Valley splitting by extended zone effective mass approximation incorporating strain in silicon

Valley splitting at the interface of Si/thermally grown SiO₂ has been studied from the early stage of silicon-based electronics and is still gaining a great deal of attention because it can be a fatal source of decoherence for silicon-based spin qubits.^1,2 The decoherence problem is almost unavoidable because the magnitude of the typical splitting in Si/SiO₂ or Si/SiGe systems is several tens of microelectron-volts and differs from device to device. Therefore, controlling valley splitting in this system is urgent. Research on valley splitting has a long history. In the integer quantum Hall effect in silicon, a splitting caused by the valley degrees of freedom was observed.^3–5 At this time, basic theories on valley splitting based on effective-mass approximation (EMA) were developed, and valley splitting at the Si/thermally grown SiO₂ interface was well explained by these theories.^6–12 

In the 1990s, papers reporting valley splitting several tens to thousands of times larger than that at the Si/thermally grown SiO₂ interface^13,14 attracted much attention, where MOSFETs were fabricated on SIMOX (001) substrates annealed at high temperatures for a long time. We have reported on this large valley splitting¹⁵ and studied its principle and applications.¹⁶ Since large valley splitting has the potential to solve the serious issue of valley degeneracy in spin qubits, several other researchers have focused on its importance and have been investigating its origin. Although the principle is different from ours, a technique to achieve large valley splitting by applying a strong electric field using a high-k dielectric has recently been demonstrated¹⁷ to overcome this problem. Large valley splitting was verified based on various theories such as tight binding models,^18,19 EMA incorporating an additional perturbation term fitted by a tight binding theory so as to avoid the problems in describing microscopic phenomena,²⁰ and EMA combined with first-principle calculations.²¹ There were also theories incorporating interface states,^22,23 abrupt interfaces,²⁴ and superlattices.^25,26 However, none of these theories have provided a clear answer to the origin of large valley splitting.

In this paper, we analyze experimental results on large valley splitting based on the theoretical framework of the extended zone EMA reported in Ref. 6. This paper incorporates the effects of strain that have not been addressed in many previous papers demonstrating that the introduction of shear strain significantly enhances valley splitting caused at the X point. Based on this theory, it is predicted that a compressive strain of approximately 5% along [110] is introduced near the buried oxide (BOX) and silicon-on-insulator (SOI) interface of the SIMOX substrate. In addition, it was confirmed that large valley splitting could be reproduced by the EMA theory incorporating the strain even in multiple devices with different film thicknesses. These results are very important findings because they may provide a solution to valley degeneracy and variation of valley splitting, which are critical issues in actualizing silicon spin-qubit quantum computers.

Here, we briefly explain the basic principle of valley splitting. Bulk silicon has six equivalent conduction-band (CB) minima on the three orthogonal ⟨100⟩ axes as shown in Figs. 1(a) and 1(b). If we introduce [001] spatial confinement to form a so-called quantum well (QW), the electrons split into two groups of valley states. The valleys with lighter EM along [001] become fourfold degenerate, and the remaining valleys with heavier EM are doubly degenerate. This doubly degenerate state is the ground state of the QW. By applying an external field or introducing a physical structure to establish even stronger confinement along [001], the doubly degenerate state exhibits further splitting. This energy splitting is referred to as valley splitting. Valley splitting is fundamentally attributed to the allowance of valley degeneracy due to the difference in wavenumber k. When a very strong confinement is introduced, the wave functions belonging to states −k₀ and +k₀ superimpose a wide range of wavenumber states in the momentum space. The probability densities of electrons in each valley eventually overlap in the momentum space as shown in Fig. 1(b). When an overlap of valley states occurs, the respective states interfere, hybridize, and exhibit energy splitting.

Figure 1(c) shows the device structure. This device uses a SIMOX (001) substrate²⁷ as the starting material. First, the SIMOX substrate is annealed at 1350 °C for 40 h in an atmosphere of 0.5% O₂ in Ar to improve the flatness of the BOX/SOI interface.^28,29 The surface oxide film formed by 40 h annealing is then removed using hydrofluoric acid, and after adjusting the thickness of the SOI (t_SOI), the front gate oxide (FOX) film is formed by thermal oxidation at 900 °C. Next, by depositing gates on both sides of the FOX and the substrate, we fabricate double-gate structure MOSFETs. Using the front gate (FG) and back gate (BG), we adjust the spatial distribution of the electron-wave function inside a thin SOI, i.e., a QW, and observe the valley splitting at the FOX or BOX interface. The thicknesses of the FOX (t_FOX) and BOX (t_BOX) are 20 and 400 nm, respectively, and that of the QWs (t_SOI) are 6 or 9 nm. The gate length (L_g) and width (W_g) are 400 and 200 μm, respectively. Note that the BOX and FOX formation methods are remarkably different because the formation of the BOX is performed around unique temperatures exceeding the glass transition of SiO₂ or temperatures near the melting point of silicon.

Figure 2(a) shows typical drain current I_D vs FG voltage V_FG characteristics of the fabricated MOSFETs (t_SOI = 9 nm). All of the following measurements were conducted at 4.5 K. For BG voltage (V_BG) > 0 and V_FG < 0, only the channel at the BOX interface is open, where a distinct kink structure is observed (indicated by arrows). This structure reflects the two-dimensional density of states (2D-DOS) in the valley-coupled condition. This structure is related to valley splitting that we focus on in this paper. A negative differential structure appears at V_FG ∼ 0 V, indicating a channel opening at the FOX interface. In contrast, for V_BG < 0, only the FOX interface channel is open and no particular structure appears. Figure 2(b) shows a contour plot of ∂²I_D/∂V_FG². For V_BG < 0, a single-line structure due to the threshold of the FOX interface channel is observed, while for V_BG > 0 a distinct splitting structure appears. This structure originates from the I_D−V_FG kink described above, which corresponds to valley splitting. The lower line of ∂²I_D/∂V_FG² represents the onset of the electron occupation of the 2D state originating from the bonding orbital of the valley-coupled state, and the upper line represents its antibonding orbital.

Figure 2(c) shows the V_BG dependence of the valley splitting for each t_SOI thickness. The sizes of valley splitting are obtained by 2D-DOS of silicon and electron densities, which are given by the capacitance of FG (C_FG= ε_SiO2 ε₀/(t_FOX+ t_SOIε_SiO2/ε_Si)) and the V_FG difference (ΔV_FG) in ∂²I_D/∂V_FG², where ε_SiO2, ε_Si, and ε₀ represent the relative permittivity of SiO₂, silicon, and the permittivity in a vacuum, respectively. This figure also shows the electric field of the SOI (F_SOI) corresponding to each V_BG. This V_BG to F_SOI conversion is obtained by comparing the V_BG dependence of the transverse optical phonon peak given by electroluminescence with the numerical simulation of the quantum confined Stark shift,¹⁶ where electroluminescence is observed using the pn contacts of the same device. In the device with t_SOI = 6 nm shown in Fig. 2(c), valley splitting is indeed several hundred times larger than that expected at the SOI/thermal-SiO₂ interface. For V_BG < 0, electrons are squeezed at the SOI/thermally grown SiO₂ interface (FOX), resulting in an energy splitting of a few hundred microelectron volts the value of which is so small that we were unable to observe a distinct structure. On the other hand, when V_BG > 0, electrons are squeezed at the BOX/SOI interface, and in this state, a huge valley splitting occurs that cannot be explained by conventional theory. Figure 2(c) shows that the difference in valley splitting due to the difference in t_SOI is small. This is because, even theoretically, the valley splitting dependence on thickness appears only near a zero electric field, and the thicker the film, the lower the film-thickness dependence. Therefore, the difference in film thickness is almost negligible on the scale we are measuring.

Comparing ΔE_Γ and ΔE_X in Eqs. (2) and (4), respectively, both equations contain overlap integral S between the envelope functions corresponding to −k₀ and +k₀. Since the real-space wave function is confined in the QW, the envelope function ξ(k) is dispersed in the wavenumber-space based on the confinement strength around −k₀ and +k₀ as shown in Fig. 1(b). The magnitude of this overlap integral is inversely proportional to the distance in the wavenumber-space (Δk) of these valleys. The distance at the Γ point is 2k₀ = (2π/a₀) × 17/10, while at the X point it is |2k₀–4π/a₀| = (2π/a₀) × 3/10, which is approximately six times closer to the X point. Consequently, S_X is approximately two orders of magnitude larger than S_Γ, and the valley splitting via the X point becomes significant when the strain along ⟨110⟩ (e_xy) is nonzero.

Since the valley splitting at the X point is obtained from the overlap integral S_x via the X point and shear strain e_xy, it is calculated using ξ(z) obtained by solving Eq. (8) in the same way as the Γ point.

Figures 5(a) and 5(b) show the calculated electric field dependence of ΔE_Γ and ΔE_X at each t_SOI, respectively. In ΔE_X, shear strain e_xy is set to 5%. Both graphs show that valley splitting increases as t_SOI decreases or F_SOI is increased. This is because under both conditions, the 2k₀ component of the overlap integral on k-space takes a large value due to the stronger confinement of the electrons in the QW. When focusing on t_SOI dependence near the zero electric field, the large t_SOI exhibits linearity in terms of F_SOI dependence, while the extremely small t_SOI exhibits nonlinearity. This is because when t_SOI is small, the barrier potential becomes the main cause of electron confinement, but as t_SOI increases, the triangular potential caused by the electric field becomes the primary factor for electron confinement. Therefore, when t_SOI is small, nonlinearity is confirmed because the effective well width changes as it transitions from well confinement to electric-field confinement. Additionally, comparing ΔE_Γ and ΔE_X in Figs. 5(a) and 5(b), ΔE_X is much larger than ΔE_Γ. This is because the shear strain is set to a large value and the Δk of each valley at the Γ and X points described above, where Δk is much closer to the X point.

Figure 5(c) shows the F_SOI dependence on each shear strain of the overall valley splitting, which is the sum of the valley splitting due to the Γ and X points expressed in Eq. (1). For a near zero electric field, valley splitting is strongly affected by the valley splitting through the X point and it exhibits a nonlinear change with respect to the electric field, but basically the size of the valley splitting is simply proportional to the shear strain.

Figure 6 shows valley splitting overlaid on a contour plot with shear strain chosen to fit the experiment. Although nonlinearities are noticeable in the theoretical value near the zero electric field, the experimental and theoretical values for 6 and 9 nm are in relatively good agreement. If a similar fitting is performed with valley splitting through only the Γ point, the electric field exceeds 1.15 GV/m, which is much larger than for the breakdown field of SiO_2.³² This would not be consistent with the experiment. We suppose that nonlinearity is caused by the transition of the wave function from the unstrained region to the strained region. We assume that a strained region exists at the BOX interface. For small positive values of V_BG from 0 V, the wave function transitions from the unstrained region to the strained BOX interface. Therefore, nonlinearity appears due to transition from the Γ point to X point induced valley splitting. Figure 6 shows that introducing a strain along [110] of 4.5% at t_SOI = 6 nm and 5% at t_SOI= 9 nm as the shear strain agrees well with the experiment results, indicating that shear strain along [110] near the BOX interface is promising as the origin of large valley splitting. This strain value is very large from a general strain viewpoint, but this would not cause silicon to fracture.^33,34

Here, we discuss the validity of considering strain as the origin of the large valley splitting. The results obtained by Niida et al.,³⁵ who measured the mobility of MOSFETs with both polarity (p/n) contacts fabricated on SIMOX substrates subjected to high-temperature annealing similar to the conditions presented herein, revealed the following characteristics. They found that the electron mobility decreased and conversely the hole mobility increased under the condition where electrons and holes were squeezed at the SOI/BOX interface compared with the condition where both carriers were squeezed at the SOI/thermally grown SiO₂ interface (FOX). According to Refs. 36 and 37, the application of compressive strain along [110] on silicon (001) results in a decrease in electron mobility and an increase in hole mobility in [110] channel MOSFETs. This behavior aligns with the electron and hole mobility observed by Niida et al. at the SOI/BOX interface. Since the mobility can vary due to various factors such as interface quality and impurity concentration, it is difficult to compare mobility by itself. However, the agreement between the Ohkawa theory and experiments, and the consistency of the changes in both electron and hole mobility when [110] directional compressive strain is applied suggest that the [110] directional compressive strain near the BOX interface is a promising candidate as the origin of large valley splitting. The origin of the massive strain near the BOX interface remains unclear, and further detailed investigation is needed to determine whether strain actually exists and, if so, its physical origin. Finally, we note that high reproducibility in the size of large valley splitting has been obtained among the multiple devices measured so far even among different wafers.

We demonstrated that large valley splitting observed in SIMOX substrates subjected to high-temperature long-duration annealing can be explained by the extended zone EMA proposed by Ohkawa et al. In the theory, valley splitting via the X point may play an important role and have a significant impact on transport phenomena in the presence of shear strain in the inversion layer of MOSFETs. Further analysis based on this theory shows that the experimental large valley splitting is well reproduced when a shear compressive strain of approximately 4%–5% along [110] is introduced into the SOI layer at the BOX interface.

We thank Professor Y. Tokura (Tsukuba University) and Professor H. Kageshima (Shimane University) for their helpful discussions.

The authors have no conflicts to disclose.

Jinichiro Noborisaka: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Resources (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Toshiaki Hayashi: Conceptualization (equal); Formal analysis (supporting); Investigation (supporting); Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Akira Fujiwara: Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Katsuhiko Nishiguchi: Funding acquisition (lead); Project administration (lead); Resources (supporting); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.