Effects of strain, electric, and magnetic fields on lateral electron-spin transport in semiconductor epilayers

We construct a spin-drift-diffusion model to describe spin-polarized electron transport in zinc-blende semiconductors in the presence of magnetic fields, electric fields, and off-diagonal strain. We present predictions of the model for geometries that correspond to optical spin injection from the absorption of circularly polarized light, and for geometries that correspond to electrical spin injection from ferromagnetic contacts. Starting with the Keldysh Green’s function description for a system driven out of equilibrium, we construct a semiclassical kinetic theory of electron-spin transport in strained semiconductors in the presence of electric and magnetic fields. From this kinetic theory we derive spin-drift-diffusion equations for the components of the spin-density matrix for the specific case of spatially uniform fields and uniform electron density. We solve the spin-drift-diffusion equations numerically and compare the resulting images with scanning Kerr microscopy data of spin-polarized conduction electrons flowing laterally in bulk epilayers of $n$-type GaAs. The spin-drift-diffusion model accurately describes the experimental observations. We contrast the properties of electron-spin precession resulting from magnetic and strain fields. Spin-strain coupling depends linearly on electron wave vector, and spin-magnetic field coupling is independent of electron wave vector. As a result, spatial coherence of precessing spin flows is better maintained with strain than with magnetic fields, and the spatial period of spin precession is independent of the applied electrical bias in strained structures whereas it is strongly bias-dependent for the case of applied magnetic fields.

Figure 1

(Color) Diagrams and corresponding experimental data of the two measurement geometries considered in this work. (a) Spin-polarized electrons, oriented along $\widehat{z}$, are optically injected into an $n:\mathrm{Ga}\mathrm{As}$ epilayer with a focused, circularly polarized laser incident along the $z$ axis (indicated by the red arrow). Lateral contacts provide an in-plane electric field $(\mathbf{E}\Vert \pm \widehat{x})$, Helmholtz coils provide an in-plane magnetic field $(\mathbf{B}\Vert \pm \widehat{y})$, and off-diagonal strain ${ϵ}_{xy}$ results from controlled uniaxial stress applied along the [110] GaAs crystal axis. Scanning polar Kerr-rotation microscopy is used to spatially image $n_z$, the $z$ component of the resulting steady-state electron-spin polarization. The drift, diffusion, and precession of the spin flow are determined (in part) by the applied electric, magnetic, and strain fields (see data below; adapted from Ref. 15). (b) Electrical injection of spin-polarized electrons into an $n:\mathrm{Ga}\mathrm{As}$ epilayer using ferromagnetic (FM) injector contacts. The contact magnetization $\left(\mathbf{M}\right)$ may be either in the $x\text{−}y$ plane, or out-of-plane $(\mathbf{M}\Vert \widehat{z})$. As shown schematically, the injected spins then flow laterally within the $n:\mathrm{Ga}\mathrm{As}$ epilayer, and $n_z$ can be spatially imaged with scanning Kerr microscopy. Experimental data, using an iron injector with $\mathbf{M}\Vert -\widehat{x}$ and $B_y=3.6\mathrm{G}$, is shown below (adapted from Ref. 16).

Figure 2

(Color) (a) and (b) show images of the calculated $z$ component of electron spin density $\left(n_z\right)$ for optically injected spins that are flowing to the right and that are subject to (a) applied [110] stress (strain ${ϵ}_{xy}=4\times {10}^{-4}$; $E_x=12\mathrm{V}∕\mathrm{cm}$), and (b) applied magnetic field ($B_y=16\mathrm{G}$; $E_x=7\mathrm{V}∕\mathrm{cm}$). Note that the color scale is modified such that black indicates negative values of $n_z$ (i.e., spins that have precessed into the sample plane). For all calculations, $D=10{\mathrm{cm}}^2∕\mathrm{s}$, $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$, and ${\tau }_s=125\mathrm{ns}$. (c) and (d) show images of corresponding experimental data under the same conditions. (e) and (f) directly compare line cuts through the center of the corresponding calculated (black line) and measured (red dots) images.

Figure 3

Calculated $z$ component of spin polarization $\left(n_z\right)$ generated by a narrow stripe excitation at $x=0$ under the influence of a fixed electric field of $E_x=15\mathrm{V}∕\mathrm{cm}$ and: top row, a magnetic field of $B_y=15\mathrm{G}$ (left) or $B_y=30\mathrm{G}$ (right); center and lower rows, strain ${ϵ}_{xy}=3.5\times {10}^{-4}$ (left) or ${ϵ}_{xy}=7\times {10}^{-4}$ (right). The solid line is for electric field alone, without magnetic field or strain. The dashed lines represent cases where, in addition to the electric field, a magnetic or strain field is applied. The bottom row is for the case of strain and strictly 1D motion. $D=10{\mathrm{cm}}^2∕\mathrm{s}$, $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$, and ${\tau }_s=125\mathrm{ns}$.

Figure 4

Calculated $z$ component of spin polarization $\left(n_z\right)$ generated by a narrow stripe excitation at $x=0$ under the influence of a lateral electric field $E_x=10\mathrm{V}∕\mathrm{cm}$ (top row) and $E_x=20\mathrm{V}∕\mathrm{cm}$ (bottom row). $B_y=15\mathrm{G}$ in the left column; ${ϵ}_{xy}=3.5\times {10}^{-4}$ in the right column. The solid line is for electric field alone, without magnetic field or strain (no precession). The dashed lines represent cases where, in addition to the electric field, a magnetic field or strain is present. $D=10{\mathrm{cm}}^2∕\mathrm{s}$, $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$, and ${\tau }_s=125\mathrm{ns}$.

Figure 5

(Color) Calculated $80\times 120\mu \mathrm{m}$ images of $n_z$ for optically injected spins, showing how diffusion directly influences the degree of spatial spin coherence for precessing spin flows. Images in the left column are for $D=1{\mathrm{cm}}^2∕\mathrm{s}$; images in the right column are for $D=20{\mathrm{cm}}^2∕\mathrm{s}$. All calculations use $E=10\mathrm{V}∕\mathrm{cm}$, $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$, and ${\tau }_s=125\mathrm{ns}$. (a) and (b) show spin flow in the absence of magnetic or strain fields. (c) and (d) show precessing spin flow in the presence of $B_y=20\mathrm{G}$. (e) and (f) show precessing spin flow in the presence of strain $({ϵ}_{xy}=7\times {10}^{-4})$. (g) and (h) show line cuts through the center of the corresponding images, with blue and red dashed lines showing the case of finite $B_y$ and ${ϵ}_{xy}$, respectively.

Figure 6

(Color) Diagram and calculated images showing how an effective magnetic field due to off-diagonal strain $\left({\mathbf{B}}_{ϵ}\right)$ can either augment or oppose a real magnetic field $B_y$, depending on the electron momentum $\mathbf{k}$. (a) Calculated $60\times 60\mu \mathrm{m}$ image of spin diffusion with no electric field, and with $B_y=6\mathrm{G}$ and strain ${ϵ}_{xy}=2\times {10}^{-4}$. Spins diffuse radially away from the point of injection, “sampling” all momenta $\mathbf{k}$ in the $x\text{−}y$ plane. Spins diffusing to the left see a net effective magnetic field and precess; spins diffusing to the right see negligible net effective magnetic field and do not precess. (b) With $E_x=-10\mathrm{V}∕\mathrm{cm}$, electrons flowing to the left show pronounced spatial precession. (c) With $E_x=+10\mathrm{V}∕\mathrm{cm}$, electrons flowing to the right show essentially no precession. (d) Corresponding line cuts through the center of the images in (b) and (c). In all images, $D=3{\mathrm{cm}}^2∕\mathrm{s}$, ${\tau }_s=125\mathrm{ns}$, and $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$.

Figure 7

(Color) $40\times 40\mu \mathrm{m}$ images of the calculated $z$, $x$, and $y$ components of optically injected spin polarization $(n_z,n_x,n_y)$. Spins are injected with ${\mathbf{n}}_0\Vert \widehat{z}$ at the center of the images, and subject to diffusion only (no electric field) in the presence of either an applied magnetic field (left column; $B_y=20\mathrm{G}$) or strain (middle column; ${ϵ}_{xy}=7\times {10}^{-4}$). $D=10{\mathrm{cm}}^2∕\mathrm{s}$ and ${\tau }_s=125\mathrm{ns}$. The rightmost column shows horizontal line cuts through the center of the corresponding images of $n_z$ and $n_x$. The diagrams at the bottom illustrate how spins precess from initially out-of-plane $({\mathbf{n}}_0\Vert \widehat{z})$ to in-plane while diffusing in the uniform magnetic field $B_y$, or in the chiral effective magnetic field ${\mathbf{B}}_{ϵ}$.

Figure 8

(Color) $100\times 200\mu \mathrm{m}$ images of the calculated $z$ component of electron-spin polarization $\left(n_z\right)$ resulting from in-plane spin injection $\left({\mathbf{n}}_0\parallel \mathbf{M}\parallel \widehat{x}\right)$ by a $40\text{−}\mu \mathrm{m}$-wide magnetized stripe contact (edges of the stripe are shown by dashed lines). Corresponding horizontal line cuts through the center of the images are shown to the right. (a) Spin diffusion alone for the case of a strained sample $({ϵ}_{xy}=1\times {10}^{-4})$. Spins diffusing to the left or right precess in opposite directions. (b) Spin diffusion alone for the case of an applied magnetic field $(B_y=1\mathrm{G})$. (c) Spin diffusion and drift for the case of applied magnetic and lateral electric fields ($B_y=1\mathrm{G}$, $E_x=4\mathrm{V}∕\mathrm{cm}$). For all images, $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$, $D=10{\mathrm{cm}}^2∕\mathrm{s}$, ${\tau }_s=125\mathrm{ns}$.

Figure 9

(Color) Calculated images of $n_z$ resulting from either out-of-plane (left column) or in-plane (right column) spin injection and subject to either magnetic or strain fields. The $300\times 200\mu \mathrm{m}$ images show the $z$ component of calculated electron-spin polarization $\left(n_z\right)$ resulting from a square $80\times 80\mu \mathrm{m}$ injection contact (edges of the source are indicated by dashed lines). In (a) and (c), the contact magnetization is out-of-plane $({\mathbf{n}}_0\Vert \widehat{z})$, and in (b) and (d) the contact magnetization is in-plane $({\mathbf{n}}_0\Vert \widehat{x})$. Measuring $n_z$ at a specific position as a function of in-plane magnetic field $B_y$ yields the “Hanle curves” (Ref. 40) shown in (e) and (f). These curves are acquired at distances of 4, 20, 40, and $80\mu \mathrm{m}$ from the right edge of the square source contact. Solid lines correspond to sweeping $B_y$ alone; dashed lines include the presence of strain $({ϵ}_{xy}=0.03\%)$, which shifts the Hanle curves to the left. In the calculations, $\mu =3000{\mathrm{cm}}^2∕\mathrm{V}\mathrm{s}$, $E_x=8\mathrm{V}∕\mathrm{cm}$, $D=3{\mathrm{cm}}^2∕\mathrm{s}$, and ${\tau }_s=125\mathrm{ns}$.