Spin transport in polarization induced two-dimensional electron gas channel in c-GaN nano-wedges

A two-dimensional electron gas (2DEG), which has recently been shown to develop in the central vertical plane of a wedge-shaped c-oriented GaN nanowall due to spontaneous polarization effect, offers a unique scenario, where the symmetry between the conduction and valence band is preserved over the entire confining potential. This results in the suppression of Rashba coupling even when the shape of the wedge is not symmetric. Here, for such a 2DEG channel, relaxation time for different spin projections is calculated as a function of donor concentration and gate bias. Our study reveals a strong dependence of the relaxation rate on the spin-orientation and density of carriers in the channel. Most interestingly, relaxation of spin oriented along the direction of confinement has been found to be completely switched off. Upon applying a suitable bias at the gate, the process can be switched on again. Exploiting this fascinating effect, an electrically driven spin-transistor has been proposed.

www.nature.com/scientificreports/ expected to be significantly high and the wurtzite lattice is non-centrosymmetric, DP is likely to dominate the relaxation process 29 . The kinetic equation for spin density of the conduction band electrons is numerically solved to obtain the relaxation times associated with spin projections along different crystalline directions as a function of donor concentration and gate bias. Relaxation rate is shown to be a strong function of not only the spin-orientation but also the density of the carriers in the channel. Most notably, the relaxation of spin oriented along the confinement direction is found to be completely shutdown. Interestingly, the phenomenon is unaffected by any deviation from the symmetrical shape of the structure. Spin relaxation can again be switched-on by increasing electron concentration in the channel through gate bias. These findings lead us to propose a spintransistor device.

Theory
Spontaneous polarization, P along −ẑ induces a net negative polarization charge (in case of Ga-polar GaN) of density ρ s = P ·n , where n is the unit vector normal to the surface, on the inclined facades of a wedge-shaped wall structure, as shown schematically in Fig. 1a. In case of n-type GaN nanowalls, polarization charges on the side facades can create a repulsive force to the conduction band electrons resulting in a confinement in the central vertical ( 1120 ) plane of the nanowall 21 . Polarization charges at the bottom surface are often compensated/ suppressed by the charges of opposite polarity resulting from the substrate polarization. Conduction band minimum for wurtzite (WZ) GaN remains spin degenerate even after considering the effects of crystal field and SOC 30 . Lack of inversion symmetry in the WZ lattice results in a k dependent term in the Hamiltonian, which can be expressed as 5,9-12,14,31,32 : where α R determines the strength of the k-linear Rashba like contribution. This term arises in bulk (even in the absence of structural inversion asymmetry) as a result of the built-in electric field due to spontaneous polarization 12 . β D and b D are the Dresselhaus parameters associated with the k 3 -terms. In case of a 2DEG confined along [112 0] direction (x-axis), one can get an expression for H SO for the conduction band electrons by replacing k x , k 2 x terms in Eq.( 1) by their expectation values [32][33][34][35] . Note that �k x � = 0 for bound eigenstates. H SO can thus be expressed as: H SO can also be expressed as H SO (k) = 2 �(k) · σ , where �(k) represents an effective magnetic field and σ is the electron-spin. In bulk WZ-GaN, always lies in xy plane (Eq. 1) and its orientation is decided by the magnitude of k x and k y . Interestingly, when 2DEG is formed in ( 1120)-plane, the effective magnetic field is always along x (+ or −) direction irrespective of the magnitude and orientation of the in-plane wave-vector k , as shown in Fig. 1b. However, the magnitude of depends upon the y-and z-components of k . Below we will see that it has a remarkable consequence on the DP spin relaxation properties of the 2DEG in this case.
The D'yakonov-Perel' spin relaxation equation for the density, S i (t) of the spin projected along î (where, i = x, y, z ) can be written as 34,[36][37][38] represents spin independent momentum scattering rate between k and k ′ � , θ is the angle between the initial and final wave-vectors, A is the box normalization factor for the free part of the wavefunction of the confined electrons and H n = 2π 0 dφ 2π H SO e −inφ . It can be shown that Ṡ x (t) = 0 [see "Supplementary"], which implies that the DP mechanism does not alter the spin projection along x i.e., the relaxation time for x component of spin τ s x is infinite. This can also be understood from the following perspective. Since H SO always commutes with σ x , S x remains a good quantum number , are the same and can be expressed as (see "Supplementary" for detailed derivation): 1 are material dependent constants. This is consistent with the fact that under the DP mechanism τ −1 z = τ −1 x + τ −1 y . Therefore, τ z = τ y is expected with 1/τ x = 0 . Note that, the phenomena of vanishing spin relaxation has also been reported earlier in various contexts [38][39][40] , however, their origin is entirely different than what we have discussed here. It should be noted that the inclusion of higher order terms [ ∼ O(k 4 ) and higher] in the expression of the Hamiltonian (Eq. 1) may in principle contribute to the spin relaxation. However, their contribution is expected to be negligible as the Fermi wave-vector ( k f ) even for the 2D electron concentration of 10 17 m −2 in the well is sufficiently close to the conduction band minimum at the Ŵ-point 40 .

Summary of main results and discussions
As a test case, we have considered a wedge-shaped c-oriented WZ-GaN nanowall with a background donor concentration ( N d ) of 1 × 10 24 m −3 and dimensions as shown in Fig. 1a. The volumetric charge density ρ v (x, y, z) and the conduction band minimum, E c (x, y, z) have been obtained by solving two dimensional (2D)-Poisson's equation with appropriate boundary as well as charge neutrality conditions as described in Ref. 21 . Note that symmetry of the problem ensures that ρ v (x, y, z) and E c (x, y, z) are invariant along y-axis. ρ v (x, y, z) is shown in Fig. 2a. Formation of 2DEG is evident from the figure. In order to find the bound energy eigenstates, one needs to solve the 2D-Schrödinger equation on xz plane. However, due to the weak dependence of E c (x, z) on z, the 2D-Schrödinger equation can be approximated as a set of one dimensional (x-dependent) Schrödinger equations each of which is associated with a specific z position 41 . These calculations are carried out at T = 10 K. Figure 2b shows the conduction band profile [ E c (x, y) ] at a depth of 20 nm from the tip. Evidently, the central part of the E c (x) profile goes below the Fermi surface ( E f ), forming a trench that extends along the y-direction. E 1 denotes the first energy eigenstate of the quantum well at that depth. We have extended the calculation for several other N d values. In Fig. 2c, E 1 at z = 20 nm and the depth of the well ( v m ) with respect to the Fermi energy are plotted as a function of N d . Evidently, the separation between E 1 and E f (also v m and E f ) increases monotonically with increasing N d , which can be directly attributed to the increasing free carrier density in the 2D channel with increasing donor concentration. It should be mentioned that the range of the donor concentration is chosen in a way that only one eigenstate exists around the Fermi level, at that depth from the wall apex. Henceforth, we have shown the calculations only for the electrons lying at a depth of 20 nm from the tip of the wall.
Next, we calculate momentum relaxation time, τ m (the details of these numerical calculations can be found in Ref. 21 ) of the quantum confined electrons limited by the neutral donor scattering, which plays the most significant role in deciding the electron mobility at low temperatures in this system 21 . It should be noted that we have considered a spherically symmetric hard wall potential profile for the neutral impurities to calculate the scattering cross-section and τ m , subsequently. Variation of τ m and mobility ( µ ) (in right y-ordinate) with N d is  Fig. 3a, which clearly shows an increase of τ m with the donor concentration. The effect can be attributed to the increasing separation between E 1 and E f [i.e., the increasing electron density] with N d . An increase of the separation leads to the enhancement of the electron's kinetic energy, which results in a lower scattering cross-section. Relaxation time for y and z components of spin τ s y,z as a function of N d is shown in Fig. 3b. As expected, the spin relaxation time decreases as τ m increases. It should be noted that τ s y,z comes out to be ∼ 100 ps for the nanowall with N d = 0.35 × 10 18 cm −3 . Interestingly, a few factor change in donor density alters the spin relaxation time by about two orders of magnitude. The spin coherence length, L s = τ s v f , where v f is the Fermi velocity, is also plotted as a function of N d in the same panel. Note that L s for the lowest donor concentration comes out to be as high as 10 µm.
One way to manipulate spin transport in this system is to control the carrier concentration in the channel through gate bias. The idea is that with increasing carrier concentration, the kinetic energy of the electrons around the Fermi level increases. This, in turn, can change both µ the electron mobility and τ s i the spin relaxation time. To calculate these changes, one needs to incorporate the effect of gate voltage in the Poisson equation solution. Gate contact and the semiconducting channel together form a capacitor [see Fig. 4a]. When the source and drain electrodes are grounded, and a positive(negative) gate voltage is applied, some amount of electrons are pumped(removed) into(from) the channel by the power supply. Since the semiconductor is no longer charge neutral, the Poisson's equation has to be solved by satisfying appropriate positive to negative charge ratio condition (instead of satisfying charge neutrality) to obtain the E c profiles. The total positive to negative charge ratio ( r ch ) should be less(greater) than 1 when sufficiently positive(negative) gate voltages are applied. The following calculations are done with N d = 1 × 10 24 m −3 . As shown in Fig. 4b, the gap between E 1 and E f decreases as r ch increases. Note that when r ch is sufficiently less than unity [high (+)ve gate voltages], more than one bound states are formed below the Fermi level.
Variation of µ and τ s y,z with r ch are shown in Fig. 4c. As the gap between E and E f decreases with the increase of r ch , µ reduces while τ s y,z enhances. As obtained earlier, the coherence time ( τ s x ) for the spin projected along x-direction is infinite as far as DP mechanism (corrected up to ∼ O(k 3 ) term) is concerned . Relaxation of the x-component of spin should thus be governed by other processes such as Eliott-Yafet (EY) mechanism 29 . EY  At an adequately high negative gate voltage, when only one subband is filled, the x-polarized spin up current injected from FM I reaches the detector electrode without losing its coherence. As a result, the spin current through the channel is high, and 'spin-resolved charge voltage' measured between FM D and NM R will be sufficiently high. Upon application of a large positive gate bias, more than one subband are filled. At this condition, the direction of the effective magnetic field ( ), which is the Larmor precession axis of electron spin, get randomized due to successive scattering events. As a result, the x projection of spin also relaxes. This should result in a rapid decay of pure spin current in the channel between FM I and FM D , as depicted in Fig. 5b. The device can thus act as a spin-transistor. Note that, in most of the spin-transistor proposals, a control over spin relaxation is achieved through electric fieldinduced change in the effective magnetic field experienced by the carriers in the channel 46,47 . Here, the goal can be achieved by changing the carrier concentration in the channel, which can turn the DP mechanism on/off by removing/introducing additional eigenstates below the Fermi level 14,44,45 . As mentioned before that the nature of quantum confinement in this system is such that it preserves the symmetry between conduction and valence band everywhere irrespective of the fact whether the nanowedge is geometrically symmetric or not. This results in the suppression of the Rashba effect, which comes as an added advantage in maintaining the spin coherence of electrons in this system. Note that the present 2DEG system is substantially different from the 2DEG, formed at the heterojunction of a-directional GaN/AlGaN as discussed in "Supplementary".

Conclusions
Spin relaxation of electrons in a 2DEG formed at the central vertical plane of a c-oriented wedge-shaped WZ-GaN nanowall is theoretically investigated. It has been found that the component of spin projected in the plane of the confinement relaxes through DP mechanism with a time-scale of a few tens of picoseconds everywhere in the channel, while the spin component along the direction of confinement never relaxes through DP process in a part of the channel, where the Fermi-level occupies only one subband. However, by applying appropriate positive gate bias, electron concentration in the well can be sufficiently enhanced, which can bring in more than one subband below the Fermi-level in most of the channel. In this situation, DP relaxation mechanism is