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Theory of impurity-induced infrared absorption in inversion layers
140
THEORY OF IMPURITY-INDUCED IN INVERSION LAYERS
INFRARED
ABSORPTION
I. Introduction Although the presence of Na + IONS in the oxide of a metal oxide semiconductor structure usually is disastrous for the performance of h4OS transistors, experiments in which such ions are purposely drifted toward\ the oxide-semiconductor interface have given much information on proper,tiea of the inversion layer. Most useful has been the experimental separation of the various scattering mechanisms for dc conduction by changing impurity COW tent, temperature, and substrate bias [I]. During the last 3 or 4 years exprriments have also been performed to investigate the influence of impurities on the conductivity at infrared frequencies. McCombe and Shafer [2] and McCombe and Cole [3] measured infrared absorption with the field of the infrared radiation perpendicular to the Interface and found an impurity shitted subChang and Koch [4] studied the band resonance. In parallel polarization, infrared conductivity and found an absorption peak on top of the Drude conductivity at very low densitiea for all the frequencies applied (3 t X me\‘i. No such peak was observed in perpendicular polarization. In this work we have calculated the infrared conductivity t>f the 4ectron.s III the inversion layer on (1OO)Si when positive iona are present near the interface First we show results for densities so low that the number of ctectrorlh Ib smaller than the number of impurities. Second we investtgate the oppohitc limit. In both cases we take the number of impurities to be so hmalt (hat multiple scattering can be neglected. 0039-6028/82,/0000-0000/$02.75
,‘:’ 1982 North-Holland
B.
vrnter / Inlpun~v-rnduced rnfrured uhsorptron
141
2. The low-density regime In this case we consider just one electron and one impurity and take the same Hamiltonian as Martin and Wallis [5] and Kramer and Wallis [6]. The wavefunctions are found [7] by expanding in the set of subband wavefunctions J,‘,,(z) obtained from the Hamiltonian without impurities: CC
wherem=O, ‘1, k2, . . . is the angular quantum number associated with the cylindrical symmetry. For energies E < E,, the bottom of the lowest subband, the states are discrete and completely bound to the impurity, whereas for E > E, the states are extended and the energy spectrum is continuous. We assume the electron to be in the ground-state bound to the impurity and study the excitation spectrum. The absorption of radiation is proportional to the oscillator strength&, for transitions from the ground-state to excited states, where f I/‘Id =21(fl~,,.Il~)12/m,,,~(E~
-a
(2)
the p-matrix element is between the initial state of energy E, and the final state of energy E,. The superscripts refer to the direction of the electric field of the radiation and m is the appropriate effective mass of Si conduction electrons.
13 _
2 g
05
% : % 0
0
1
I
z,=lO
0;
2
-hw [Ry’l
Fig. I. Oscillator strengths df/dE for transitions from the lowest bound state to excited Upper row, parallel polarization: lower row. perpendicular polarization. z0 is the distance impurity from the interface. Height of arrows below the continuum indicates strength 6 function. Inversion layer potential corresponds to a depletion layer density of 5X IO” u; =21.7 A. Ry*=42.6 meV.
states. of the of the cm ?
In fig. 1 we compare oscillator strengths for the two polarizations and for various impurity positions z. from the interface. The most remarkable feature in parallel polarization is the very strong absorption for the transition to the lowest-lying m = 1 bound state. 70-80% of the total oscillator strength is exhausted by this transition. There can hardly be doubt that this is what gives rise to the absorption peak observed by Chang and Koch [4]. The fact that the peak is seen at several frequencies is naturally understood as a distribution of impurity positions. For the perpendicular polarization the behavior is quite different. In general the oscillator strength is more evenly distributed for small 2,) and the transition energies are higher. Increasing 2,) does not give rise to absorption at lower frequencies; rather it reduces the oscillator strength at the low frequencies and concentrates it at higher energies corresponding to transitions to resonances in the continuum. Thus a distribution in z. will not help to give absorption at the frequencies used in [4]. Our results lend support to the interpretation of the measurements in refs. [2] and [3] as impurity-shifted subband resonances. Quantitatively. however, use of the single-electron model may be suspect at the relatively high impurity and electron densities used in those experiments.
02
I
I
,
I
‘\
il, \ I
I
I
\
!
\ I/
\
I’/, / t \, ‘\
1 N
-T
01
\
3
z
/x\ ,‘jE2 \\ \
Drude
-’ WT
\
i
0 0
I
1
I
I
2
hid/E, Fig. 2. Calculated frequency dependence of real part of dynamical conductivity U( w ) for parallel polarization. Inversion layer density N, = 3 X 10” cm ‘. depletion layer density 3.6X IO” cm ‘, acparation number of impurities JV>,,,~= IO” cmm2, E, = IX.85 meV. Arrow indicates intcrsubband E, -E,,. The dc scattering time 7= 1.85X IO ” s. ho(O)/e’ ~3.37
143
3. The high-density limit When the number of electrons is much greater than the number of impurities, screening and effects of exchange and correlation play very important roles both for the subbands and for the impurity scattering. We have earlier treated this problem self-consistently employing the density-functional method in which the Hartree potential is treated exactly and many-body effects are taken into account through a local exchange-correlation potential [7,8]. The results showed one very shallow bound state just below the bottom of the lowest subband and they gave a good description of the dc impurity-limited mobility. The real part of the dynamical conductivity is taken as a sum over transitions from occupied states below the Fermi energy E, to states above E, weighted by the oscillator strength of each transition. In fig. 2 we show results of such a calculation. Two features are worth mentioning. First, even though the impurity potential makes intersubband transitions possible, this does not lead to intersubband resonances in parallel polarization. Second, there is a pronounced threshold just above the Fermi energy originating from transitions from the bound state just below the bottom of the lowest subband. Its strength is rather remarkable and although it must be exaggerated in the present model, we feel that it should be experimentally observable. Certainly such an observation would give much support to the self-consistent theory. This work was supported 128.
by the Deutsche
Forschungsgemeinschaft
via SFB
References [I] A. Hartstein. A.B. Fowler and M. Albert. Surface Sci. 9X (19X0)I8 I. [2] B.D. McCombe and D.E. Schafer, in: Proc. 14th Intern. Conf. on the Physics of Semiconductors, Edinburgh, 197X, Ed. B.L.H. Wilson, Inst. Phys. Conf. Ser. 43 (Inst. Phys.. London. 1979) p. 1227. [3] B.D. McCombe and T. Cole. Surface Sci. 98 (1980)469. [4] H.R. Chang and F. Koch, unpublished and private communication. [5] B.G. Martin and R.F. Wallis, Phys. Rev. Bl8 (1978) 5644. [h] G.M. Kramer and R.F. Wallis, in: Proc. 14th Intern. Conf. on the Physics of Semiconductors. Edinburgh. 1978, Ed. B.L.H. Wilson. Inst. Phys. Conf. Ser. 43 (Inst. Phys.. London. 1979) p. 1243. [7] B. Vinter. Solid State Commun. 28 (1978) 861. [8] B. Vinter, Surface Sci. 98 (19X0) 197.
THEORY OF IMPURITY-INDUCED IN INVERSION LAYERS
INFRARED
ABSORPTION
I. Introduction Although the presence of Na + IONS in the oxide of a metal oxide semiconductor structure usually is disastrous for the performance of h4OS transistors, experiments in which such ions are purposely drifted toward\ the oxide-semiconductor interface have given much information on proper,tiea of the inversion layer. Most useful has been the experimental separation of the various scattering mechanisms for dc conduction by changing impurity COW tent, temperature, and substrate bias [I]. During the last 3 or 4 years exprriments have also been performed to investigate the influence of impurities on the conductivity at infrared frequencies. McCombe and Shafer [2] and McCombe and Cole [3] measured infrared absorption with the field of the infrared radiation perpendicular to the Interface and found an impurity shitted subChang and Koch [4] studied the band resonance. In parallel polarization, infrared conductivity and found an absorption peak on top of the Drude conductivity at very low densitiea for all the frequencies applied (3 t X me\‘i. No such peak was observed in perpendicular polarization. In this work we have calculated the infrared conductivity t>f the 4ectron.s III the inversion layer on (1OO)Si when positive iona are present near the interface First we show results for densities so low that the number of ctectrorlh Ib smaller than the number of impurities. Second we investtgate the oppohitc limit. In both cases we take the number of impurities to be so hmalt (hat multiple scattering can be neglected. 0039-6028/82,/0000-0000/$02.75
,‘:’ 1982 North-Holland
B.
vrnter / Inlpun~v-rnduced rnfrured uhsorptron
141
2. The low-density regime In this case we consider just one electron and one impurity and take the same Hamiltonian as Martin and Wallis [5] and Kramer and Wallis [6]. The wavefunctions are found [7] by expanding in the set of subband wavefunctions J,‘,,(z) obtained from the Hamiltonian without impurities: CC
wherem=O, ‘1, k2, . . . is the angular quantum number associated with the cylindrical symmetry. For energies E < E,, the bottom of the lowest subband, the states are discrete and completely bound to the impurity, whereas for E > E, the states are extended and the energy spectrum is continuous. We assume the electron to be in the ground-state bound to the impurity and study the excitation spectrum. The absorption of radiation is proportional to the oscillator strength&, for transitions from the ground-state to excited states, where f I/‘Id =21(fl~,,.Il~)12/m,,,~(E~
-a
(2)
the p-matrix element is between the initial state of energy E, and the final state of energy E,. The superscripts refer to the direction of the electric field of the radiation and m is the appropriate effective mass of Si conduction electrons.
13 _
2 g
05
% : % 0
0
1
I
z,=lO
0;
2
-hw [Ry’l
Fig. I. Oscillator strengths df/dE for transitions from the lowest bound state to excited Upper row, parallel polarization: lower row. perpendicular polarization. z0 is the distance impurity from the interface. Height of arrows below the continuum indicates strength 6 function. Inversion layer potential corresponds to a depletion layer density of 5X IO” u; =21.7 A. Ry*=42.6 meV.
states. of the of the cm ?
In fig. 1 we compare oscillator strengths for the two polarizations and for various impurity positions z. from the interface. The most remarkable feature in parallel polarization is the very strong absorption for the transition to the lowest-lying m = 1 bound state. 70-80% of the total oscillator strength is exhausted by this transition. There can hardly be doubt that this is what gives rise to the absorption peak observed by Chang and Koch [4]. The fact that the peak is seen at several frequencies is naturally understood as a distribution of impurity positions. For the perpendicular polarization the behavior is quite different. In general the oscillator strength is more evenly distributed for small 2,) and the transition energies are higher. Increasing 2,) does not give rise to absorption at lower frequencies; rather it reduces the oscillator strength at the low frequencies and concentrates it at higher energies corresponding to transitions to resonances in the continuum. Thus a distribution in z. will not help to give absorption at the frequencies used in [4]. Our results lend support to the interpretation of the measurements in refs. [2] and [3] as impurity-shifted subband resonances. Quantitatively. however, use of the single-electron model may be suspect at the relatively high impurity and electron densities used in those experiments.
02
I
I
,
I
‘\
il, \ I
I
I
\
!
\ I/
\
I’/, / t \, ‘\
1 N
-T
01
\
3
z
/x\ ,‘jE2 \\ \
Drude
-’ WT
\
i
0 0
I
1
I
I
2
hid/E, Fig. 2. Calculated frequency dependence of real part of dynamical conductivity U( w ) for parallel polarization. Inversion layer density N, = 3 X 10” cm ‘. depletion layer density 3.6X IO” cm ‘, acparation number of impurities JV>,,,~= IO” cmm2, E, = IX.85 meV. Arrow indicates intcrsubband E, -E,,. The dc scattering time 7= 1.85X IO ” s. ho(O)/e’ ~3.37
143
3. The high-density limit When the number of electrons is much greater than the number of impurities, screening and effects of exchange and correlation play very important roles both for the subbands and for the impurity scattering. We have earlier treated this problem self-consistently employing the density-functional method in which the Hartree potential is treated exactly and many-body effects are taken into account through a local exchange-correlation potential [7,8]. The results showed one very shallow bound state just below the bottom of the lowest subband and they gave a good description of the dc impurity-limited mobility. The real part of the dynamical conductivity is taken as a sum over transitions from occupied states below the Fermi energy E, to states above E, weighted by the oscillator strength of each transition. In fig. 2 we show results of such a calculation. Two features are worth mentioning. First, even though the impurity potential makes intersubband transitions possible, this does not lead to intersubband resonances in parallel polarization. Second, there is a pronounced threshold just above the Fermi energy originating from transitions from the bound state just below the bottom of the lowest subband. Its strength is rather remarkable and although it must be exaggerated in the present model, we feel that it should be experimentally observable. Certainly such an observation would give much support to the self-consistent theory. This work was supported 128.
by the Deutsche
Forschungsgemeinschaft
via SFB
References [I] A. Hartstein. A.B. Fowler and M. Albert. Surface Sci. 9X (19X0)I8 I. [2] B.D. McCombe and D.E. Schafer, in: Proc. 14th Intern. Conf. on the Physics of Semiconductors, Edinburgh, 197X, Ed. B.L.H. Wilson, Inst. Phys. Conf. Ser. 43 (Inst. Phys.. London. 1979) p. 1227. [3] B.D. McCombe and T. Cole. Surface Sci. 98 (1980)469. [4] H.R. Chang and F. Koch, unpublished and private communication. [5] B.G. Martin and R.F. Wallis, Phys. Rev. Bl8 (1978) 5644. [h] G.M. Kramer and R.F. Wallis, in: Proc. 14th Intern. Conf. on the Physics of Semiconductors. Edinburgh. 1978, Ed. B.L.H. Wilson. Inst. Phys. Conf. Ser. 43 (Inst. Phys.. London. 1979) p. 1243. [7] B. Vinter. Solid State Commun. 28 (1978) 861. [8] B. Vinter, Surface Sci. 98 (19X0) 197.