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Remarks on the theory of electron capture into shallow donors in GaAs
Solid State Communications, Vol. 18, pp. 991—993, 1976.
Pergamon Press.
Printed in Great Britain
REMARKS ON THE THEORY OF ELECTRON CAPTURE INTO SHALLOW DONORS IN GaAs J. GoIka’~ International Centre for Theoretical Physics, Trieste, Italy and J. Mostowski Institute of Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland (Received 7 November 1975 by M. Cardona) A new model for low-temperature recombination of conduction electrons to ionized donors in GaAs is proposed. It assumes that the conduction electron is captured into the ground state of shallow impurity by simultaneous emission of two LA phonons. From a comparison of computed capture rate with recent experimental results the estimated value of the two-phonon deformation potential is d2LA = 0.02 eV. 1. INTRODUCTION THE THEORY of free-carrier recombination with hydrogenic ionized impurities in semiconductors was first discussed by Lax,1 and developed subsequently by many authors.2 Its latest formulation was given by Brown and Rodriguez.3 The theoretical model used by Brown and Rodriguez3 considers initial capture of an electron into excited, hydrogenic s states, followed by cascade phonon-assisted transitions to either the impurity ground state or back to the conduction band. Each transition occurs with the emission (or absorption) of a single acoustic phonon. Ulbrich4 has recently studied the capture of thermalized electrons into shallow donor ground state in GaAs at low temperature. Using the time resolved spectroscopy of donor related luminescence, he estimated a lower bound on the rate at which electrons enter the donor ground states. The observed transition rates are more than two orders of magnitude higher than the values predicted by the theory of Brown and Rodriguez.3 A similar discrepancy between experiment and theory was observed by Norton eta!. who measured the recombination cross-sections for electrons at ionized group-V donors in silicon at temperatures ranging from 1.5 to 20K. Above 4K the temperature dependence of their results agrees quite well with theory, but the absolute values of the experimental cross-sections are larger by a factor 125 than theoretically computed crosssections. However, Norton et a!.5 suggested that the
dependence of cross-section magnitude on very high powers of some parameters can explain this discrepancy. In this paper we want to point out another possible explanation of the considerable quantitative discrepancy between the existing capture theories and experimental values. Because of momentum and energy conservation, probabilities of one-phonon transitions between the lowest shallow impurity states are very small, and the last step in the relaxation cascade, 2s -÷ ls, bridging the largest energy gap is the one limiting the net total transition rate into the donor ground state. Therefore, it seems that two-phonon processes, in which an electron can relax its energy with small (or even zero) net momentum change, should be considered as an important possibility. In the present work we discuss only the simplest and, probably, the most important situation, assuming that the conduction band electron goes directly into the donor ground state, without the help of the excited states It seems, however, that thestates cascade phononladder. transitions through the excited are twoalso possible. 2. TWO-PHONON PROCESSES AND TRANSITION
,~
*
Permanent address: Institute of Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland.
PROBABILITY The interaction Hamiltonian between electron and phonons is obtained by expanding the electron energy in powers of atom displacements from the equilibrium positions. An electron interaction with two phonons can be described by either a term bilinear in atomic displacements (the so-called anharmonic term) or by a repeated application of the linear term, through intermediate states. Ngai6 has recently proposed the effective carrier-
991
992
ELECTRON CAPTURE INTO SHALLOW DONORS IN GaAs
two-phonon interaction Hamiltonian, H2, in which both contributions are included: ~
“2
~
=
,~
~q,q,_
~2
k’ k q’
q
x c~’c “ ~ k krqrq
~ ~.
—
k + ‘C,. ~
(P ~.
j
Equation (1) describes the coupling of a carrier to two phonons with arbitrary wave vectors q and q’ and frequencies c~q and = q + q’, tron operator, fl iswe’. theQvolume of ck therepresents crystal of the the elecmass density p, and a 1 is the lattice constant. ~ = a + a~ is the field operator for phonons, andfD is the ~wo- q phonon effective deformation potential. The probability of transition of the conduction electron to the ls donor state with emission of two LA phonons is W~,
=
J
(2ir/h) Kqiq
2 2 is 11121k>1
x ~(h~o~+ h~o
9 2—E~—Ek)w(Ek/kBT)(27r)
Vol. 18, No.8
dispersion for LA phonons, wq = v5IqI, v~being the sound velocity in the crystal. We also assume that the captured electron had zero kinetic energy in the initial state, i.e. we put k= Oin equation (2).~ Thus, introducing the spherical coordinate system with polar axis parallel to one ofq’s, we obtain from equations (2), (4) and (5) the following expression for the transition probability: 2a2hvSpc~a~)d~LA Wc is =x (2/37r (4x3/3y3)+(x3/2y2)+(x/2y2)+(3x3/4y) [—
2 1) tan’xJ, (6) (x/4y)+ *(3x E~a/hv 2. 5andy = 1 + x 3. NUMERICAL RESULTS AND DISCUSSION —
where x
=
Numencal computations were performed for Unidentified donors in GaAs, with the same values of parameters as used by Ulbrich:4 E~= 5.82 meV, a =98.4 A, v 5cm/sec, p = 5.31 g/cm3 and a 8 = 5.22 x l0 1 = 5.65 A. 1measured by Comparing the transition the experimental value 5 xrate 108given sec~by equation (6) with Ulbrich,4 we obtain the following estimate of the two-
x d3q1 d 3q2 d ~ k ~2’~‘ ‘ where Ek is the energy of electron in the conduction band state k, E~is the ground state binding energy, q 12 LA phonon deformation potential for GaAs: d2LA are wave vectors of emitted phonons of frequencies 0.02 eV. Provided the proposed mechanism is really w1,2, and w(Ek/kBT) denotes the distribution of the responsible for the electron capture, this would provide conduction electron energies. the first estimate of two-phonon deformation potential Inspection of the integral on the right hand side of for GaAs. equation (2) shows that for IEk I ~ IEi8I the relevant conEstimates of the two-phonon deformation potentribution to the transition rate is given by phonons with tials have beenvalues obtained various semiconductorsby 6 These rangefor from 0.5 to 5eV, depending 1q1 + q2 1/a, where a is the effective Bohr radius. By Ngai. energy conservation, Iqi I + Iq~I ~ 1/a. Hence, q 1 q2 on the material and also on the type of experiment con• and c~~ = E~/2h.This situation is formally sidered. This is due to that by theNgai,6 two-phonon de6 for the critical points formation potential dz,theas fact defined contains similar to that discussed Ngai Thus following Ngai6 bywe reformulate the Hamilimplicit information about the density of two-phonon tonian (1) so as to resemble the usual one-phonon destates and one can expect that for different experiformation potential interaction, introducing the twomental situations it can have considerably different phonon deformation potential d2LA as values. i 3 Our estimated value of d2LA is lower than those d2LA ~jpVBZI87r pa2 —~1/2cn —
—
—
— —
1w,
.L1,
~
where p denotes the fraction of the total volume VBZ of the Brillouin which contains the wave vectors of phonons emittedzone in the transition process. Using H 2 from equation (1) with (3), and a plane wave exp (1k r) as the wave function of an electron in the conduction band, one gets for the transition matrix element the following formula: .
—
2 1/2
(qi q2 is 11(2k> = 2(h/2p~Za1) d2LAF(k qi q2~, where 2aSI’2(1+ IKI2a2)2 (4) (5) F(ic) = 87rl~’ is the Fourier transform of the is envelope function. To simplify our calculations we assume a linear —
—
proposed by Ngai for other materials. However, the effective deformation potential publishedby 6 have been estimated fromvalues experiments such as Ngai, the resonant Raman scattering or magnetophonon effect, which favour the critical points in the Brillouin zone, where the joint density of two-phonon states is large. In our case the situation is quite different. Wave vectors of emitted phonons belong to this region of the Brillouin zone, where there are no critical points and the joir.t density of two-phonon states is relatively low. Correspondingly, effective deformation one can potential expectwill thatbeinlower, our case andthe so it appears that the value we estimated is quite reasonable. It seems, therefore, that two-phonon processes can be
Vol. 18, No.8
ELECTRON CAPTURE INTO SHALLOW DONORS IN GaAs
responsible for the conduction electron capture into shallow donors in GaAs. It follows from the above that for low temperatures the capture rate is temperature-independent and that the temperature dependence of the capture cross-section is due to the dependence of an average velocity of conduction electron on T. Thus, the capture cross-section should be proportional to T~’2.
993
Acknowledgements The authors are grateful to Professor N.H. March, Professor J. Mycielski, Professor M. Suffczydski and Dr J. Szymai%ski for helpful discussions. One them (J.G.) wishes to thank Professor Abdus Salamofthe International Atomic Energy Agency and UNESCO, for hospitality at the International Centre for Theoretical Physics, Trieste. —
REFERENCES 1.
LAXM.,Phys.Rev. 119, 1502 (1960).
2.
ASCARELLI G. & RODRIGUEZ S.,Phys. Rev. 124, 1321 (1961) and Phys. Rev. 127, 167 (1962); HAMMAN D.R. & MCWHORTER A.L.,Phys. Rev. 134, A250 (1964); BELEZNAY F. & PATAKJ G.,Phys. Status Solidi 13,499 (1966).
3.
BROWN R.A. & RODRIGUEZ S., Phys. Rev. 153, 890 (1967).
4.
ULBRJCH R., Proc. 12th mt. Conf Phys. Semicond. Stuttgart, p. 376. Tuebner Stuttgart (1974).
5. 6.
NORTON P., BRAGGINS T. & LEVINSTEIN H., Phys. Rev. Lett. 30,488(1973). NGAI K.L., Proc. 12th mt. Conf. Phys. Semicond. Stuttgart, p. 489. Tuebner Stuttgart (1974); see also: NGAJ KL. & GANGULY A.K., Proc. NA TO Advanced Study Institute on Elementary Excitations in Solids Moleculesand Atoms, Antwerp 1973, (Edited by DEVREESE J.), p. 295. Plenum Press, NY (1974).
7.
Uibrich8 has shown that in his experimental conditions the intraband relaxation of electron energy (“thermalization”) is much faster than capture transition rate. Therefore, for very low temperatures, for which 1c 8 T ~ E~,this assumption is justified quite well. ULBRICH R.,Phys. Rev. B8, 5719 (1973).
8.
Pergamon Press.
Printed in Great Britain
REMARKS ON THE THEORY OF ELECTRON CAPTURE INTO SHALLOW DONORS IN GaAs J. GoIka’~ International Centre for Theoretical Physics, Trieste, Italy and J. Mostowski Institute of Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland (Received 7 November 1975 by M. Cardona) A new model for low-temperature recombination of conduction electrons to ionized donors in GaAs is proposed. It assumes that the conduction electron is captured into the ground state of shallow impurity by simultaneous emission of two LA phonons. From a comparison of computed capture rate with recent experimental results the estimated value of the two-phonon deformation potential is d2LA = 0.02 eV. 1. INTRODUCTION THE THEORY of free-carrier recombination with hydrogenic ionized impurities in semiconductors was first discussed by Lax,1 and developed subsequently by many authors.2 Its latest formulation was given by Brown and Rodriguez.3 The theoretical model used by Brown and Rodriguez3 considers initial capture of an electron into excited, hydrogenic s states, followed by cascade phonon-assisted transitions to either the impurity ground state or back to the conduction band. Each transition occurs with the emission (or absorption) of a single acoustic phonon. Ulbrich4 has recently studied the capture of thermalized electrons into shallow donor ground state in GaAs at low temperature. Using the time resolved spectroscopy of donor related luminescence, he estimated a lower bound on the rate at which electrons enter the donor ground states. The observed transition rates are more than two orders of magnitude higher than the values predicted by the theory of Brown and Rodriguez.3 A similar discrepancy between experiment and theory was observed by Norton eta!. who measured the recombination cross-sections for electrons at ionized group-V donors in silicon at temperatures ranging from 1.5 to 20K. Above 4K the temperature dependence of their results agrees quite well with theory, but the absolute values of the experimental cross-sections are larger by a factor 125 than theoretically computed crosssections. However, Norton et a!.5 suggested that the
dependence of cross-section magnitude on very high powers of some parameters can explain this discrepancy. In this paper we want to point out another possible explanation of the considerable quantitative discrepancy between the existing capture theories and experimental values. Because of momentum and energy conservation, probabilities of one-phonon transitions between the lowest shallow impurity states are very small, and the last step in the relaxation cascade, 2s -÷ ls, bridging the largest energy gap is the one limiting the net total transition rate into the donor ground state. Therefore, it seems that two-phonon processes, in which an electron can relax its energy with small (or even zero) net momentum change, should be considered as an important possibility. In the present work we discuss only the simplest and, probably, the most important situation, assuming that the conduction band electron goes directly into the donor ground state, without the help of the excited states It seems, however, that thestates cascade phononladder. transitions through the excited are twoalso possible. 2. TWO-PHONON PROCESSES AND TRANSITION
,~
*
Permanent address: Institute of Physics, Polish Academy of Sciences, 02-668 Warszawa, Poland.
PROBABILITY The interaction Hamiltonian between electron and phonons is obtained by expanding the electron energy in powers of atom displacements from the equilibrium positions. An electron interaction with two phonons can be described by either a term bilinear in atomic displacements (the so-called anharmonic term) or by a repeated application of the linear term, through intermediate states. Ngai6 has recently proposed the effective carrier-
991
992
ELECTRON CAPTURE INTO SHALLOW DONORS IN GaAs
two-phonon interaction Hamiltonian, H2, in which both contributions are included: ~
“2
~
=
,~
~q,q,_
~2
k’ k q’
q
x c~’c “ ~ k krqrq
~ ~.
—
k + ‘C,. ~
(P ~.
j
Equation (1) describes the coupling of a carrier to two phonons with arbitrary wave vectors q and q’ and frequencies c~q and = q + q’, tron operator, fl iswe’. theQvolume of ck therepresents crystal of the the elecmass density p, and a 1 is the lattice constant. ~ = a + a~ is the field operator for phonons, andfD is the ~wo- q phonon effective deformation potential. The probability of transition of the conduction electron to the ls donor state with emission of two LA phonons is W~,
=
J
(2ir/h) Kqiq
2 2 is 11121k>1
x ~(h~o~+ h~o
9 2—E~—Ek)w(Ek/kBT)(27r)
Vol. 18, No.8
dispersion for LA phonons, wq = v5IqI, v~being the sound velocity in the crystal. We also assume that the captured electron had zero kinetic energy in the initial state, i.e. we put k= Oin equation (2).~ Thus, introducing the spherical coordinate system with polar axis parallel to one ofq’s, we obtain from equations (2), (4) and (5) the following expression for the transition probability: 2a2hvSpc~a~)d~LA Wc is =x (2/37r (4x3/3y3)+(x3/2y2)+(x/2y2)+(3x3/4y) [—
2 1) tan’xJ, (6) (x/4y)+ *(3x E~a/hv 2. 5andy = 1 + x 3. NUMERICAL RESULTS AND DISCUSSION —
where x
=
Numencal computations were performed for Unidentified donors in GaAs, with the same values of parameters as used by Ulbrich:4 E~= 5.82 meV, a =98.4 A, v 5cm/sec, p = 5.31 g/cm3 and a 8 = 5.22 x l0 1 = 5.65 A. 1measured by Comparing the transition the experimental value 5 xrate 108given sec~by equation (6) with Ulbrich,4 we obtain the following estimate of the two-
x d3q1 d 3q2 d ~ k ~2’~‘ ‘ where Ek is the energy of electron in the conduction band state k, E~is the ground state binding energy, q 12 LA phonon deformation potential for GaAs: d2LA are wave vectors of emitted phonons of frequencies 0.02 eV. Provided the proposed mechanism is really w1,2, and w(Ek/kBT) denotes the distribution of the responsible for the electron capture, this would provide conduction electron energies. the first estimate of two-phonon deformation potential Inspection of the integral on the right hand side of for GaAs. equation (2) shows that for IEk I ~ IEi8I the relevant conEstimates of the two-phonon deformation potentribution to the transition rate is given by phonons with tials have beenvalues obtained various semiconductorsby 6 These rangefor from 0.5 to 5eV, depending 1q1 + q2 1/a, where a is the effective Bohr radius. By Ngai. energy conservation, Iqi I + Iq~I ~ 1/a. Hence, q 1 q2 on the material and also on the type of experiment con• and c~~ = E~/2h.This situation is formally sidered. This is due to that by theNgai,6 two-phonon de6 for the critical points formation potential dz,theas fact defined contains similar to that discussed Ngai Thus following Ngai6 bywe reformulate the Hamilimplicit information about the density of two-phonon tonian (1) so as to resemble the usual one-phonon destates and one can expect that for different experiformation potential interaction, introducing the twomental situations it can have considerably different phonon deformation potential d2LA as values. i 3 Our estimated value of d2LA is lower than those d2LA ~jpVBZI87r pa2 —~1/2cn —
—
—
— —
1w,
.L1,
~
where p denotes the fraction of the total volume VBZ of the Brillouin which contains the wave vectors of phonons emittedzone in the transition process. Using H 2 from equation (1) with (3), and a plane wave exp (1k r) as the wave function of an electron in the conduction band, one gets for the transition matrix element the following formula: .
—
2 1/2
(qi q2 is 11(2k> = 2(h/2p~Za1) d2LAF(k qi q2~, where 2aSI’2(1+ IKI2a2)2 (4) (5) F(ic) = 87rl~’ is the Fourier transform of the is envelope function. To simplify our calculations we assume a linear —
—
proposed by Ngai for other materials. However, the effective deformation potential publishedby 6 have been estimated fromvalues experiments such as Ngai, the resonant Raman scattering or magnetophonon effect, which favour the critical points in the Brillouin zone, where the joint density of two-phonon states is large. In our case the situation is quite different. Wave vectors of emitted phonons belong to this region of the Brillouin zone, where there are no critical points and the joir.t density of two-phonon states is relatively low. Correspondingly, effective deformation one can potential expectwill thatbeinlower, our case andthe so it appears that the value we estimated is quite reasonable. It seems, therefore, that two-phonon processes can be
Vol. 18, No.8
ELECTRON CAPTURE INTO SHALLOW DONORS IN GaAs
responsible for the conduction electron capture into shallow donors in GaAs. It follows from the above that for low temperatures the capture rate is temperature-independent and that the temperature dependence of the capture cross-section is due to the dependence of an average velocity of conduction electron on T. Thus, the capture cross-section should be proportional to T~’2.
993
Acknowledgements The authors are grateful to Professor N.H. March, Professor J. Mycielski, Professor M. Suffczydski and Dr J. Szymai%ski for helpful discussions. One them (J.G.) wishes to thank Professor Abdus Salamofthe International Atomic Energy Agency and UNESCO, for hospitality at the International Centre for Theoretical Physics, Trieste. —
REFERENCES 1.
LAXM.,Phys.Rev. 119, 1502 (1960).
2.
ASCARELLI G. & RODRIGUEZ S.,Phys. Rev. 124, 1321 (1961) and Phys. Rev. 127, 167 (1962); HAMMAN D.R. & MCWHORTER A.L.,Phys. Rev. 134, A250 (1964); BELEZNAY F. & PATAKJ G.,Phys. Status Solidi 13,499 (1966).
3.
BROWN R.A. & RODRIGUEZ S., Phys. Rev. 153, 890 (1967).
4.
ULBRJCH R., Proc. 12th mt. Conf Phys. Semicond. Stuttgart, p. 376. Tuebner Stuttgart (1974).
5. 6.
NORTON P., BRAGGINS T. & LEVINSTEIN H., Phys. Rev. Lett. 30,488(1973). NGAI K.L., Proc. 12th mt. Conf. Phys. Semicond. Stuttgart, p. 489. Tuebner Stuttgart (1974); see also: NGAJ KL. & GANGULY A.K., Proc. NA TO Advanced Study Institute on Elementary Excitations in Solids Moleculesand Atoms, Antwerp 1973, (Edited by DEVREESE J.), p. 295. Plenum Press, NY (1974).
7.
Uibrich8 has shown that in his experimental conditions the intraband relaxation of electron energy (“thermalization”) is much faster than capture transition rate. Therefore, for very low temperatures, for which 1c 8 T ~ E~,this assumption is justified quite well. ULBRICH R.,Phys. Rev. B8, 5719 (1973).
8.