Electric tuning of direct-indirect optical transitions in silicon

Electronic band structures in semiconductors are uniquely determined by the constituent elements of the lattice. For example, bulk silicon has an indirect bandgap and it prohibits efficient light emission. Here we report the electrical tuning of the direct/indirect band optical transition in an ultrathin silicon-on-insulator (SOI) gated metal-oxide-semiconductor (MOS) light-emitting diode. A special Si/SiO2 interface formed by high-temperature annealing that shows stronger valley coupling enables us to observe phononless direct optical transition. Furthermore, by controlling the gate field, its strength can be electrically tuned to 16 times that of the indirect transition, which is nearly 800 times larger than the weak direct transition in bulk silicon. These results will therefore assist the development of both complementary MOS (CMOS)-compatible silicon photonics and the emerging “valleytronics” based on the control of the valley degree of freedom.

In this supplementary information, we describe device fabrication and the data analysis method. We also describe some valley related physics for a conventional Si/SiO2 interface based on effective mass (EM) theory, where the valley splitting is an order of magnitude smaller than the results at a BOX/Si interface. Nevertheless, the underlying physics related to valley splitting and optical transition seems to be fairly well explained by the theory. We start with device fabrication and data analysis and finally turn to valley related physics including a field dependent direct optical transition.

I. Device fabrication
The devices are fabricated on SIMOX (separation by implantation of oxygen) (001) wafer [S1] annealed at 1350 °C for 40 hours to minimize the influence of interfacial roughness at the Si and buried oxide (BOX) [S2-S4]. In a previous work it was reported that this treatment results in the formation of lattice steps and terraces at the silicon/buried-oxide (BOX) [S3, S4] interface and the interfacial flatness improves within the terraces while it deteriorates over a long range of about 1 μm due to the steps, and the average period of the steps aligned with <110> was several 100 nm. When fabricating our device, after this thermal treatment, we thinned the silicon-on-insulator (SOI) layer by thermal oxidation and etching with dilute hydrofluoric acid solution. This was followed by dry gate oxidation at 700˚C to form the front SiO2/Si interface and etching to define the device geometry. We prepared samples with two SOI thicknesses (tSOI), nominally 3 and 6 nm (measured by ellipsometry). The front poly-Si gates were then formed to define a channel with a width (W) of 200 m and a length (L) of 400 m. The nominal thicknesses of the front-gate oxide (tOX) and the buried oxide (tBOX) were about 20 and 400 nm, respectively. After forming the poly-Si front gate, we formed the p + and n + contacts by the ion implantation of boron and phosphorous, respectively, where the SOI thickness of these contact regions was 50 nm. To activate implanted dopants, thermal treatment was performed at 1000 ˚C for 20 minutes in a hydrogen atmosphere. This thermal treatment sometime causes the diffusion of dopants from a poly-gate into the SOI layer for a thin gate oxide [S5] but it is negligible in the present case thanks to use of a thicker front gate oxide.

II. Data analysis A. Valley related structure in Ipn
In the main text we plot the first derivative of Ipn with respect to VFG in Fig. 3a. The second derivative is shown in Fig. S1. The figure shows a similar structure to the second derivative of IDn (Fig. 2b) and the line structure is attributed to valley splitting because of its splitting size. In the first derivative of Ipn, the threshold of the color plot is determined by the peak position of the second derivative. Thus, we identified that the line structure in Fig. 3a is caused by valley splitting.

B. EL peak assignments
The peaks were assigned by multi-peak Gaussian fitting. The EL spectrum for tSOI = 6 nm at zero gate bias is decomposed into four peaks (P1: 1.04, P2: 1.10, P3: 1.14, P4: 1.16 eV) where the most intense peak is P2 as shown in Fig. S2 (a). With undoped Si at a low temperature, the dominant radiative recombination process is commonly caused by a TO phonon mediated free-exciton transition with a typical peak energy of 1.097 eV [S6]. Therefore we attribute P2 to the TO peak. Although there is some variation in the energy difference ( between P2 (TO) and P4 ( Fig.  S2 (b)) probably due to error related to peak deconvolution, it is almost constant and close to the TO phonon energy, and so we conclude that P4 originates from a non-phonon (NP) peak. The other possibilities such as the emission related to the QW excited state can be excluded because shows significant VBG dependence in theoretical calculations. Similarly P1 and P3 can be assigned to TO + O  and a TA phonon mediated transition, respectively. Figure S1| Second derivative of Ipn. Second derivative of Ipn at the bias conditions for (Vn, Vp) = (-1.5, 2.5) V Figure S2| EL spectrum analysis. a, Deconvolution of the EL spectrum taken at VBG = 0. All NP and TO peaks are fitted by Gaussian distributions. b, Peak energy differences between NP and TO associated with VBG variation. The symbols indicate the average energy difference  between NP and TO. The bars cover the larger standard deviation between NP and TO at a given VBG.

C. Estimation of valley splitting energy from EL peaks
Because the emission peak energy should reflect the valley splitting energy as well as the energy shift due to the quantum confined Stark effect (QCSE) [S7], we can estimate the valley splitting energy from EL data, as shown in Fig. 4b (main text). First, we carried out an analysis of the Stark shift for a negative VBG where electrons were squeezed to a thermal Si/SiO2 interface thus the valley splitting was negligible. Figure S3 shows the experimental Stark shift compared with the calculated result based on the standard single-valley EM theory (the parameters used are shown in the following section). Good agreement between the calculation and experimental results suggests strongly that the energy shift for a negative VBG is well explained by QCSE. The lever factor  which defines the relation between V BG and F SOI as FSOI =  VBG, is extracted from this analysis. The obtained  values are respectively 5.5 x10 5 m -1 for tSOI =6 nm and 6.6 x10 5 m -1 for tSOI = 4.3 nm. The fields are consistent with those estimated from the device geometries (tFOX = 20 nm, tBOX = 400 nm). From these results, the contribution of the valley splitting to peak shifts is extracted by comparing the energy shifts for positive and negative VBG and subtracting one from the other. We thus estimated the valley splitting energies and plotted them in Fig. 4b (main text).

D. Relation between valley splitting and direct optical transition
The theory of valley splitting based on effective-mass (EM) approximation [S8-S10] was developed by Ohkawa et al., where the splitting energy is given by Here  is the energy difference between the 15 and 1 u bands, k0 (|k0|= 0.85x2.0/a0) is the wavenumber for the conduction band minima, and  el (z) is an electron envelope function in a quantum confinement potential. The qualitative interpretation of the equation (S1) is as follows: (i) Due to strong electron confinement, the envelope function tends to have large high frequency components. (ii) As a result, the electron wave numbers distribute around -k0 and k0 and have a significantly large component at -point. (iii) Because these two valley states are both expanded by the 15 and 1 u basis thus mixed at -point with an energy gap of   (see the band diagram in Fig. 1), Figure S3| Stark shift analysis. The symbols indicate experimental TO peaks at VBG<0. The sold lines are Stark shifts calculated by EMA. A lever factor is obtained by comparing the quadratic fitting parameter for experimental data with that for EMA results.
the valley splitting energy is given by   and the 2k0 component of the Fourier coefficient of the squared electron envelope function. The envelope function in eq. (S1) is obtained by solving a single-valley effective-mass equation.
where m * is electron effective mass (0.98m0), and U0(z) is a potential barrier with a value of 3.10 eV for |z| > tSOI/2 and otherwise 0, q is the electron charge, (z) is the dielectric constant, and FSOI is the field at the SOI layer. By combining eq. (S1) with eq. (S2) we can easily estimate the valley splitting in bulk Si-MOSFETs. The constants for the following calculations are listed in Table S1. The simplest case is an infinite square well where the analytic solution of valley splitting is known [S11]. The valley splitting at a zero electric field as a function of tSOI is shown in Fig. S4. For the numerical calculation, an infinite barrier is simulated by setting the U0(z) at a high value (~30 eV). In Fig. S4, the magnitude of the valley splitting oscillates as a function of well width (tSOI) because of interference between |k0| = 0.85x2/a0 and the minimum k step (= 2/tSOI) in the QW. Essentially, those values are inversely proportional to tSOI due to an increase in the higher frequency Fourier components of the envelope function. The field dependence of valley splitting for a finite-barrier QW with various well widths is shown in Fig. S5. For a small FSOI, an analytic solution of the valley splitting gives values ranging from 0 to   / tSOI 3 k0 3 due to the interference. Even at FSOI = 0, some cases have finite valley splitting 0 (zero-field valley splitting) due to the interference. With increasing fields the splitting approaches a universal line and the thickness dependence disappears. This is because the electrons are likely to be confined in the triangular potential at the interface for higher field. Although there have been several reports on valley splitting using other sophisticated calculations [S12-S14], the results are basically similar to those given by eq. (S1). Therefore, we use eq. (S1) in the present work. For example, the coefficient describing the relationship between valley splitting and fields defined as = 2/FSOI (eV/(V/m)) is 2.6 x10 -11 in the present calculation while it is 2.74 (±0.2) x 10 -11 in Ref. S12. The slight deviation may be attributed to the different shape of the envelope function near the Si/SiO2 interface, which strongly depends on the parameters used for the numerical simulation. The use of the simple EM theory is attractive because we can easily calculate the relative EL intensities. The EL intensity of a direct optical transition (NP) is determined by Fermi's golden rule. The intensity of an inter-band transition can be described as where the term (Eel-hh-E ħ photon) assures energy conservation, uhh(r) is the periodic part of the valence Bloch function, the uel(r) is that of the conduction Bloch function at the -point Cardona [S15] 15 x 15 band kp perturbation (We omit the details of the calculation, it is noted in Ref. S15.). According to the envelope function approximation [S16, S17], the electron wavefunction can be approximated by the product of a periodic part of the Bloch function (uel,k uhh,k) and the envelope function. In eq. (S3), the modules of each component of the Bloch function are determined by the Fourier coefficient of the envelope function at given momenta. So the NP rate is essentially proportional to the dispersion of the envelope function. Therefore, we set the perturbation term for the periodic part of the Bloch function as a constant (MOP) in the main text. The hole envelope function hh(z) is obtained by solving eq. (S2) with m* = 0.28 m0 and U0 (z) = 4.8 eV at a given field.
The second term of eq. (S3) extracts the k0 component of the Fourier coefficient for the product of heavy-hole and electron envelope functions k0, which increases approximately linearly with the field due to real-space confinement. On the other hand a TO phonon mediated transition is proportional to , which corresponds to the DC components of the Fourier coefficient for the product of the hole and electron envelope function 0. This transition decreases monotonically with the field and is commonly inversely proportional to tSOI and FSOI. Thus the NP/TO ratio is approximated by the following In Fig. S6 (b), INP/ITO compared with a bulk-like condition (NP/TO at FSOI =0 and tSOI =6 nm), which corresponds to the NP rate, is plotted as a function of FSOI, where the perturbation term MOP is constant for both thicknesses. As a rough approximation, we introduce a constant offset to allow us to ignore the zero-field valley splitting caused by the QW confinement, which is not observed for our case of large valley splitting, and thereby highlight the field dependence. . According to the calculation, INP/ITO reaches 5,000 compared with that for FSOI = 0 V/m and tSOI = 6 nm (nearly bulk) conditions in the SiO2 breakdown field (FSOI ~ 300 MV/m) and it becomes large for a thinner QW. These results imply that for a high electric field the significant NP transition induced by strong