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Oscillator strengths for excitons bound to impurities and quantum wells
Solid State Communications, Printed in Great Britain. OSCILLATOR
Vol. 54, No. 4, pp. 343-345,
STRENGTHS
1985.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
FOR EXCITONS BOUND TO IMPURITIES
AND QUANTUM WELLS
DC. Herbert and J.M. Rorison Royal Signals and Radar Establishment, (Received
10 September
St Andrews Road, Malvern, UK
1984 by C. W. McCombie)
It is argued that electron-hole correlation can have a large effect on the oscillator strengths of bound excitons, and may help to explain the observations of shallow excited states for donor bound excitons in direct gap semiconductors. For quantum wells it is found that compression of the exciton wave function yields a l/L dependence of oscillator strength on well width L for narrow wells. THE OSCILLATOR STRENGTHS for bound exciton absorption in semiconductors can be many orders of magnitude greater than the free exciton values. This effect was first explained by Rashba and Gurgenishvili [l] and studied for CdS by Henry and Nassau [2]. In this letter we extend the theory to consider the explicit effects of correlation on the oscillator strengths, and also extend the theory to free excitons in a quantum well. For a free exciton in a pure crystal the exciton wave function in effective mass approximation can be written in the form
where s is a vector in the plane of the well, A is the area, and the axis of the well is taken in the z direction. We take axzD to have a two dimensional exciton form for very narrow wells. Bastard, Mendez, Chang and Esaki [3] considered a wave function in the form
cos ($)cos
.
Gk
=
-“& Gx3D(re - bl),
(1)
where k represents the wave-vector for translational motion, @x3D is the three dimensional hydrogenic correlated exciton component and re - rh represents the electron-hole separation. The oscillator strength is proportional to the expression
R = I(flpli)l* N
’
IIPII~ l$~~~~(O>l*%,
+(:-“)*I.
(5)
They found that for L/a, < 1 where a, is the three dimensional exciton Bohr radius, the term (Z, -Zh)* has very little effect so that we can use the twodimensional exciton form as in (4). If we evaluate the quantity R for the quantum well and take the ratio with the three dimensional value (3) we find that the oscillator strength is enhanced by the factor
If we take L = 5OA then using the results of Bastard et [3] and a Bohr radius of 145 A for the three dimensional exciton [4], we find that for GaAs an enhancement of 13 is predicted. This value is consistent with the experimental results of Christen, Bimberg, Stechenburn and Weimann [4]. We note that the electron-hole exchange will also be enhanced by the same factor (6). For a neutral donor bound exciton involving two electrons and one hole bound to a positively charged impurity core we consider two models for the wavefunction. If we neglect electron-hole correlation and use a Hartree approximation we have that the total wavefunction J, is a product wavefunction rl, = +i 9: $h. The hole has equal probability to combine with either electron so we have
al.
where lfi, Ii> represent the final and initial states respectively, p is the momentum operator derived from the Asp electron-photon coupling and N is the number of atoms in the crystal. Following the usual approximation we consider p to operate on the Bloch components of the wave functions and set k = 0, when (2) can be reduced to evaluating the electron-hole overlap. The result is R -
@)exp [-“*
no = V/N,
(3)
where llpll is the numerical value of the momentum matrix element between Bloch functions and a0 is the atomic volume. For a quantum well the free exciton wave function takes the form . \kk -_ “; (px*D (Se - %u-@df@d (4)
RD
343
= l(flpli)l*
- 211~11~1~ $fY#YW3~12
(7)
OSCILLATOR
344
STRENGTHS
In (7)~ operates on the Bloch components of the wavefunction and the integral is over the electron-hole hydrogenie wavefunction overlap. We can approximate the electron and hole wavefunctions in the form [5]
r
-40s 43 @h = -&;-re-flr
,
(8)
where y and /3 are variational parameters set by minimizing the total energy. The 4’ function is of a 1s form while I#J so that the wavefunction is h is constructed forced to vanish at the origin due to repulsion from the impurity core. Whereas y will be closely related to mf (y = Zm ,*/e in the hydrogenic model) fi will depend on both m,h and rn:. In particular if rnz > m,* the @ orbital will be pinned at the @” radius since it is repelled by the donor charge and attracted by the electron charge. Inserting these wavefunctions into the integral in equation (7) leads to the simple analytic expression RD - 2llpll* - 192 y3p5 (Y + P)* Putting y = /3, pinning radius yields RD -
(94
the hole radius at the electron
1.511Pl12(Y = 0).
(9b)
For strong electron-hole correlation we consider a model of a free exciton orbiting a neutral donor [6,7]. In this case we find that the wavefunction may be represented as $ = @x3D(X - R)q5h(Rh) where for m,l* - IIPl121v3D(~)12 Using the wavefunctions
(10)
given above in (8) we have
(114
RD - lIplIz * 192; Putting y = & pinning radius yields
I!^ 4h @9a3r1*
the hole radius at the electron
RD - lb112 192 (7 = p) .
(1 lb)
Comparing (1 lb) and (9b) shows that the electron hole correlation can yield large increases in bound exciton strength for the hole radius pinned at the electron radius. Both expressions are very sensitive to the ratio y/P and our estimate fi = y, pinning the hole at the electron radius probably over estimates R IJ by maximizing the electron-hole overlap. This would seem qualitatively to be a good approximation for m E > m ,*which is the situation in the tetrahedral semiconductors. We
FOR EXCITONS
Vol. 54, No. 4
have used a better @h wavefunction than that in (8) which allows for a decay of the tail as a 1s hydrogenic wavefunction while still maintaining the peaking at r = (l/r) and find that there is very little change in the numbers derived using the wavefunctions in (8) with y = /3. If we evaluate (3) for GaAs we find R - 2 x 10e6 llpllz so that both expressions (11) and (9) exhibit the giant oscillator strength effect. Finally we consider how these results can help to explain experimental observations of shallow excited states of neutral donor bound excitons in InP. It has been suggested that the excited states observed by White et el. [8] and Ruhle and Klingenstein [9] could be understood as P-like angular momentum states of the hole [S] . The calculation using a quasi-acceptor approximation yielded the energy sequence Sa,*, P3,2, P5,*, PI,*. The calculation was performed in a Hartree approximation using the spherical model of Baldereschi and Lipari [lo] for the hole. The inter-particle interactions were adjusted to allow for correlation and obtain reasonable energies for the exciton ground state. The theoretical energy sequence is compatible with results for normal acceptors for all values of the valence band parameters [lo]. It is found that the magnetic properties of the bound exciton are very anisotropic indicating that cubic corrections are strong so that the P,,, states observed by Ruhle et al could derive from a crystal field splitting of the P,,, state, alternatively the correlation energy would have to vary considerably between the different excited states to bring the P,,, level to the required energy. The P-like character of the first excited state has been confirmed experimentally from 2-electron transition spectra in ZnTe [6], but it has remained difficult to understand how the oscillator strengths of the parity forbidden P-like states could be comparable with the Slike ground state as observed by Ruhle and Klingenstein. These states are very extended so that the finite wave vector of the photon can allow appreciable oscillator strengths, but to increase this contribution to a comparable magnitude with the S,, ground state it seems necessary to conclude that the electron-hole correlation is increased in the P-like excited states. In high magnetic fields the ground state of the neutral donor bound exciton in InP is proposed to be a P-like hydrogenic state [7] . Again relative intensity of the 2-electron transitions also suggests that the electron-hole correlation is strong in the P-like state. The accurate calculation of correlation in these excited states remains a formidable numerical problem as the ground state properties have only recently been satisfactorily explained for a simple effective mass model, neglecting valence band degeneracy [ 1 l] .
Vol. 54, No. 4
OSCILLATOR
STRENGTHS
Acknowledgements - The authors are grateful to MS Skolnick and PJ Dean for several very helpful discussions. Note: A related calculation on the oscillator strength of bound excitons in quantum wells employing the models used here was presented at the international conference on superlattices, microstructures and microdevices held at Champaign-Urbana Illinois Aug 13-16 1984 and will appear in the new journal “Superlattices and Microstructures”. REFERENCES 1. 2. 3.
E.I. Rashba & G.E. Gurgenishvili, Sov. Phys. Solid State 4,256 (1962). C.H. Henry & K. Nassau, Phys. Rev. Bl 1628 (1970). G. Bastard, E.E. Mendez. L.L. Chang & L. Esaki, Phys. Rev. B26, 1974 (1980).
4. 5.
6. 7. 8. 9. 10. 11.
FOR EXCITONS
345
J. Christen, D. Bimberg, A. Steckenburn & G. Weimann, Appl. Phys. Lett. 44,84 (1984). D.C. Herbert, J. Phys. C 10 3327 (1970). [It should be noted that curve (a) in Figure (1) of this paper is incorrect due to a coding error in a computer programme]. P.J. Dean, D.C. Herbert & A.M. Lahee, J. Phys. C 13,507l (1980). J. Rorison, D.C. Herbert, P.J. Dean & M.S. Skolnick, J. Phys. C 17,6435 (1984). A.M. White, P.J. Dean & B. Day, J. Phys. C 7, 1400 (1974). W. Rtihle & W. Klingenstein, Phys. Rev. B18, 7011 (1978). A. Balderschi & N.O. Lipari, Phys. Rev. B8, 2697 (1973). B. Stebe & G. Munschy, Solid State Commun. 35, 557 (1980).
Vol. 54, No. 4, pp. 343-345,
STRENGTHS
1985.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
FOR EXCITONS BOUND TO IMPURITIES
AND QUANTUM WELLS
DC. Herbert and J.M. Rorison Royal Signals and Radar Establishment, (Received
10 September
St Andrews Road, Malvern, UK
1984 by C. W. McCombie)
It is argued that electron-hole correlation can have a large effect on the oscillator strengths of bound excitons, and may help to explain the observations of shallow excited states for donor bound excitons in direct gap semiconductors. For quantum wells it is found that compression of the exciton wave function yields a l/L dependence of oscillator strength on well width L for narrow wells. THE OSCILLATOR STRENGTHS for bound exciton absorption in semiconductors can be many orders of magnitude greater than the free exciton values. This effect was first explained by Rashba and Gurgenishvili [l] and studied for CdS by Henry and Nassau [2]. In this letter we extend the theory to consider the explicit effects of correlation on the oscillator strengths, and also extend the theory to free excitons in a quantum well. For a free exciton in a pure crystal the exciton wave function in effective mass approximation can be written in the form
where s is a vector in the plane of the well, A is the area, and the axis of the well is taken in the z direction. We take axzD to have a two dimensional exciton form for very narrow wells. Bastard, Mendez, Chang and Esaki [3] considered a wave function in the form
cos ($)cos
.
Gk
=
-“& Gx3D(re - bl),
(1)
where k represents the wave-vector for translational motion, @x3D is the three dimensional hydrogenic correlated exciton component and re - rh represents the electron-hole separation. The oscillator strength is proportional to the expression
R = I(flpli)l* N
’
IIPII~ l$~~~~(O>l*%,
+(:-“)*I.
(5)
They found that for L/a, < 1 where a, is the three dimensional exciton Bohr radius, the term (Z, -Zh)* has very little effect so that we can use the twodimensional exciton form as in (4). If we evaluate the quantity R for the quantum well and take the ratio with the three dimensional value (3) we find that the oscillator strength is enhanced by the factor
If we take L = 5OA then using the results of Bastard et [3] and a Bohr radius of 145 A for the three dimensional exciton [4], we find that for GaAs an enhancement of 13 is predicted. This value is consistent with the experimental results of Christen, Bimberg, Stechenburn and Weimann [4]. We note that the electron-hole exchange will also be enhanced by the same factor (6). For a neutral donor bound exciton involving two electrons and one hole bound to a positively charged impurity core we consider two models for the wavefunction. If we neglect electron-hole correlation and use a Hartree approximation we have that the total wavefunction J, is a product wavefunction rl, = +i 9: $h. The hole has equal probability to combine with either electron so we have
al.
where lfi, Ii> represent the final and initial states respectively, p is the momentum operator derived from the Asp electron-photon coupling and N is the number of atoms in the crystal. Following the usual approximation we consider p to operate on the Bloch components of the wave functions and set k = 0, when (2) can be reduced to evaluating the electron-hole overlap. The result is R -
@)exp [-“*
no = V/N,
(3)
where llpll is the numerical value of the momentum matrix element between Bloch functions and a0 is the atomic volume. For a quantum well the free exciton wave function takes the form . \kk -_ “; (px*D (Se - %u-@df@d (4)
RD
343
= l(flpli)l*
- 211~11~1~ $fY#YW3~12
(7)
OSCILLATOR
344
STRENGTHS
In (7)~ operates on the Bloch components of the wavefunction and the integral is over the electron-hole hydrogenie wavefunction overlap. We can approximate the electron and hole wavefunctions in the form [5]
r
-40s 43 @h = -&;-re-flr
,
(8)
where y and /3 are variational parameters set by minimizing the total energy. The 4’ function is of a 1s form while I#J so that the wavefunction is h is constructed forced to vanish at the origin due to repulsion from the impurity core. Whereas y will be closely related to mf (y = Zm ,*/e in the hydrogenic model) fi will depend on both m,h and rn:. In particular if rnz > m,* the @ orbital will be pinned at the @” radius since it is repelled by the donor charge and attracted by the electron charge. Inserting these wavefunctions into the integral in equation (7) leads to the simple analytic expression RD - 2llpll* - 192 y3p5 (Y + P)* Putting y = /3, pinning radius yields RD -
(94
the hole radius at the electron
1.511Pl12(Y = 0).
(9b)
For strong electron-hole correlation we consider a model of a free exciton orbiting a neutral donor [6,7]. In this case we find that the wavefunction may be represented as $ = @x3D(X - R)q5h(Rh) where for m,
(10)
given above in (8) we have
(114
RD - lIplIz * 192; Putting y = & pinning radius yields
I!^ 4h @9a3r1*
the hole radius at the electron
RD - lb112 192 (7 = p) .
(1 lb)
Comparing (1 lb) and (9b) shows that the electron hole correlation can yield large increases in bound exciton strength for the hole radius pinned at the electron radius. Both expressions are very sensitive to the ratio y/P and our estimate fi = y, pinning the hole at the electron radius probably over estimates R IJ by maximizing the electron-hole overlap. This would seem qualitatively to be a good approximation for m E > m ,*which is the situation in the tetrahedral semiconductors. We
FOR EXCITONS
Vol. 54, No. 4
have used a better @h wavefunction than that in (8) which allows for a decay of the tail as a 1s hydrogenic wavefunction while still maintaining the peaking at r = (l/r) and find that there is very little change in the numbers derived using the wavefunctions in (8) with y = /3. If we evaluate (3) for GaAs we find R - 2 x 10e6 llpllz so that both expressions (11) and (9) exhibit the giant oscillator strength effect. Finally we consider how these results can help to explain experimental observations of shallow excited states of neutral donor bound excitons in InP. It has been suggested that the excited states observed by White et el. [8] and Ruhle and Klingenstein [9] could be understood as P-like angular momentum states of the hole [S] . The calculation using a quasi-acceptor approximation yielded the energy sequence Sa,*, P3,2, P5,*, PI,*. The calculation was performed in a Hartree approximation using the spherical model of Baldereschi and Lipari [lo] for the hole. The inter-particle interactions were adjusted to allow for correlation and obtain reasonable energies for the exciton ground state. The theoretical energy sequence is compatible with results for normal acceptors for all values of the valence band parameters [lo]. It is found that the magnetic properties of the bound exciton are very anisotropic indicating that cubic corrections are strong so that the P,,, states observed by Ruhle et al could derive from a crystal field splitting of the P,,, state, alternatively the correlation energy would have to vary considerably between the different excited states to bring the P,,, level to the required energy. The P-like character of the first excited state has been confirmed experimentally from 2-electron transition spectra in ZnTe [6], but it has remained difficult to understand how the oscillator strengths of the parity forbidden P-like states could be comparable with the Slike ground state as observed by Ruhle and Klingenstein. These states are very extended so that the finite wave vector of the photon can allow appreciable oscillator strengths, but to increase this contribution to a comparable magnitude with the S,, ground state it seems necessary to conclude that the electron-hole correlation is increased in the P-like excited states. In high magnetic fields the ground state of the neutral donor bound exciton in InP is proposed to be a P-like hydrogenic state [7] . Again relative intensity of the 2-electron transitions also suggests that the electron-hole correlation is strong in the P-like state. The accurate calculation of correlation in these excited states remains a formidable numerical problem as the ground state properties have only recently been satisfactorily explained for a simple effective mass model, neglecting valence band degeneracy [ 1 l] .
Vol. 54, No. 4
OSCILLATOR
STRENGTHS
Acknowledgements - The authors are grateful to MS Skolnick and PJ Dean for several very helpful discussions. Note: A related calculation on the oscillator strength of bound excitons in quantum wells employing the models used here was presented at the international conference on superlattices, microstructures and microdevices held at Champaign-Urbana Illinois Aug 13-16 1984 and will appear in the new journal “Superlattices and Microstructures”. REFERENCES 1. 2. 3.
E.I. Rashba & G.E. Gurgenishvili, Sov. Phys. Solid State 4,256 (1962). C.H. Henry & K. Nassau, Phys. Rev. Bl 1628 (1970). G. Bastard, E.E. Mendez. L.L. Chang & L. Esaki, Phys. Rev. B26, 1974 (1980).
4. 5.
6. 7. 8. 9. 10. 11.
FOR EXCITONS
345
J. Christen, D. Bimberg, A. Steckenburn & G. Weimann, Appl. Phys. Lett. 44,84 (1984). D.C. Herbert, J. Phys. C 10 3327 (1970). [It should be noted that curve (a) in Figure (1) of this paper is incorrect due to a coding error in a computer programme]. P.J. Dean, D.C. Herbert & A.M. Lahee, J. Phys. C 13,507l (1980). J. Rorison, D.C. Herbert, P.J. Dean & M.S. Skolnick, J. Phys. C 17,6435 (1984). A.M. White, P.J. Dean & B. Day, J. Phys. C 7, 1400 (1974). W. Rtihle & W. Klingenstein, Phys. Rev. B18, 7011 (1978). A. Balderschi & N.O. Lipari, Phys. Rev. B8, 2697 (1973). B. Stebe & G. Munschy, Solid State Commun. 35, 557 (1980).