Boundtobound intersubband transitions in a δdoped ptype Si/Si x Ge1−x /Siquantum wellS. K. Chun, D. S. Pan, and K. L. Wang Citation: Applied Physics Letters 62, 1119 (1993); doi: 10.1063/1.108761 View online: http://dx.doi.org/10.1063/1.108761 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/62/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Light emission lifetimes in p-type δ-doped GaAs/AlAs multiple quantum wells near the Mott transition J. Appl. Phys. 112, 043105 (2012); 10.1063/1.4745893 Bound-to-bound midinfrared intersubband absorption in carbon-doped GaAs ∕ AlGaAs quantum wells Appl. Phys. Lett. 87, 091116 (2005); 10.1063/1.2037859 Measurement of the excited-state position of bound-to-bound quantum-well infrared detectors J. Appl. Phys. 90, 2045 (2001); 10.1063/1.1388575 Intersubband absorption in Sb δdoped Si/Si1−x Ge x quantum well structures grown on Si (110) Appl. Phys. Lett. 60, 2264 (1992); 10.1063/1.107049 Intersubband absorption in Sb δdoped molecular beam epitaxy Si quantum well structures J. Vac. Sci. Technol. B 10, 992 (1992); 10.1116/1.586110

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Bound-to-bound inters&band transitions in a S-doped p-type Si/Si,Ge, -,/Si quantum well

S. K. Chun, D. S. Pan, and K. L. Wang Department of Electrical Engineering, University of California, Los Angeles, California 90024

(Received 6 October 1992; accepted for publication 8 December 1992)

The absorption spectrum in p-type Si/Sic6Gec.4/Si structure with a S-doped quantum well grown on Si(OO1) substrate is calculated using a multiband model. We have generalized the previous treatment of the depolarization effect in n-type Si to include subband multiplicity, nonparabolicity, and valence band anisotropy. An implicit formula for the effective plasma frequency was used to conveniently include these effects in the calculation. We found that it is necessary to treat the depolarization effect in the complicated couplings among valence bands in order to-explain the observed spectrum.

Intersubband transitions in quantum-well structures have attracted a great deal of interest for making infrared detectors”2 and modulators. Much progress has been made in the GaAs/Al,Ga,_fis system.3” Recently, the quantum-well-type structures of Si,Ge, -,/‘Si have shown a great potential for such applications. Park et al6 have re- ported experimental absorption spectra of p-type SiXGel +/Si quantum-well structures. In their paper, most of the features of the absorption spectra can be reasonably explained by the involved subband structures. The ob- served prominent peak at about 6 pm for the 45” angle incident light was explained as the transition from the heavy hole ground subband to its first excited subband. However, the unusual large width cannot be attributed to the bound-to-bound transitions, even with the consider- ation of nonparabolicity and depolarization effect along.

The purpose of this letter is to extend the treatise of intersubband transition to include three valence bands based on one-band theory. We show that the origin of the 6 pm peak is mainly due to multiple band effect of the valence bands and their depolarization effect. We have car- ried out an analysis of the intersubband optical transition using the k l p method, including three valence bands along with the depolarization effect. The result of the calculation including subband nonparabolicity and anisotropy shows two peaks around 9 and 7 ,um, the former due to HHl to HH2 and the latter due to HHl to LH2. With the subband multiplicity, nonparabolicity, valence band anisotropy, and impurity broadening included in the calculation, the depo- larization effect due to high carrier densities makes the two peaks merge into a broad one with the peak location at about 6 pm. The calculation result agrees with the exper- imental data.

In calculation, we use the quantum-well structure of p-type Si/Si0.6Gee&Si grown on a Si(OO1) buffer layer as used in the experiment by Park et ai. with 40 A quantum well, in which the center 30 A of the well is the S-doped region of 5 X lOI9 cmm3. The doping is assumed to be uni- formly distributed in the doped regions. The subbands of the structure is shown in Fig. 1. The potential discontinu- ity between the relaxed Si and strained S&,:iGee, layers is obtained from the band offset calculation by Van de Walle and Martin7 and People’ as also shown in Fig 1. Here, the

change of band bending for each valence band due to the doping is assumed to be 20 meV and the quantum well is considered as a square type for the simplicity of the calcu- lation.’ From the experimentally obtained inverse mass band parameters (IMBP) and deformation potentials of bulk Si and Ge, ‘“’ ’ the IMBPs of SiGe alloy are obtained by Lawaetz’s method, l2 whereas the deformation potential of the SiGe alloy is linearly interpolated assuming it is strain independent. In the framework of single particle the- ory, the 3 X 3 k. p and strain Hamiltonian is formulated using the above band parameters.13 Two-dimensional sub- band dispersions are then obtained within the scheme of Wessel and Altarelli’s formulation. 14*15 No self-consistency of the single particle potential is strictly followed.

When the doping concentration is higher than 1 X lOI cme3, as in our case of the &doped quantum wells, the depolarization effect is expected to be dominant in shifting the resonance frequency. As shown in the n-type inversion layer, l6 the resonance frequency o, for a two-level system, where bands are isotropic and parabolic, is shifted to a higher frequency c+ and can be simply expressed as

+=o$+o$, (1)

where wp is the plasma frequency. When the multiplicity, nonparabolicity, and anisotropy of the valence subbands are included for accurate evaluation of depolarization ef- fect, the above simple formulation is invalid. To date to our knowledge, the inclusion of these effects has not been treated in the literature. We have generalized the formula- tion of the depolarization effect by Ando17 and Allen et al. l6 to include these effects of the valence bands. Treat- ing the effective perturbation as by Ando,17 we obtain a similar set of linear equations, but with more complicated coupling matrix elements. In doing this, we choose the simple particle Bloch states as the basis functions without considering the random distribution of the acceptors. The effect of the random distribution, sometimes referred as band tailing effect, is included as a damping lifetime 7. To facilitate the calculation in this case, we define an effective plasma frequency which can be derived from the effective coupling matrix equation in an implicit form as follows

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P- 40A --I

(.400)S0 . . . . . . . . . . . . . i ,.... Si S’0.6Ge0.4 Si

FIG. 1. Band diagram of the subbands in the strained S&,6Gq,a quantum well grown on Si(OO1) with a 40 8, well width, where the center 30 %, are doped with a doping concentration of 5 x 10” cm 3.

$n,Jk,, I= 2 z; n’k;

--o ,t,$ Whk ll ,n’,$ ) X

zndk;, 1 d&q 1 +o&,#,, ) --~~fjdr,,

znm(k~~ j w;r,(k; 1 +co;nt,O$, > --z~+j~/rntm’ (2)

where m and n are the subbands of the initial and final states, l o and n, are the dielectric constant and refractive index, respectively, S(n, kll , n’, ki ) is a function of the overlap of the involved wavefunctions,’ z,, is the dipole matrix element, and fi/rR,, represents a level broadening. To simplify the calculation, the square of S(n, kll , n’, ki ) is evaluated as the product of S( n, kll , n, kll )S(n’, ki , n’, ki ) since the evaluation of S(n, kll ,n’, ki ) is much more time consuming. We have compared these two calculations and they are very close. Thus for all subsequent calcula-

-; -5 0.6

z

G 0.6 .$ E 8

0 0.4 5 “4 5 2 0.2

3 4 5 6 7 6 9 10 Wavelength (pm )

FIG. 2. Absorption coefficient of the bound-to-bound intersubband tran- sitions for the y- and z-polarized fields at room temperature. The dotted curve is the absorption coefficient without the depolarization effect. The curve with symbol A is the experimental data (Ref. 6). The solid and dashed curves are the ones including the depolarization effect for the multi- and two-level system, respectively.

tions, we use the latter to evaluate S(n, kll , n’, kfi ). It is noted that the above plasma frequency for the n and m states at kll is atIected by other plasma frequencies due to the coupling among oscillators. Equation (2) can be easily solved by iteration, beginning the evaluation without cou- pling. For the damping factor smaller than 30 meV, the convergence is good only near the resonance frequency. However, for the damping factor larger than 30 meV, which is close to our case, the convergence is fast and the coupling among oscillators is easily taken into account.

The absorption can be represented as a collection of all the responses of the damped oscillators shifted with the corresponding plasma frequency due to the depolarization elfect. The absorption is written as,

%b) = c e2 n+iceon&j L 4 d2k,, (f??z--fn) I (htl (z*P),l &Z) I2

%moq )/~nrn ” I w&(kll 1 +w&Jkll > --~~-kjd~,~ 1 2

d”k,, (fm-fn) I (Qnl wPL~,,lJhiJ I2 ~nm Oq )/~nm

1 o:,(k,, 1 --02+jw/r,, 1 2’ (3)

where i is the unit polarization vector of the incident pho-

I

the coupling interaction due to the lower subband effec- tively decreases the damping constant. In other words, it decreases the width and increases the absorption coehi- cient. Likewise, the coupling interaction with the higher subbands effectively broadens the absorption peak.

ton, L is the quantum well width, and f is the Fermi distribution function. Here, we assume that only the ab- sorption due to the polarization field in the growth direc- tion experiences the depolarization effect.’ As can be shown in Eq. (2)) the plasma frequency for the same sub- band at near the resonance frequency is mainly real. The real value of the plasma frequency causes a shift of the resonance peak position similar to that of Eq. ( 1). The coupling interaction from the lower subbands (smaller res- onance energy) induces a large negative imaginary part, whereas the higher subband (larger resonance energy) produces a positive value. From Eq. (3 ) it can be seen that

1120 Appl. Phys. Lett., Vol. 62, No. 10, 8 March 1993

Figure 2 shows the absorption coefficient for the bound-to-bound intersubband transitions for the structure considered. In calculating this, the multiplane wave vector directions (29 directions between [ 1001 and [OlO] ) are used for including the valence band anisotropy. The Fermi en- ergy level is calculated to be 107 meV below the top of the valence band edge.

The dotted curve is the absorption coefficient without

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level broadenings (assuming infinite scattering time) and the depolarization effect. In this case, two large peaks are clearly shown. The peak at 9.3 pm is mainly due to the transitions involving the same Bloch state (HHl -+HH2). And the other peak near at 6.8 pm is the transition from the ground heavy to the excited light hole subbands (HHl + LH2). This peak is forbidden at kll = 0. However, kll #0 induces band mixing and makes the transition pos- sible. The peak intensity at 9.3 ,um (HHl + HH2) is higher than that at 6.8 pm (HHl +LH2). However, as shown, the experimental data do not show the two peaks; the ex- planation follows.

Including the depolarization effect, the broadened ab- sorption spectra are obtained for the isolated two- and cou- pled multilevel cases. Here, the absorption only involves the transitions from the ground state (HH 1) because most carriers occupy this ground state at room temperature. The broadening #i/r,, for each subband is assumed to be kll independent. Here, we adopt the broadening of 50 meV. For the isolated two-level case as shown in dashed curve, the assigned two peaks are shifted mainly due to the depo- larization effect and they are also broadened, resulting in an apparent single broad peak at near the 8 p,rn (FWHM-100 meV> shifted from the heavy-to-heavy transition. However, the peak position is too far from the experimental peak (67 meV). In treating the coupled mul- tilevel cases (solid curve), we use five confined subbands. As mentioned earlier, the coupling of other subbands sup- presses the oscillator strength of the HHl -+HH2 transition (the lower resonance energy), but enhances the HHl +LH2 (the higher resonance energy j . At the same time, the FWHM is broadened somewhat to 95 meV. The slight difference of the peak position ( 17 meV) may be due to the lack of self-consistency. The self-consistent calcula- tion by Park et aL6 shows that the subband separation is 13-17 meV larger than ours. The uncertainty of the doping concentration and quantum-well width may be another

reason for the small discrepancy. The higher peak value in theoretical result may also be due to the same above rea- sons. On the other hand, the use of a larger broadening may reduce this discrepancy.

In summary, the absorption spectra for the intersub- band transition in the p-type S-doped quantum well are calculated with and without inclusion of the depolarization effects. The location and the width of the experimental peaks can only be verified by including the multiband de- polarization effect which takes the complicated valence band couplings into consideration. The detailed calculation of the absorption spectrum shows a good agreement with the experimental data.

One of the authors (S.K.C.) would like to thank Dr. R. P. G. Karunasiri for helpful discussions. This letter is in part supported by Office of Naval Research and Army Research Office.

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