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Scattering of excitons by excitons in semiconducting quantum well structures
J. Phys. Chem. Solids Vol. 48, No. 12, pp. 1191-1196, 1987 Printed in Great Britain.
0
SCATTERING OF EXCITONS SEMICONDUCTING QUANTUM
0022-3697187 1987 Pergamon
S3.00 + 0.00 Journals Ltd.
BY EXCITONS IN WELL STRUCTURES
YUAN-PING FENG and HAROLD N. SPECTOR Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, U.S.A. (Received
19 December
1986; accepted in revised form 15 April 1987)
Abstract-We have theoretically investigated the scattering of excitons by excitons in a two-dimensional semiconducting quantum well system. The scattering cross sections have been calculated using the Born approximation for both the elastic and inelastic scattering of the excitons by excitons. The threshold for inelastic scattering is increased over the value in a bulk semiconductor because of the enhancement of the exciton binding energy by its confinement. The behavior of the scattering cross section as a function of the energy of relative motion of the excitons is different than in the bulk and the cross section is a more sensitive function of the ratio of the electron and hole masses than in the bulk. Keyworh:
Excitons, quantum well structures, scattering cross section.
1. INTRODUCTION Exciton-exciton (Exe-Exe) collisions have been observed [l-5] in highly excited bulk semiconductors such as ZnO, CdS and CdSe. For the interpretation of these observations, the exciton-exciton elastic and inelastic cross sections are of interest and these cross sections have been calculated for different semiconductors in the bulk case [6, 71. With the development of crystal growth techniques such as molecular beam epitaxy (MBE), it has become possible to grow high quality semiconducting quantum well and superlattice structures [8]. These structures have become of great interest technically because of their enhanced mobilities when they are modulation doped [9-l 1] and the appearance of strong room temperature exciton peaks [12-151 when the width of the layer in which the exciton is confined is less than its Bohr radius. Because of the great interest in the optical properties of these structures, theoretical calculations have been performed of the excitonic contribution to the optical absorption [16] and the effect of the confinement on the exciton binding energy [17-191. In addition to the effect of the carrier confinement on the binding energy of excitons in quantum well structures, there is interest in calculating and measuring the exciton linewidth. Calculations have been performed which take into account the effect of scattering from acoustic [20] and optical phonons [21] and also alloy scattering [22] on the exciton linewidth in quantum well structures. However, there are also other possible scattering mechanisms which might play a role in determining the exciton linewidth which have not as yet been considered such as the elastic and inelastic scattering of excitons from other excitons and from free carriers. In a recent work [23], we theoretically investigated the scattering of excitons by free electrons and holes in a two-dimensional semiconducting quantum well system. We found that PCS
48,12--c
the cross sections for such scattering mechanisms were appreciable for certain ranges of the energy of the relative motion of the excitons and free carriers, and, that there were some interesting features which appeared as the result of the confinement in the quantum well structure. In the present work, we wish to present theoretical calculations of the cross sections for exciton-exciton elastic and inelastic scattering in quantum well systems. In our calculations, we will assume a twodimensional gas of excitons interacting with each other and use the central field approximation [24] and the Born approximation [25] to calculate the cross sections due to collisions between the excitons for the simple transitions Exc(ls/B) + Exe (Is/A)-+Exc (Is/B) + Exc(m/A) where m = 1s (elastic scattering), 2s, 2p, 3s, 3p and 3d (inelastic scattering). The use of the two-dimensional model should be valid for very narrow wells and should show whether the confinement of the excitons in a quantum well system has an appreciable effect on their scattering cross sections. The use of the two-dimensional model also enables us to use the exact two-dimensional wave functions and energy eigenvalues for the excitons instead of those approximate wave functions and energies obtained using a variational approach in quantum wells of finite width. The cross sections for different semiconductors are functions of the ratio of the electron and hole effective masses, Q = me/m,. In our calculations the values of these masses which are used are those for the bulk semiconductor. The cross sections are given for different available values of 0 for the semiconductors ZnO [6,26-281, CdS [26-281, CdSe [4,26,27,29] as well as GaAs [ 191.The reason for the calculations being done using the parameters of GaAs is that this is the prototype material for most quantum well structures, while the calculations using the parameters of the other semiconductors are per-
1191
YUAN-PINGFENGand
1192
formed in order to facilitate the comparison with the calculations done for these materials in the bulk case [6,7]. This mechanism might play an important role in determining the exciton linewidth in experiments in which a high density of excitons is optically generated. 2. EXCITON-EXCITON COLLISIONS IN TWO DIMENSIONS The energies and wave functions of purely twodimensional excitons are given by [30] E, = R/(n - ;)*, n = 1, 2, 3
(1)
and
where L;(p) is the associated Laguerre polynomial, the exciton Bohr radius and Rydberg energy are defined as a = (ch2/pe2) and R = (e2/2a), respectively where t is the dielectric constant of the semiconductor and ~1= m,m,,/(m, + mh) is the reduced mass of the exciton. The principal quantum number n can take on any positive integer value while the angular momentum quantum number m = 0, + 1, f 2, +3, . . . , +(n - 1) for a given n. For convenience we adopt a notation for the exciton states similar to that used in the three-dimensional case and call the states 1s (n= 1, m = 0), 2s (n = 2, m = 0), 2p (n = 2, m = *l), 3s (n = 3, m = 0),3p (n = 3, m = * l), 3d (n =3, m = +2), etc. Consider an encounter between an exciton A, comprised of a hole a and an electron 1 and another exciton B, which consists of a hole b and an electron 2. From our previous work on exciton-free carrier scattering in a two-dimensional quantum well system, the cross section, within the Born approximation [25], associated with the reaction in which exciton A is excited from its ground state to an excited state while exciton B remains in its ground state is Q,+=
s2nL,@W,
HAROLDN. SPECTOR where ~,(r,,) and ,q(r,,) are the wave functions of the electron 1 bound to the hole a before and after the collision, x,(rhZ) is the ground state wave function of the electron 2 bound to the hole b, R is the relative position vector of the exciton, V(r,, ,rh2, R) is the interaction potential between the excitons and M is the reduced mass of the system. The reduced mass of the system is given by M = fm,(1 + 0 -I), where the mass ratio is defined as D = me/m,. Also, here k, = (MVJh), k,= (Myf/h), where V, and V, are the initial and final velocities of relative motion between the two excitons and K = k, - K,. conservation of energy we have From k_f= kf - (2M/m,)(EfE,) where E, and E, are the sum of the internal energies of the two excitons before and after the collision. It is convenient to use the Rydberg energy R. = (m,e4/2c2h2) and Bohr radius a, = (th*/e’m,) as the basic units of energy and length, respectively. Using these units, we have kj=k;--#+a -‘)(E,-E,) where E, = (-S/l + a). For the transitions Is -+2s and Is -+2p, E,= t-40/9 (1 + a)] while for the transitions 1s -+3s, Is +3p and Is +3d, Ef = [ - 104/25 (1 + a)]. For elastic collisions, Is-+ls, Ef= E, and k, = kf. The interaction potential between the excitons, using the central field approximation. is given by V(r,, ,r,,,R’l = (e*/t)(rG’
- r,z’ - rh’ + r fi’), (5)
where rxy is the distance between particle x and particle y, which can be expressed in terms of ro, ,rhZ and R as shown in Fig. 1. Here R is the relative position vector of the two excitons. Using the exciton wave functions given in eqn (2) and completing the integrations over R and rh2, we obtain the following expression for the different cross section: I,,/(Q) =;a(1 + a~‘)2(k,K2)-‘l{[l + (aK/4)*]-“’ -[l
+ (K/4)‘]-3’2} J(i+jy,
(6)
where
-expW
+a)~‘Kr,,I}~;(r,,)~,(r~,).
(7)
(3)
0
where 0 is the scattering angle in the center of mass frame of the excitons. The differential cross section l,_,(e) is given by I,,f(0)
= (M2/2xh2ki)ljdrll, drb,dR between the vectors rob, r12, ry2, rb, , gives the relative positions of the and holes in the two excitons and the positions of their centers of mass.
Fig. 1. The relationship
x expiKR k(‘b2)Izi2?
V@,,
) rb27
W
xf
(cl
r,, , rb2 and R which
hi@,,)
electrons
(4)
Scattering of excitons by excitons 2.1 Elastic scattering For elastic collisions, both the excitons remain in the ground state after the collision. Therefore k,= ki and K = 2ki jsin(8/2)] where B is the scattering angle. J(ls-+ls) is readily calculated and is given by
Here K is defined by K2 = kj+ kf- 2kik,cos@, which is a function of the scattering angle. From eqns {10~~4), the exciton inelastic cross section Q is given in terms of a, as Q,,=
J(ls+ls)
= [l +((TK/4)q-3’7-
[l +(K/4)2]-“2.
1193
n(1 + a-‘)2k;‘l
(8)
‘d&C-*{[l
+ (~rK,‘4)7-~‘~
s0 -
The total elastic cross section in units of a, is given
[1 + (K/4)2]-3”) J(i-q-p.
(15)
by 3.
Q*=n(I
+~-‘)~k;’
‘d&C-‘{[I s0 -[I
+
+(~rK/4)~]-“’
(K/4)*]-3'2j4. (9)
2.2 Inelastic scattering if there is enough energy in the relative motion of the excitons, one of the excitons can be excited through the collision into a higher energy state. The threshold energy for this transition to take place is Ef - Ei. Here we will carry out the calculation when the final states for the exciton are the 2s, 2p, 3s, 3p and 3d states. For the transitions ls+2.r and ls-+2p, the threshold energy is (32/9)(1 f a)-’ and the energy conservation relation becomes kj= kf-(16/9) o-l. For the transitions Is-*3s, ls-+3p and ls-+3d, the threshold energy is (96/25)( 1 + a)-’ and we have kj = k: - (48/25)0 -‘. J(i-*f) can be calculated for these transitions and the results are given by: J(ls -2s)
= (37’2/2g)K2{a~l + (~cTK/~)*]-~‘* - [l + (3K/8)7]-“2),
J(1s-r2p)
(10)
= - (35~2i/2’3~2)X(a[l+ (3aK/8)2]-S’2 + [l + (3K/8)2]-5’2}1
RESULTS AND DISCUSSION
We now wish to present the numerical results of our calculations of the elastic and inelastic exciton-exciton cross sections in two-dimensional quantum well systems. The integrals in eqns (9) and (15) cannot be performed analytically so we have performed numerical integrations to obtain the cross sections. In Fig. 2, we show the cross sections for elastic exciton-exciton scattering using the same values of the mass ratio u used in bulk semiconductors [6,7] and in our previous calculations of exciton-free carrier scattering [23]. Our results for the elastic case show that as a function of the relative energy of motion of the two excitons, k2, the cross sections initially increase rapidly but reach a peak value and then fall off with increasing energy beyond the peak. The peak value of the cross section and the value of the energy at which the peak occurs depend upon the mass ratio cr of the electron to the hole in the exciton. With increasing u, the cross section decreases until it reaches a minimum when o = 1 and then starts increasing again with further increase in e. The reason for the minimum in the elastic cross section is the same as in the case of exciton-free carrier scattering 1231.The peak value of the cross section shifts to lower energy as u increases and the peak value itself for elastic section cross decreases. The
(11) 1.8
J(ls-+k)
= (57’2)/253J)K2(~2[l+ 2(5aK/12)*][1 f (5aK/12)7]-7’z - [1+2(X/12)2] x [I + (sK/l2)7]-“2f,
J(Is-.~P)
= -(55’2i/2”239’2)K{a[l
(12) m
+ 2(5~K/12)~]
0.9
a x [l + pIJK/l2)2f-“* + [l -t 2(5K/12)2] x fl + (5K/l2)3-7”), J(ls+3d)
(13) 0.0 0
= - (57’2/21’~23”‘2)K2{u2[l
20
30
k'(a-2)
+ (saK/12)*]-“* - [1 + (sK/12)q-“9.
10
(14)
Fig. 2. The elastic cross sections for exciton-exciton scatterinn ” are shown for different mass ratios Q.
1194
YUAN-PING FENG
and
exciton-exciton scattering is larger than that calculated for electron-exciton elastic scattering in twodimensional quantum well systems using a similar approach to that used here [23] but is an order of that calculated for magnitude smaller than holeexciton elastic scattering. Elkomos et al. [6] have calculated the elastic cross section for exciton-exciton scattering in bulk semiconductors using the central field approximation [24] with and without the symmetry effect in the collision of the two identical partners [31]. In these calculations, the elastic cross section was generally found to decrease from some finite initial value at k = 0, decrease with increasing k2, reach a minimum value and then increase to a maximum value before starting to decrease with a further increase. in k’. The positions of the minima and maxima as a function of k2 were found to depend upon the value of cr. The cross section was obtained by calculating the phase shifts due to the effective interaction potential between the excitons. Their results taking account of the symmetry effect due to the fact that the collisions are between two identical excitons showed the same behavior as the results they obtained neglecting the symmetry effect, except that the maxima in the cross sections were slightly displaced towards higher values of k’. Our results have used the central field approximation but have neglected the symmetry effect and have been done within the Born approximation. Because the results for elastic exciton-exciton scattering in bulk semiconductors showed almost the same behavior with and without the symmetry effect, we feel that our neglect of symmetry effect for collisions between two-dimensional excitons will not lead to a drastically different behavior of the cross sections. Our results differ from those obtained for the bulk in that the cross section increases from zero at k = 0 and increases until it reaches a maximum at a value of k* which depends upon G before decreasing with a further increase in k’. Whether the difference between our results and those in the bulk for the elastic cross section is due to the confinement of the exciton or to our use of the Born approximation is not clear since the kind of behaviour we have found is very similar to that found in the bulk for D = 0.5. Both our calculations and those of Elkomos et al. [6,7] have neglected the effect of exchange between the electrons and holes of the two excitons. However, in an earlier paper in which Elkomos et al. [27] treated electron-exciton elastic scattering both including and neglecting the effect of exchange, they found that the effects of exchange were not important for the mass ratio u larger than 0.2 1, and, that their theoretical calculations in a bulk semiconductor having such a large mass ratio neglecting the exchange effect were in good agreement with experiment. Therefore, because the calculations would involve much more computer
with exchange time than with-
out the exchange effect, we feel justified that the use of the central field approximation without exchange
HAROLD
N.
SPECTOR
0
20
40
60
k’(a ?
Fig. 3. The inelastic cross sections for exciton+zxciton scattering for the transition 1s -+2s are shown for different mass ratios.
would still give satisfacto~ results for the cross sections in the two-dimensional case since it did so in the bulk case for the large values of 0. The inelastic cross section for exciton-exciton scattering corresponding to the transitions Is+23, ls+2p, ls-+3s, ls+3p and ls-+3d for the different values of u used in our calculations of the elastic cross section are shown in Figures 3-7. The general tendency for transitions to s and p states is for the inelastic cross sections to monotonically decrease with increasing relative energy of the two excitons, with some bumps appearing at certain values of the mass ratio 0 for transitions to s states. For transitions to d states, the cross sections initially rapidly increase with energy until they reach a peak value and then decrease with a further increase in energy. The peak value decreases and shifts to lower energies as the
a: b: c: d: e: f:
U = 0.1333 0=0.15 u = 0.178 0 = 0.21 D = 0.24 (7= 0.5
k2(a ‘1
Fig. 4. The inelastic cross sections for exciton-exciton scattering for the transition IS+3s are shown for different mass ratios.
Scattering of excitons by excitons 81
1.0
1 a: b: c: d: e: f:
1195
v=o.1333 u=O.15 e=0.178 ST= 0.21 o= 0.24 0=0.5
0.8-
h: cr=1.435
e: o = 0.24
0.6-
3 0 lo 6
?
0.4 -
0.0 0
20
0
60
40
20
40 k'(a
k2(am2)
60 *)
Fig. 5. The inelastic cross sections for exciton+xciton scattering for the transition 1s-+2p are shown for different mass ratios.
Fig. 7. The inelastic cross sections for exciton+xciton scattering for the transition 1s +3d are shown for different
mass ratio Q increases. The cross section decreases as a function of increasing mass ratio c until it reaches a minimum value when cr = 1 and then increases again. This is because the differential cross section given by eqn (6) vanishes when cr = 1, irrespective of the final state of the exciton when one of the excitons remains in the ground state. This is quite different from the results obtained for inelastic scattering of excitons by free carriers, where the minimum in the cross section as a function of the mass ratio occurred when the final state was an s or d state but not a p state. Presumably, different results would have been obtained if we considered scattering events in which both excitons were excited out of the ground state. The inelastic cross sections are at least an order of magnitude smaller than the elastic cross sections over
the range of energies of relative motion considered. The cross sections become smaller as the quantum number n of the final state increases which indicates that the cross sections for transitions to states with n 2 4 are probably negligible compared to those calculated here except possibly for the case of ionizing exciton-exciton collisions. The cross sections for transitions to p states are larger than those to either s or d states having the same quantum number n. Elkomos et al. [7] have calculated the inelastic cross section for exciton-exciton scattering in bulk semiconductors using the same approach we have used here. The analytical results they obtained for ]J(i-j)] differ from our results because of the difference between the two- and three-dimensional exciton wave functions. Some of the general tendencies are the same in the bulk and in two-dimensional systems such as the decrease in the cross section with increasing value of the principal quantum number n, the decrease in the cross section with increasing mass ratio Q with a minimum in the cross section when cr = 1, the existence of a threshold for inelastic scattering, which however is smaller in the bulk case than it is in the two-dimensional case because of the increase in the exciton energy because of confinement and the fact that for a final state of given n, the cross section is larger when the final state is a p state than when it is an s or d state. However, there is a difference between the bulk and two-dimensional results in that except for the transition ls-+3d, the cross section in the two-dimensional limit starts out at the threshold with a finite value and then decreases with increasing energy while in the three-dimensional limit, the cross section starts out from zero at the threshold and reaches a peak value before starting to decrease with increasing energy. Since both calculations have been done within the Born approximation [25], the differences between the results for the
ia: b: c: d: e: f:
,r= I,= ,r= CT= ,r= cr=
0.1333 0.15 0.178 0.21 0.24 0.5
h: CT= 1.435
z 2 e 20
40
60
k2W2)
Fig. 6. The inelastic cross sections for exciton-exciton scattering for the transition 1s+3p are shown for different mass ratios.
mass ratios.
YUAN-PING FENG and HAROLDN. SPECTOR
1196
cross sections are due to the two-dimensional nature of the excitons in a quantum well system. Finally, using the parameters of GaAs, we find that the cross sections for the scattering of light holes (a = 0.8) are much smaller than that for heavy holes (0 = 0.15). Therefore, in quantum well systems fabricated from GaAs, where the degeneracy of the light and heavy hole valence bands is removed by the confinement of the carriers, the light hole exciton will be scattered by other excitons to a lesser extent than heavy hole excitons. This is also similar to what we have calculated for exciton-free carrier scattering [23]. Acknowledgemenr-This work was submitted by one of us (Y.F.) in partial fulfilment of the requirements for the Ph.D. degree at Illinois Institute of Technology. This work was supported in part by GTE Laboratories. Waltham, MA 02254, U.S.A.
REFERENCES
1. Benoit A., La Guillaume 2. 3. 4. 5. 6. 7. 8. 9. IO.
C., Debever J. and Salvan F.. Phys. Rev. 177, 567 (1969). Magde D. and Mahr H., Phys Rev. Left. 24,890 (1970). Hvam J. M., Phys. Reu. E 4, 4459 (1971). Braun W., Bille J., Fischer T. and Huber G., Phys. Sratus Solidi (a) 58, 759 (1973). Levy R. and Grun J. B., Phvs. Status Solidi (a) 22, 1I (1974). Elkomos S. G. and Munschy G., J. Phys. Chem. Solids 42, 1 (1981). Elkomos S. G. and Munschy G., J. Phys. Chem. Solids 45, 345 (1984). Esaki L. and Tsu R., IBM J. Res. Deu. 14, 61 (1970). Stormer H. L.. Dinele R.. Gossard A. C.. Wieemann W. and Sturge M: D., ?. Vat. Sci. Technol. i6, 15’7 (1979). Drummond T. S., Morkoc H., Hess K. and Cho A. Y., J. appl. Phys. 32, 5231 (1981).
II. Paalenen M. A.. Tsui D. C., Gossard A. C. and Hwang J. C. M., Phys. Rev. B 29, 6003 (1984). 12. Dingle R.. Wiegmann W. and Henry C. H., Phys. Rev. Lett. 33, 827 (1974). D. A., Tsang W. T. and 13. Miller R. C., Kleinman Gossard A. C., Phys. Rev. B 24, 1134 (1982). 14. Miller D. A. B., Chemla D. S., Eilenberger D. J., Smith P. W.. Gossard A. C. and Tsane. W. T.. Aunl. . . Phw. Lzetr. 41, 679 (1982). 15. Chemla D. S., Miller D. A. B., Smith P. W., Gossard A. C. and Wiegmann W., IEEE J. Quant. Electron. QE-20, 265 (1984). S. and Altarelli M., Phvs. Reo. E 23, 2977 16. Satpathy (1981). 17. Bastard G.. Mendez E. E., Change L. 1. and Esaki L., Phys. R~u. B 26, 1974 (1982). M., Phys. Rev. B 28,4878 18. Shinozuka Y. and Matsuuara (1983); 29, 3719 (1984). 19. Greene R. L., Bajaj K. K. and Phelps D. E.. Phys. Ret>. B 29, 1807 (1984). 20. Lee J., Koteles E. S. and Vassell M. 0.. Phys. Rer B 33, 5512 (1986). 21. Spector H. N., Lee J. and Melman P., Phys. Reo. B 34, 2554 (1986). 22. Singh J. and Bajaj K. K.. Appl. Phys. Len. 44, 1075 (1984). 23. Feng Y. and Spector H. N., J. Phys. Chem. Solids 48, 593 (1987). 24. Massey H. S. W. and Moiseiwitsch B. L., Proc. R. Sot. (Land.) A205, 483 (1951). 25. Bates D. R. and Gritting G., Proc. Phys. Sot. 66, 961 (1953). 26. Elkomos S. G., Phys. Rev. B4, 3411 (1971). 27. Elkomos S. G. and Munschy G., J. Phys. Chem. Solids 38, 557 (1977). 28. Elkomos S. G. and Munschy G., J. Phys. Chem. Solids 40, 431 (1979). 29. Elkomos S. G., Phys. Rev. B 6, 3913 (1972). 30. Lee Y. C. and Lin D. L., Phys. Rev. B 19, 1982 (1979). 31. Massey H. S. W. and Mohr C. B. 0.. Proc. R. Sot. (Lond.) A141, 434 (1933).
0
SCATTERING OF EXCITONS SEMICONDUCTING QUANTUM
0022-3697187 1987 Pergamon
S3.00 + 0.00 Journals Ltd.
BY EXCITONS IN WELL STRUCTURES
YUAN-PING FENG and HAROLD N. SPECTOR Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, U.S.A. (Received
19 December
1986; accepted in revised form 15 April 1987)
Abstract-We have theoretically investigated the scattering of excitons by excitons in a two-dimensional semiconducting quantum well system. The scattering cross sections have been calculated using the Born approximation for both the elastic and inelastic scattering of the excitons by excitons. The threshold for inelastic scattering is increased over the value in a bulk semiconductor because of the enhancement of the exciton binding energy by its confinement. The behavior of the scattering cross section as a function of the energy of relative motion of the excitons is different than in the bulk and the cross section is a more sensitive function of the ratio of the electron and hole masses than in the bulk. Keyworh:
Excitons, quantum well structures, scattering cross section.
1. INTRODUCTION Exciton-exciton (Exe-Exe) collisions have been observed [l-5] in highly excited bulk semiconductors such as ZnO, CdS and CdSe. For the interpretation of these observations, the exciton-exciton elastic and inelastic cross sections are of interest and these cross sections have been calculated for different semiconductors in the bulk case [6, 71. With the development of crystal growth techniques such as molecular beam epitaxy (MBE), it has become possible to grow high quality semiconducting quantum well and superlattice structures [8]. These structures have become of great interest technically because of their enhanced mobilities when they are modulation doped [9-l 1] and the appearance of strong room temperature exciton peaks [12-151 when the width of the layer in which the exciton is confined is less than its Bohr radius. Because of the great interest in the optical properties of these structures, theoretical calculations have been performed of the excitonic contribution to the optical absorption [16] and the effect of the confinement on the exciton binding energy [17-191. In addition to the effect of the carrier confinement on the binding energy of excitons in quantum well structures, there is interest in calculating and measuring the exciton linewidth. Calculations have been performed which take into account the effect of scattering from acoustic [20] and optical phonons [21] and also alloy scattering [22] on the exciton linewidth in quantum well structures. However, there are also other possible scattering mechanisms which might play a role in determining the exciton linewidth which have not as yet been considered such as the elastic and inelastic scattering of excitons from other excitons and from free carriers. In a recent work [23], we theoretically investigated the scattering of excitons by free electrons and holes in a two-dimensional semiconducting quantum well system. We found that PCS
48,12--c
the cross sections for such scattering mechanisms were appreciable for certain ranges of the energy of the relative motion of the excitons and free carriers, and, that there were some interesting features which appeared as the result of the confinement in the quantum well structure. In the present work, we wish to present theoretical calculations of the cross sections for exciton-exciton elastic and inelastic scattering in quantum well systems. In our calculations, we will assume a twodimensional gas of excitons interacting with each other and use the central field approximation [24] and the Born approximation [25] to calculate the cross sections due to collisions between the excitons for the simple transitions Exc(ls/B) + Exe (Is/A)-+Exc (Is/B) + Exc(m/A) where m = 1s (elastic scattering), 2s, 2p, 3s, 3p and 3d (inelastic scattering). The use of the two-dimensional model should be valid for very narrow wells and should show whether the confinement of the excitons in a quantum well system has an appreciable effect on their scattering cross sections. The use of the two-dimensional model also enables us to use the exact two-dimensional wave functions and energy eigenvalues for the excitons instead of those approximate wave functions and energies obtained using a variational approach in quantum wells of finite width. The cross sections for different semiconductors are functions of the ratio of the electron and hole effective masses, Q = me/m,. In our calculations the values of these masses which are used are those for the bulk semiconductor. The cross sections are given for different available values of 0 for the semiconductors ZnO [6,26-281, CdS [26-281, CdSe [4,26,27,29] as well as GaAs [ 191.The reason for the calculations being done using the parameters of GaAs is that this is the prototype material for most quantum well structures, while the calculations using the parameters of the other semiconductors are per-
1191
YUAN-PINGFENGand
1192
formed in order to facilitate the comparison with the calculations done for these materials in the bulk case [6,7]. This mechanism might play an important role in determining the exciton linewidth in experiments in which a high density of excitons is optically generated. 2. EXCITON-EXCITON COLLISIONS IN TWO DIMENSIONS The energies and wave functions of purely twodimensional excitons are given by [30] E, = R/(n - ;)*, n = 1, 2, 3
(1)
and
where L;(p) is the associated Laguerre polynomial, the exciton Bohr radius and Rydberg energy are defined as a = (ch2/pe2) and R = (e2/2a), respectively where t is the dielectric constant of the semiconductor and ~1= m,m,,/(m, + mh) is the reduced mass of the exciton. The principal quantum number n can take on any positive integer value while the angular momentum quantum number m = 0, + 1, f 2, +3, . . . , +(n - 1) for a given n. For convenience we adopt a notation for the exciton states similar to that used in the three-dimensional case and call the states 1s (n= 1, m = 0), 2s (n = 2, m = 0), 2p (n = 2, m = *l), 3s (n = 3, m = 0),3p (n = 3, m = * l), 3d (n =3, m = +2), etc. Consider an encounter between an exciton A, comprised of a hole a and an electron 1 and another exciton B, which consists of a hole b and an electron 2. From our previous work on exciton-free carrier scattering in a two-dimensional quantum well system, the cross section, within the Born approximation [25], associated with the reaction in which exciton A is excited from its ground state to an excited state while exciton B remains in its ground state is Q,+=
s2nL,@W,
HAROLDN. SPECTOR where ~,(r,,) and ,q(r,,) are the wave functions of the electron 1 bound to the hole a before and after the collision, x,(rhZ) is the ground state wave function of the electron 2 bound to the hole b, R is the relative position vector of the exciton, V(r,, ,rh2, R) is the interaction potential between the excitons and M is the reduced mass of the system. The reduced mass of the system is given by M = fm,(1 + 0 -I), where the mass ratio is defined as D = me/m,. Also, here k, = (MVJh), k,= (Myf/h), where V, and V, are the initial and final velocities of relative motion between the two excitons and K = k, - K,. conservation of energy we have From k_f= kf - (2M/m,)(EfE,) where E, and E, are the sum of the internal energies of the two excitons before and after the collision. It is convenient to use the Rydberg energy R. = (m,e4/2c2h2) and Bohr radius a, = (th*/e’m,) as the basic units of energy and length, respectively. Using these units, we have kj=k;--#+a -‘)(E,-E,) where E, = (-S/l + a). For the transitions Is -+2s and Is -+2p, E,= t-40/9 (1 + a)] while for the transitions 1s -+3s, Is +3p and Is +3d, Ef = [ - 104/25 (1 + a)]. For elastic collisions, Is-+ls, Ef= E, and k, = kf. The interaction potential between the excitons, using the central field approximation. is given by V(r,, ,r,,,R’l = (e*/t)(rG’
- r,z’ - rh’ + r fi’), (5)
where rxy is the distance between particle x and particle y, which can be expressed in terms of ro, ,rhZ and R as shown in Fig. 1. Here R is the relative position vector of the two excitons. Using the exciton wave functions given in eqn (2) and completing the integrations over R and rh2, we obtain the following expression for the different cross section: I,,/(Q) =;a(1 + a~‘)2(k,K2)-‘l{[l + (aK/4)*]-“’ -[l
+ (K/4)‘]-3’2} J(i+jy,
(6)
where
-expW
+a)~‘Kr,,I}~;(r,,)~,(r~,).
(7)
(3)
0
where 0 is the scattering angle in the center of mass frame of the excitons. The differential cross section l,_,(e) is given by I,,f(0)
= (M2/2xh2ki)ljdrll, drb,dR between the vectors rob, r12, ry2, rb, , gives the relative positions of the and holes in the two excitons and the positions of their centers of mass.
Fig. 1. The relationship
x expiKR k(‘b2)Izi2?
V@,,
) rb27
W
xf
(cl
r,, , rb2 and R which
hi@,,)
electrons
(4)
Scattering of excitons by excitons 2.1 Elastic scattering For elastic collisions, both the excitons remain in the ground state after the collision. Therefore k,= ki and K = 2ki jsin(8/2)] where B is the scattering angle. J(ls-+ls) is readily calculated and is given by
Here K is defined by K2 = kj+ kf- 2kik,cos@, which is a function of the scattering angle. From eqns {10~~4), the exciton inelastic cross section Q is given in terms of a, as Q,,=
J(ls+ls)
= [l +((TK/4)q-3’7-
[l +(K/4)2]-“2.
1193
n(1 + a-‘)2k;‘l
(8)
‘d&C-*{[l
+ (~rK,‘4)7-~‘~
s0 -
The total elastic cross section in units of a, is given
[1 + (K/4)2]-3”) J(i-q-p.
(15)
by 3.
Q*=n(I
+~-‘)~k;’
‘d&C-‘{[I s0 -[I
+
+(~rK/4)~]-“’
(K/4)*]-3'2j4. (9)
2.2 Inelastic scattering if there is enough energy in the relative motion of the excitons, one of the excitons can be excited through the collision into a higher energy state. The threshold energy for this transition to take place is Ef - Ei. Here we will carry out the calculation when the final states for the exciton are the 2s, 2p, 3s, 3p and 3d states. For the transitions ls+2.r and ls-+2p, the threshold energy is (32/9)(1 f a)-’ and the energy conservation relation becomes kj= kf-(16/9) o-l. For the transitions Is-*3s, ls-+3p and ls-+3d, the threshold energy is (96/25)( 1 + a)-’ and we have kj = k: - (48/25)0 -‘. J(i-*f) can be calculated for these transitions and the results are given by: J(ls -2s)
= (37’2/2g)K2{a~l + (~cTK/~)*]-~‘* - [l + (3K/8)7]-“2),
J(1s-r2p)
(10)
= - (35~2i/2’3~2)X(a[l+ (3aK/8)2]-S’2 + [l + (3K/8)2]-5’2}1
RESULTS AND DISCUSSION
We now wish to present the numerical results of our calculations of the elastic and inelastic exciton-exciton cross sections in two-dimensional quantum well systems. The integrals in eqns (9) and (15) cannot be performed analytically so we have performed numerical integrations to obtain the cross sections. In Fig. 2, we show the cross sections for elastic exciton-exciton scattering using the same values of the mass ratio u used in bulk semiconductors [6,7] and in our previous calculations of exciton-free carrier scattering [23]. Our results for the elastic case show that as a function of the relative energy of motion of the two excitons, k2, the cross sections initially increase rapidly but reach a peak value and then fall off with increasing energy beyond the peak. The peak value of the cross section and the value of the energy at which the peak occurs depend upon the mass ratio cr of the electron to the hole in the exciton. With increasing u, the cross section decreases until it reaches a minimum when o = 1 and then starts increasing again with further increase in e. The reason for the minimum in the elastic cross section is the same as in the case of exciton-free carrier scattering 1231.The peak value of the cross section shifts to lower energy as u increases and the peak value itself for elastic section cross decreases. The
(11) 1.8
J(ls-+k)
= (57’2)/253J)K2(~2[l+ 2(5aK/12)*][1 f (5aK/12)7]-7’z - [1+2(X/12)2] x [I + (sK/l2)7]-“2f,
J(Is-.~P)
= -(55’2i/2”239’2)K{a[l
(12) m
+ 2(5~K/12)~]
0.9
a x [l + pIJK/l2)2f-“* + [l -t 2(5K/12)2] x fl + (5K/l2)3-7”), J(ls+3d)
(13) 0.0 0
= - (57’2/21’~23”‘2)K2{u2[l
20
30
k'(a-2)
+ (saK/12)*]-“* - [1 + (sK/12)q-“9.
10
(14)
Fig. 2. The elastic cross sections for exciton-exciton scatterinn ” are shown for different mass ratios Q.
1194
YUAN-PING FENG
and
exciton-exciton scattering is larger than that calculated for electron-exciton elastic scattering in twodimensional quantum well systems using a similar approach to that used here [23] but is an order of that calculated for magnitude smaller than holeexciton elastic scattering. Elkomos et al. [6] have calculated the elastic cross section for exciton-exciton scattering in bulk semiconductors using the central field approximation [24] with and without the symmetry effect in the collision of the two identical partners [31]. In these calculations, the elastic cross section was generally found to decrease from some finite initial value at k = 0, decrease with increasing k2, reach a minimum value and then increase to a maximum value before starting to decrease with a further increase. in k’. The positions of the minima and maxima as a function of k2 were found to depend upon the value of cr. The cross section was obtained by calculating the phase shifts due to the effective interaction potential between the excitons. Their results taking account of the symmetry effect due to the fact that the collisions are between two identical excitons showed the same behavior as the results they obtained neglecting the symmetry effect, except that the maxima in the cross sections were slightly displaced towards higher values of k’. Our results have used the central field approximation but have neglected the symmetry effect and have been done within the Born approximation. Because the results for elastic exciton-exciton scattering in bulk semiconductors showed almost the same behavior with and without the symmetry effect, we feel that our neglect of symmetry effect for collisions between two-dimensional excitons will not lead to a drastically different behavior of the cross sections. Our results differ from those obtained for the bulk in that the cross section increases from zero at k = 0 and increases until it reaches a maximum at a value of k* which depends upon G before decreasing with a further increase in k’. Whether the difference between our results and those in the bulk for the elastic cross section is due to the confinement of the exciton or to our use of the Born approximation is not clear since the kind of behaviour we have found is very similar to that found in the bulk for D = 0.5. Both our calculations and those of Elkomos et al. [6,7] have neglected the effect of exchange between the electrons and holes of the two excitons. However, in an earlier paper in which Elkomos et al. [27] treated electron-exciton elastic scattering both including and neglecting the effect of exchange, they found that the effects of exchange were not important for the mass ratio u larger than 0.2 1, and, that their theoretical calculations in a bulk semiconductor having such a large mass ratio neglecting the exchange effect were in good agreement with experiment. Therefore, because the calculations would involve much more computer
with exchange time than with-
out the exchange effect, we feel justified that the use of the central field approximation without exchange
HAROLD
N.
SPECTOR
0
20
40
60
k’(a ?
Fig. 3. The inelastic cross sections for exciton+zxciton scattering for the transition 1s -+2s are shown for different mass ratios.
would still give satisfacto~ results for the cross sections in the two-dimensional case since it did so in the bulk case for the large values of 0. The inelastic cross section for exciton-exciton scattering corresponding to the transitions Is+23, ls+2p, ls-+3s, ls+3p and ls-+3d for the different values of u used in our calculations of the elastic cross section are shown in Figures 3-7. The general tendency for transitions to s and p states is for the inelastic cross sections to monotonically decrease with increasing relative energy of the two excitons, with some bumps appearing at certain values of the mass ratio 0 for transitions to s states. For transitions to d states, the cross sections initially rapidly increase with energy until they reach a peak value and then decrease with a further increase in energy. The peak value decreases and shifts to lower energies as the
a: b: c: d: e: f:
U = 0.1333 0=0.15 u = 0.178 0 = 0.21 D = 0.24 (7= 0.5
k2(a ‘1
Fig. 4. The inelastic cross sections for exciton-exciton scattering for the transition IS+3s are shown for different mass ratios.
Scattering of excitons by excitons 81
1.0
1 a: b: c: d: e: f:
1195
v=o.1333 u=O.15 e=0.178 ST= 0.21 o= 0.24 0=0.5
0.8-
h: cr=1.435
e: o = 0.24
0.6-
3 0 lo 6
?
0.4 -
0.0 0
20
0
60
40
20
40 k'(a
k2(am2)
60 *)
Fig. 5. The inelastic cross sections for exciton+xciton scattering for the transition 1s-+2p are shown for different mass ratios.
Fig. 7. The inelastic cross sections for exciton+xciton scattering for the transition 1s +3d are shown for different
mass ratio Q increases. The cross section decreases as a function of increasing mass ratio c until it reaches a minimum value when cr = 1 and then increases again. This is because the differential cross section given by eqn (6) vanishes when cr = 1, irrespective of the final state of the exciton when one of the excitons remains in the ground state. This is quite different from the results obtained for inelastic scattering of excitons by free carriers, where the minimum in the cross section as a function of the mass ratio occurred when the final state was an s or d state but not a p state. Presumably, different results would have been obtained if we considered scattering events in which both excitons were excited out of the ground state. The inelastic cross sections are at least an order of magnitude smaller than the elastic cross sections over
the range of energies of relative motion considered. The cross sections become smaller as the quantum number n of the final state increases which indicates that the cross sections for transitions to states with n 2 4 are probably negligible compared to those calculated here except possibly for the case of ionizing exciton-exciton collisions. The cross sections for transitions to p states are larger than those to either s or d states having the same quantum number n. Elkomos et al. [7] have calculated the inelastic cross section for exciton-exciton scattering in bulk semiconductors using the same approach we have used here. The analytical results they obtained for ]J(i-j)] differ from our results because of the difference between the two- and three-dimensional exciton wave functions. Some of the general tendencies are the same in the bulk and in two-dimensional systems such as the decrease in the cross section with increasing value of the principal quantum number n, the decrease in the cross section with increasing mass ratio Q with a minimum in the cross section when cr = 1, the existence of a threshold for inelastic scattering, which however is smaller in the bulk case than it is in the two-dimensional case because of the increase in the exciton energy because of confinement and the fact that for a final state of given n, the cross section is larger when the final state is a p state than when it is an s or d state. However, there is a difference between the bulk and two-dimensional results in that except for the transition ls-+3d, the cross section in the two-dimensional limit starts out at the threshold with a finite value and then decreases with increasing energy while in the three-dimensional limit, the cross section starts out from zero at the threshold and reaches a peak value before starting to decrease with increasing energy. Since both calculations have been done within the Born approximation [25], the differences between the results for the
ia: b: c: d: e: f:
,r= I,= ,r= CT= ,r= cr=
0.1333 0.15 0.178 0.21 0.24 0.5
h: CT= 1.435
z 2 e 20
40
60
k2W2)
Fig. 6. The inelastic cross sections for exciton-exciton scattering for the transition 1s+3p are shown for different mass ratios.
mass ratios.
YUAN-PING FENG and HAROLDN. SPECTOR
1196
cross sections are due to the two-dimensional nature of the excitons in a quantum well system. Finally, using the parameters of GaAs, we find that the cross sections for the scattering of light holes (a = 0.8) are much smaller than that for heavy holes (0 = 0.15). Therefore, in quantum well systems fabricated from GaAs, where the degeneracy of the light and heavy hole valence bands is removed by the confinement of the carriers, the light hole exciton will be scattered by other excitons to a lesser extent than heavy hole excitons. This is also similar to what we have calculated for exciton-free carrier scattering [23]. Acknowledgemenr-This work was submitted by one of us (Y.F.) in partial fulfilment of the requirements for the Ph.D. degree at Illinois Institute of Technology. This work was supported in part by GTE Laboratories. Waltham, MA 02254, U.S.A.
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