Spin-dependent tunneling in modulated structures of (Ga,Mn)As

A model of coherent tunneling, which combines multiorbital tight-binding approximation with Landauer-Büttiker formalism, is developed and applied to all-semiconductor heterostructures containing (Ga,Mn)As ferromagnetic layers. A comparison of theoretical predictions and experimental results on spin-dependent Zener tunneling, tunneling magnetoresistance, and anisotropic magnetoresistance is presented. The dependence of spin current on carrier density, magnetization orientation, strain, voltage bias, and spacer thickness is examined theoretically in order to optimize device design and performance.

Figure 1

(Color online) Dependence of tunneling current in arbitrary units and (b) its spin polarization (color scale) on the in-plane wave vector in $p\text{−}{\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}∕n\text{−}\mathrm{Ga}\mathrm{As}$ Zener-Esaki diode in the limit of low bias $\left(0.01\mathrm{V}\right)$. The calculation was performed for the hole concentration $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$, electron concentration $n={10}^{19}{\mathrm{cm}}^{-3}$, and the saturated magnetization (along the [110] crystallographic axis) corresponding to the Mn content $x=0.08$.

Figure 2

(Color online) Schematic of the tunneling process from the valence band of $p$-type (Ga,Mn)As to the conduction band of $n$-type GaAs.

Figure 3

(Color online) Cross section of the valence bands at the Fermi energy for various hole concentrations $p$ in $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$. Color scale denotes spin polarization. Magnetization is taken along the [110] direction.

Figure 4

(Color online) (a) Dependence of Zener tunneling current in arbitrary units and (b) its spin polarization on the direction of in-plane magnetization without strain and for deformations up to 1% applied along the [110] axis. Calculations performed for Mn content $x=0.08$, hole concentration $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$, and electron concentration $n={10}^{19}{\mathrm{cm}}^{-3}$ in the limit of low bias $\left(0.01\mathrm{V}\right)$.

Figure 5

(Color online) (a) Dependence of Zener tunneling current in arbitrary units and (b) its spin polarization on the direction of in-plane magnetization for various hole concentrations; Mn content $x=0.08$ and electron concentration $n={10}^{19}{\mathrm{cm}}^{-3}$ in the limit of low bias $\left(0.01\mathrm{V}\right)$.

Figure 6

(Color online) The bias dependence of the relative change in the tunneling resistance in Zener-Esaki $p\text{−}{\mathrm{Ga}}_{0.94}{\mathrm{Mn}}_{0.06}\mathrm{As}∕n\text{−}\mathrm{Ga}\mathrm{As}$ diode when magnetization is rotated out of the plane, for various hole concentrations; $n={10}^{19}{\mathrm{cm}}^{-3}$. The black squares are experimental points from Ref. 12.

Figure 7

(Color online) The calculated bias dependence of the TMR effect in $p\text{−}{\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}∕{\left(\mathrm{Ga}\mathrm{As}\right)}_4∕p\text{−}{\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ trilayer (with the GaAs barrier width $d=4$ monolayers) (a) for various hole concentrations $p$ and Mn ions content $x=0.08$ and (b) for various values of $x$ and $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$.

Figure 8

(Color online) The schematic band structure of the (a) $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}∕\mathrm{Ga}\mathrm{As}∕(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ and (b) $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}∕\mathrm{Al}\mathrm{As}∕(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ heterostructures for the hole concentration $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$.

Figure 9

(Color online) (a) The bias dependence of the TMR ratio for $p\text{−}{\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}∕{\left(\mathrm{Al}\mathrm{As}\right)}_4∕p\text{−}{\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ with $x=0.08$ and various hole concentrations and (b) for various values of $x$ and $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$.

Figure 10

(Color online) The bias dependence of the TMR ratio for various thicknesses $d$ of the barrier layer in (a) $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}∕{\left(\mathrm{Ga}\mathrm{As}\right)}_d∕p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$ and (b) $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}∕{\left(\mathrm{Al}\mathrm{As}\right)}_d∕p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$.

Figure 11

(Color online) Dependence of the TMR ratio on the direction of in-plane magnetization for various hole concentrations in the $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}∕{\left(\mathrm{Ga}\mathrm{As}\right)}_4∕p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$ trilayer structure.

Figure 12

(Color online) Dependence of tunneling current [in arbitrary units, in the limit of low $\left(0.01\mathrm{V}\right)$ bias] on the direction of the in-plane wave vector for antiparallel configuration of magnetizations (AFM) along the (a) [110] and (b) [010] crystallographic directions. Two different values of hole concentration were assumed: $1\times {10}^{19}{\mathrm{cm}}^{-3}$ (left panels) and $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$ (right panels).

Figure 13

(Color online) Dependence of tunneling current (in arbitrary units) on the direction of in-plane magnetization for (a) parallel (FM) and (b) antiparallel (AFM) magnetization orientations in $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}∕{\left(\mathrm{Ga}\mathrm{As}\right)}_4∕p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$ trilayer structures trigonally distorted along the [110] crystallographic axis; $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$; applied bias $0.01\mathrm{V}$.

Figure 14

(Color online) Dependence of tunneling current in arbitrary units on the direction of in-plane magnetization structures with parallel magnetization configuration (FM) for various hole concentrations in the $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}∕{\left(\mathrm{Ga}\mathrm{As}\right)}_4∕p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$ trilayer structure; $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$; applied bias $0.01\mathrm{V}$.

Figure 15

(Color online) Bias dependence of the ${\mathrm{TAMR}}_{\perp }$ ratio for TMR structure consisting of two magnetic $p\text{−}{\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$ layers separated by (a) GaAs and (b) AlAs barrier layer; $p=3.5\times {10}^{20}{\mathrm{cm}}^{-3}$.