) using a semi-classical Monte-Carlo method for II-VI semiconductors for nanowire structure in an attempt to investigate their spin transport properties. The effect of variation of temperature and of nanowire width on the spin relaxation length is studied. From the studies it can be inferred that with an increase in the temperature, the SRL decreases as a result of the increased acoustic and polar optical phonon scattering. Also, with a reduction in cross-sectional dimensions, the SRL decreases owing to the increased surface roughness scattering. With the observed behaviour pattern, one can choose the suitable region of operation. The results obtained can be treated more on a qualitative rather than a quantitative basis, as more detailed analysis with additional scatterings and approximations is required to accurately predict the spin behaviour in the nanowires.

Figure (3)  Table (13) Reference (19) Relative (20)

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