Origin of giant valley splitting in silicon quantum wells induced by superlattice barriers

We investigate here the Si QW and GeSi SL interaction effects on the valley coupling between the two lowest Si valley states using a direct diagonalization of the band Hamiltonian $-{1\over 2}\nabla^{2}+V({\bf r})$ for the QW/SL hybrid structure described by its potential $V({\bf r})$. We use for the potential of the hybrid structure a superposition of overlapping, spherical screened pseudopotential $v_{\alpha}(r)$ of the constituent atom Wang and Zunger (1995); Wang et al. (1999),

where $\hat{v}_{\alpha}({\bf r}-{\bf R}_{n}-{\bf d}_{\alpha})$ is the screened pseudopotential containing spin-orbit interaction of atom type $\alpha$ at site ${\bf d}_{\alpha}$ in the $n$th primary cell ${\bf R}_{n}$. Considering the 4.2% lattice mismatch between Si and Ge, a large strain exists between atomic thin Si and Ge layers in GeSi SL. We use the atomistic valence force field (VFF) method Pryor et al. (1998); Williamson et al. (2000) to minimize the strain energy finding the atomic equilibrium positions. We diagonalize $-{1\over 2}\nabla^{2}+V({\bf r})$ within a plane-wave basis Wang and Zunger (1995) whose size is sufficiently selected such that the weak valley coupling is accurately considered. The supercell approach combined with the periodic boundary conditions is implemented. The screened pseudopotentials $\{\hat{v}_{\alpha}\}$ are fitted Wang and Zunger (1995) so as to remove the ”LDA error” in the bulk crystal; they reproduce well not only the band gaps throughout the zone, but also the electron and hole effective-mass tensors, as well as the valence band and conduction band offsets between well and barrier materials, spin-orbit splittings, and GW spin-splitting in bulk materialsLuo et al. (2009a). This is described in Ref. Wang and Zunger, 1995; Luo et al., 2009a. This approach has been previously applied to superlattices Magri and Zunger (2002); Wang et al. (1997), colloidal quantum dots An et al. (2006), and Stranski-Krastanow (SK) quantum dots Luo et al. (2009b).

The hybrid system consists of [001]-oriented Si QW and (Ge_mSi_m)_n SL barrier. These two components possess $D_{2d}$, $C_{2v}$ point symmetries. The whole system is of $C_{1}$ symmetry in the QW growth direction. Symmetry allows the couplings between the QW-VS and the SL-MBS because these states share the single representation of the point group (in terms of the double group, including spin-orbit coupling). These couplings may influence the VS of the Si QW subunit. To access this influence, we expand the Hilbert space of the lowest two Si valley states Ohkawa and Uemura (1977); Boykin et al. (2004); Nestoklon et al. (2006); Friesen et al. (2007); Valavanis et al. (2007); Saraiva et al. (2009) to include SL-MBS of the SL barrier and have the effective Hamiltonian for the hybrid system by introducing the coupling between QW-VS and SL-MBS states (off-diagonal elements) as follows:

where the diagonal element $\epsilon_{0}^{\textrm{QW}}$ is the average of the lowest two degenerate valley states in the Si QW. This two-fold degeneracy is lifted in the energy of $2\delta$ by the confinement potential. Therefore, the energy levels of the two Si valleys are $\epsilon_{1}^{\textrm{QW}}=\epsilon_{0}^{\textrm{QW}}-\delta$, and $\epsilon_{2}^{\textrm{QW}}=\epsilon_{0}^{\textrm{QW}}+\delta$, respectively. $\epsilon_{i}^{SL}$ is the energy level of $i$th state in (Ge_mSi_m)_n SL ($i=1,2,...,2n$). Note that each period has $\Delta_{z}$ and $\Delta_{-z}$ two valley states, and thus SL with n periods has $2n$ valley states. The remaining $\Delta_{x}$ and $\Delta_{y}$ valley states are neglected due to weak coupling between $\Delta_{z}$ and $\Delta_{x,y}$. $\lambda_{i}^{1,2}$ depict the interaction strengths between two Si valley states and the $i^{th}$ state of the (Ge_mSi_m)_n SL barrier. Figs. 3(a) (b) schematically illustrate these SL-MBS and QW-VS as well as their couplings.

The substantial space confinement in the short-period SL barrier yields energy levels of the SL-MBS much higher than that of the QW-VS, with an energy difference being far more extensive than $\delta$. We, therefore, can reduce the $(2+2n)$-dimension effective Hamiltonian Eq. (2) to a $2\times 2$ Hamiltonian for the lowest two Si valley states taking into account the effects of SL-MBS as first-order perturbation based on the quasi-degenerate perturbation theory using the Löwding partitioning method Winkler (2003). The reduced effective Hamiltonian is thus as follows:

where,

The reduced Hamiltonian can now be diagonalized directly, and the corresponding eigenvalues of two valley states read,

and the VS of the hybrid system is,

If we learn $\lambda_{i}^{1,2}$, $\epsilon_{0}^{\textrm{QW}}$, and $\epsilon_{i}^{\textrm{SL}}$, we can then reproduce $E_{\textrm{VS}}$ predicted from atomistic calculations according to Eq. (6), which will tell the factors’ contribution to the enhancement of VS due to coupling between QW-VS and SL-MBS ($\lambda_{i}^{1,2}$).

To understand the computationally calculated results as mentioned above, we have developed an effective Hamiltonian model as presented in the method section. However, it requires assessing $\lambda_{i}^{1,2}$, $\epsilon_{0}^{\textrm{QW}}$, and $\epsilon_{i}^{\textrm{SL}}$, which are difficult to obtain. In the following, we take further approximations. Firstly, we expect the coupling strengths between each SL state and two Si valley states to be approximately the same, i.e., $\lambda^{1}_{i}\sim\lambda^{2}_{i}$, regarding two Si valley states have a similar envelop wavefunction (see FIG. 4(a)) and $2\delta\ll\epsilon_{i}^{SL}-\epsilon_{0}^{QW}$. For the sake of simplicity, we make a rough assumption that all SL-MBS have the same coupling strength to the QW-VS, i.e., $\lambda^{1}_{i}\sim\lambda^{2}_{i}\sim\Lambda$. As a result, we can further simplify Eq. (5) and Eq. (6) as,

where

Where $\Lambda$ quantifies the averaged coupling strength between SL-MBS and QW-VS, and $T$ is an energy factor representing the energy distribution of the SL-MBS relative to QW-VS. Therefore, $\Lambda^{2}T$ quantifies the total QW/SL coupling strength.

It is straightforward to read from Eq. (7a) that, if $\Lambda^{2}T\leqslant\delta$, $E_{+}=\epsilon_{0}^{\textrm{QW}}+\delta-2\Lambda^{2}T$ and $E_{-}=\epsilon_{0}^{\textrm{QW}}-\delta$, otherwise $E_{+}=\epsilon_{0}^{\textrm{QW}}-\delta$ and $E_{-}=\epsilon_{0}^{\textrm{QW}}+\delta-2\Lambda^{2}T$. Interestingly, we can learn that the QW/SL coupling will not alter the energy level of the lower one of the two valley states in Si QW but push down the upper one. This model result explains exactly the above observation in the atomistic calculations (FIG. 2) that there is one energy level that stays almost constant with a small fluctuation as varying the number of periods $n$ as well as the period thickness $m$. It has also been schematically illustrated in Fig. 3(c). As increasing the QW/SL strength $\Lambda^{2}T$ from zero to $\delta$, the upper valley approaches toward the lower valley, resulting in the reduction in $E_{\textrm{VS}}$. At $\Lambda^{2}T=\delta$, the upper valley passes the lower valley, eliminating the VS ($E_{\textrm{VS}}=0$). Afterward, further increasing the QW/SL strength $\Lambda^{2}T$, the original upper valley (now becomes lower in energy) continuously goes down in energy, raising again the VS, but being still smaller than $2\delta$ the VS of the isolated Si QW. The enhancement in VS (i.e., $E_{\textrm{VS}}>2\delta$) occurs only when $\Lambda^{2}T>2\delta$. Consequently, the presence of the SL barrier is, surprisingly, suppressing the VS instead of enhancing it unless the QW/SL coupling strength is strong enough.This finding explains why the VS of hybrid systems for all $m\neq 4$ SL barriers is even smaller than that of the isolated Si QW, as shown in FIG. 1(c).

Fig. 3(d) sketches that we can divide the QW/SL coupling into two regions: strong coupling ($\Lambda^{2}T>2\delta$) and weak coupling ($\Lambda^{2}T\leqslant 2\delta$). From the results presented in FIG. 1(b)(c) we learn that the (Ge₄Si₄)_n SL barrier provides a strong QW/SL coupling and the remaining $m\neq 4$ (Ge_mSi_m)_n SL barriers are in the weak coupling region. In the following, we attempt to unravel the physics causing the $m=4$ SL barrier to stand out clearly from the remaining $m\neq 4$ SL barriers by examining the coupling matrix $\Lambda$ and energy factor $T$ separately.

Following the definition of the coupling matrix $\Lambda\sim\left\langle\psi^{\textrm{QW}}|\Delta V|\psi^{\textrm{SL}}\right\rangle$, we can thus assess its magnitude by examining the wave function overlaps around the interfaces, as shown in Fig. 4(a). Ref. Zhang et al., 2013 has shown that the SL barriers always make wave functions of the lowest two valley states strongly localized inside the Si QW layer with a much smaller leakage compared with the SiGe alloy barrier. However, there is no one-to-one relationship between VS and wave function leakage. The difference in (Ge_mSi_m)_n SL barriers introduces hardly a sizable change in wave functions of two valley states, as shown in Fig. 4(a). Here, we go stead to examine the penetration of the high-lying states of the SL barriers into the Si QW layer. We roughly approximate the coupling matrix $\Lambda$ as the evanescent integral of the (Ge_mSi_m)_n SL wave functions: $\Lambda\sim\int_{z_{0}}^{z_{1}}\left|\Psi_{i}^{\textrm{SL}}(z)\right|^{2}dz$ (the integral is running from $z_{0}$ inside the SL barrier to $z_{1}$ inside the Si QW, indicated by red dashed lines in FIG. 4(a)). The evanescent integrals for different (Ge_mSi_m)₅ SL barriers are given in Fig. 4(b). One can see that the magnitude of $\Lambda$ decreases as the SL period thickness $m$ increases. Note that, in SLs made by semiconductors, the electron states in neighboring QWs can interact as the barrier width decreases to a sufficient narrow thickness. The corresponding discrete energy levels confined inside QWs broaden into energy bands known as minibands, which are very narrow in energy compared to bulk energy bands Yu and Cardona (2010). In the Si/Ge heterostructures, it is well known that the electron states are localized inside the Si layer d’Avezac et al. (2012). Thus, thicker Ge sublayers of the (Ge_mSi_m)_n SL cause SL miniband states to be less expanded and creates narrower minibands. Furthermore, the thicker Si_m sublayer will also give the electron states more space confinement, making their wave functions more localized inside Si sublayers. Fig. 4 (a) indeed shows that the wave function in (Ge₆Si₆)_n SL is the most localized while the wave function in (Ge₂Si₂)_n SL is the most extended. Therefore, electron states of SL with larger $m$ have less probability of tunneling into the Si QW region. In this respect, a smaller coupling matrix $\Lambda$ is expected for (Ge_mSi_m)_n SL with larger $m$, as shown in Fig. 4(b) based on a rough evaluation from the evanescent integral.

We now turn to evaluate the dependence of the energy factor $T$ on the period thickness $m$ of the (Ge_mSi_m)_n SL barrier in hybrid systems. To do so, we have also computed the unperturbed energy levels $\epsilon_{0}^{\textrm{QW}}$ and $\epsilon_{i}^{\textrm{SL}}$ of isolated Si QW and five-period (Ge_mSi_m)₅ SL, respectively, by carrying out atomistic SEPM calculations. Both isolated Si QW and five-period (Ge_mSi_m)₅ SL are separately embedded within the pure Ge matrix. FIG. 4(c) shows the atomistic calculations predicted results. One can see that, as the period thickness is getting thinner (or $m$ decreases), the SL energy levels $\epsilon_{i}^{\textrm{SL}}$ raise and are going far away from the Si QW valley level $\epsilon_{0}^{\textrm{QW}}$. This raising in the SL energy levels is attributed to the enhanced quantum confinement effect. The energy spacing between SL energy levels $\epsilon_{i}^{\textrm{SL}}$ is also getting larger, responsible for a wider miniband width. We sum up the SL energy levels relative to $\epsilon_{0}^{\textrm{QW}}$ according to Eq. (8), giving rise to the energy factor $T$, which is shown in FIG. 4(d) as a function of SL period thickness $m$. One can see that $T$ gets larger almost linearly as increasing the SL period thickness $m$. It is straightforward to learn that this linear increase in $T$ is due to going down in energy of all $\epsilon_{i}^{\textrm{SL}}$ relative to $\epsilon_{0}^{\textrm{QW}}$.

Furthermore, if the number of SL periods $n$ is infinite, the discrete energy levels in the SL form minibands. The summation in $T$ term can then be approximated by the integral over a fixed energy interval from miniband bottom ($\epsilon_{1}^{\textrm{SL}}$) to miniband top ($\epsilon_{2n}^{\textrm{SL}}$): $T\equiv\sum_{i}\frac{1}{\epsilon_{i}^{\textrm{SL}}-\epsilon_{0}^{\textrm{QW}}}\propto\int_{\epsilon_{1}^{\textrm{SL}}}^{\epsilon_{2n}^{\textrm{SL}}}\frac{1}{\epsilon_{i}^{\textrm{SL}}-\epsilon_{0}^{\textrm{QW}}}d\epsilon=\ln(\frac{\epsilon_{2n}^{\textrm{SL}}-\epsilon_{0}^{\textrm{QW}}}{\epsilon_{1}^{\textrm{SL}}-\epsilon_{0}^{\textrm{QW}}})$. This approximation tells that, in addition to the absolute energy level positions, the miniband width is also an important parameter. Wider miniband width $\epsilon_{2n}^{\textrm{SL}}-\epsilon_{1}^{\textrm{SL}}$ occurred in smaller period thickness will give larger $T$ and thus larger VS, which compensates partially the reduction in T and thus VS induced by energy level shifting up due to strong confinement in narrow period thickness.

So far, we have shown that by decreasing the periodic thickness of the SL barriers but keeping the number of periods constant, the coupling matrix $\Lambda^{2}$ is going to become bigger, and the energy factor $T$ will get smaller for hybrid systems. FIG. 4(e) shows that the opposite trends in $\Lambda^{2}$ and $T$ render their product the total coupling strength $\Lambda^{2}T$ to be a non-monotonic function as varying the periodic thickness $m$ with a sharp peak occurring at $m=4$. This result is in excellent agreement with the atomistic calculation predicted $E_{\textrm{VS}}$ for the hybrid systems, as shown in FIG. 1(a). Therefore, we have illustrated that the competition between the coupling matrix and energy factor makes $m=4$ SL barrier defeating the remaining $m\neq 4$ SL barriers to possess alone the strong coupling. In contrast, the hybrid systems associated with the remaining $m\neq 4$ SL barriers are in the weak coupling region, with $E_{\textrm{VS}}$ being smaller than the VS of the isolated Si QW, which is 0.96 meV for an isolated 40-ML Si QW embedded in the pure Ge matrix.

We can now explain why only the (Ge₄Si₄)_n SL barrier will continuously enhance the VS by increasing the number of SL periods $n$; whereas the remaining $m\neq 4$ SL barriers lack such enhancement, as shown in FIG. 1(c). Because the number of SL periods quantifies the total number of quantum states in the miniband Gerald (1990), the number of the valley states composing the SL miniband grows doubly as increasing the number of SL periods $n$, which has been explained in the text above Eq. (2). According to Eq. (8), a more significant number of SL periods $n$ gives a more considerable energy factor $T$. On the other hand, increasing the SL periods is expected to reduce the component of wavefunctions penetration into the Si QW, which will suppress the coupling matrix $\Lambda$. FIG. 5 shows the spatial distributions of the two lowest SL-MBS in (Ge_mSi_m)_n SL with varying $m$ and $n$. One can observe a unique feature of the $m=4$ SL: the lowest energy ground state is $p$-like as it has one node in its envelope function. Whereas, remaining $m\neq 4$ SLs have an $s$-like ground state as usual. FIG. 5 exhibits that the p-like ground state in the $m=4$ SL guarantees more distributions of wavefunctions at the two terminated interfaces. That, in turn, ensures the component penetration into the Si QW, equally the coupling matrix $\Lambda$, is unchanged as increasing $n$. In contrast, the $s$-like ground state in the remaining $m\neq 4$ SLs is distributed mainly inside the SL with tails on the two terminated interfaces. In these $m\neq 4$ SLs, increasing the number of SL periods will increase the component inside the SL and thus reduce the component penetration into the Si QW. Subsequently, as the SL period increases, the coupling matrix $\Lambda$ decreases for $m\neq 4$ SLs. The rise of $T$ and decrease of $\Lambda$ may make the $E_{\textrm{VS}}$ a product of $\Lambda$ and $T$ unchanged in $m\neq 4$ SLs, as shown in FIG. 1(c). However, the increased $T$ and unchanged $\Lambda$ make the $E_{\textrm{VS}}$ enhanced continuously as increasing $n$ in $m=4$ SL.