Spin rotation, spin filtering, and spin transfer in directional tunneling through barriers in noncentrosymmetric semiconductors

We discuss possible tunneling phenomena associated with complex wave vectors along directions where the spin degeneracy is lifted in noncentrosymetric semiconductors. We show that the result drastically depends on the direction. In the [110] direction, no solution can be calculated in the usual way assuming that the wave function and its derivative are continuous. A method for obtaining physical solutions is given and consequences are drawn. As a result, there is no spin filtering in such a direction but the spin undergoes a precession through the barrier with the rotation angle being proportional to the barrier thickness. In a direction close to [001] we find a spin-filter effect in close agreement with the model discussed by Perel’ et al. [Phys. Rev. B 67, 201304(R) (2003)].

Figure 1

Sketch of the tunnel geometry with definition of notations. The spin-orbit-split barrier material of thickness $a$ (medium II) is located between two free-electronlike materials (media I and III). The tunnel axis, normal to the barrier, is the $z$ axis. In the free-electronlike materials, the real electron wave vector in the $z$ direction is referred to as $\mathbf{q}$. In the barrier material, the evanescent wave vector along the $z$ axis is referred to as $\mathbf{Q}+i\mathbf{K}$, where $\mathbf{Q}$ and $\mathbf{K}$ are real quantities. The transverse wave-vector component, in the barrier plane, $\mathbf{\xi }$ is conserved in the tunnel process. Then, the overall wave vectors in the three media are, respectively, ${\mathbf{k}}_{\text{I}}={\mathbf{k}}_{\text{III}}=\mathit{\xi }+\mathbf{q}$ and ${\mathbf{k}}_{\text{II}}=\mathit{\xi }+\mathbf{Q}+i\mathbf{K}$.

Figure 2

This figure illustrates transformations which, starting from a state of wave vector $\mathbf{k}$ and with a mean value of the Pauli operator $⟨\hat{\mathit{\sigma }}⟩$, construct degenerate states. ${\widehat{K}}_0$ is the complex conjugation and $\widehat{K}=-i{\sigma }_y{\widehat{K}}_0$ is the Kramers time-reversal operator. $\widehat{K}$ yields a state with the wave vector $-{\mathbf{k}}^{\ast }$ and with the mean spin $-⟨\hat{\mathit{\sigma }}⟩$. The state of wave vector ${\mathbf{k}}^{\ast }$ may be associated to another spin state, corresponding to the mean value ${⟨\hat{\mathit{\sigma }}⟩}^{\prime }$. Applying $\widehat{K}$ to this state, we form a degenerate state with the wave vector $-\mathbf{k}$ and associated to the mean spin $-{⟨\hat{\mathit{\sigma }}⟩}^{\prime }$. Four states are finally obtained.

Figure 3

Plot of the real-energy lines inside the gap for $\mathbf{k}=\left[\xi ,0,iK\right]$, where $K$ and $\xi$ are real and positive and $\text{tan}\theta =\xi /K$. The calculation is performed using a $14\times 14$ $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian. The loops are drawn versus $\Vert \mathbf{k}\Vert$ in $2\pi /a_0$ units, where $a_0$ is the cubic lattice parameter. In all these directions, the spin degeneracy is lifted. Their shape and extension sharply depend on $\theta$. For $\theta =43.2°$, the two branches are too close to be resolved at this scale. The parameters used in the calculation are $P=9.88\text{eV}\text{Å}$, $P^{\prime }=0.41\text{eV}\text{Å}$, $E_G=1.519\text{eV}$, ${\Delta }_c=E_{\Gamma 8c}-E_{\Gamma 7c}=0.171\text{eV}$, $P_X=8.68\text{eV}\text{Å}$, $\Delta =0.341\text{eV}$, $E_{\Delta }=E_{\Gamma 7c}-E_{\Gamma 6c}=2.969\text{eV}$, and ${\Delta }^{\prime }=-0.17\text{eV}$ (see Ref. 30 for a complete discussion). Inset: real-band structure (left, dashed line; for clarity, only a valence band which is connected to the evanescent branch is drawn) and evanescent band across the band gap (right, full line) along the [001] direction $\left(\theta =0\right)$ where the DP exchange field is zero (no spin splitting).

Figure 4

Mathematical plot of the real-energy lines for $\mathbf{k}$ along [110] as a function of the real part of the wave vector $Q$ in the barrier. The calculation is performed for a ratio $\gamma /{\gamma }_c=0.438\text{Å}$. We are only concerned with negative energies, which refer to evanescent states. More precisely, the physical states are located within a very small energy domain below the origin. The domain $Q>0$ refers to up-spin states, whereas the domain $Q<0$ refers to down-spin states. In each case, the imaginary component of the wave vector can take the values $\pm K$. Thus, at a given energy, we have exactly the four possible states $\left(Q\pm iK\right)\uparrow$ and $\left(-Q\pm iK\right)\downarrow$. The down-spin states are Kramers conjugates of the up-spin states.

Figure 5

This figure is a special case of Fig. 2, when the DP field $\mathit{\chi }$ lies along a real direction $\mathbf{n}$. Following the same procedure, four degenerate states are constructed, which now have their spins quantized along the same direction $\mathbf{n}$ (i.e., $⟨\hat{\mathit{\sigma }}⟩={⟨\hat{\mathit{\sigma }}⟩}^{\prime }$).

Figure 6

(Color online) The lower part of this figure illustrates the spin-dependent tunneling scheme in the case of a [001]-oriented barrier (Perel’s case). The horizontal plane describes the electron wave vector in the barrier; $\mathbf{K}$ is taken along the [001] axis and $\mathit{\xi }$ lies in the barrier plane, along [100]. The upper part of the figure $\left(E>0\right)$ corresponds to the real conduction band—the wave vectors are real quantities—and the parabolalike curves describing spin-split states along the [101] direction are drawn. An up-spin state (full line, open circle) with the wave vector ${\mathbf{q}}^{\prime }$ is degenerate with a down-spin state at the wave vector $\mathbf{q}$ (dotted line, dark circle) and also with up- and down-spin states at the wave vectors $-\mathbf{q}$ and $-{\mathbf{q}}^{\prime }$, respectively. This is useful for the calculation of a quantum well, given in Appendix , Sec. . Concerning the evanescent states, in a naive effective-mass picture, one may think of evanescent states being mirrors of these real states (in the $E<0$ domain) with imaginary wave vectors. Then up- and down-spin electrons at the energy $E$ would tunnel with the two different wave vectors $i{\mathbf{q}}^{\prime }$ and $i\mathbf{q}$, thus resulting in a spin-filter effect. However, our calculation shows that, concerning evanescent states (lower part of the figure, $E<0$), the situation is not so simple. In the negative-energy region, the $\mathbf{K}$ axis refers to the imaginary wave-vector component and $\mathit{\xi }$ refers to the real wave-vector component. Real-energy lines are found only when $\text{tan}\theta =\xi /K<1$. These real-energy lines, when drawn for a given $\theta$, consist of loops connecting nearly-opposite spin states at the zone center (“up” spin: full curve and “down” spin: dotted curve). Obviously, when going off the zone center, the spin no longer remains a good quantum number—in fact, it can be calculated that its average value rotates along the loop—but it has to be pointed out that, in the D’yakonov and Perel’ description, the energy eigenvectors are pure spin states which depend on the $\theta$ ratio. Two of these loops are drawn here. Let us consider a tunneling process at the energy $E$ (horizontal gray plane or yellow plane in the online edition) of an electron with the wave-vector component $\mathit{\xi }$ in the barrier plane, which has to be conserved in the tunneling process. It can be observed here that the two states marked on the loops by a dark circle $\left({\mathbf{K}}^{\prime }\right)$ and an open circle $\left(\mathbf{K}\right)$—which are energy degenerate—are associated to the same real wave-vector component $\mathit{\xi }$. However, they correspond to two different $\theta$ as they are, respectively, associated to the imaginary components $i\mathbf{K}$ and $i{\mathbf{K}}^{\prime }$, along the tunneling direction. The difference between $K$ and $K^{\prime }$ results in a spin-filter effect. Inset (upper left): top view of the plane at energy $E$ showing the intercepts with the loops which determine the relevant wave vectors $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$.