- Email: [email protected]
Energy transfer in semiconductors
SESSION
G:
the photoconductivity peak at 500 rnp and the peak of the yellow band, we fmd that a monochromatic radiation of 575 rnp with the same intensity as that of the yellow luminescence will give the same photoconductivity. Indeed, 575 rnp is exactly the wavelength of the band maximum of the yellow luminescence. Measurements in the region of liquid-helium temperature have recently been performed. With the crystals investigated up to now no measurable diffusion of photoconductivity could be observed. This may be also an indirect argument for the light scattering theory, as in this case no excitation of photoconductivity with direct light in the long wavelength tail of the absorption band could be found, and therefore, neither the luminescent nor the scattered light could produce photoconductivity in the not directly illuminated part of the crystal. Summarizing the results of our measurements, we may state that there is no cogent reason to assume exciton diffusion in order to explain the propagation of photoconductivity in photo-
J. Phys.
Chem. Solids
Pergamon
ENERGY
G.4
conductors. On the other hand, it cannot be concluded from our investigations that exciton diffusion in the crystals does not occur at all, but if there is any energy transport by excitons, it will be masked by the much more effective mechanism of scattering and reabsorption of luminescent light. REFERENCES 1. BALKANSKI M. and BROSER I., paper given at International Conference on Semiconductors and Garmisch-Partenkirchen, Germany, Phosphors, August 2%September 1 (1956). 2. DIEMER G. and HOOCENSTRAATENW., J. Phys. Chem. Solids 2, 119 (1957). 3. BALKANSKIM. and BROSER I., 2. Elektrochem. 61, 715 (1957). 4. BROSER I. and BROSER-WARMINSKYR., J. Phys. Chem. Solids 6, 386 (1958). 5. BALKANSKIM. and WAL~RON R. D., Tech. Rep. 123, Laboratory for Insulation Research. Massachusetts Institute of Technology, Cambridge, Mass., U.S.A. (Nov. 1957). 6. BROSER I. and WARMINSKY R., X Naturf. 6a, 85 (1951). 7. DIEMER G., Private communication.
Press 1959. Vol. 8. pp. 179-181.
TRANSFER
179
EXCITONS
Printed in Great Britain
IN SEMICONDUCTORS
M. BALKANSKI Laboratoire
de Physique,
Ecole Norrnale SupCrieure, Paris, France
an exciton produced in a semiconductor annihilates after propagation by returning in its fundamental state emitting a photon, this photon can be reabsorbed creating a new exciton if its frequency corresponds to the resonance energy. The processes can repeat themselves as long as the photon emission takes place in the interior of the crystal at a few h beneath the surface. The energy propagates by alternative creation and annihilation of excitons coupled by the radiation field until the excited state reaches the surface where the photon is emitted out of the system and detected. This mechanism has as a model the multiple diffusion demonstrated for the first time in the case of gases WHEN
in the Laboratory of Prof. Kastler by J. E. Blamont. The idea for introducing multiple diffusion in the treatment of exciton diffusion was given by P. Aigrain and is based on two experimental facts observed in cadmium sulfide. (1) Migration of the excitation in the solid for a large distance from the point where it has been created. (2) The lifetime of the macroscopically observed excitation is much longer than the lifetime of a single excited state. The mechanism we are proposing to describe the energy transfer at great distances includes two successive steps repeated alternately.
180
SESSION
G:
(a) Absorption of light and creation of an electron-hole pair strongly interacting between them whose zeroth order excited wave function can be constructed as(l)
@c---K, k) = _Ic(-)pp’I’(k-K, v’(~N)!
k)Px
p
where @(k-K,
k) = ufi,(q) . . . z@(q) . . . u~,~(Q).
PCk,(TN+l)...U~_K(rN+Z)...UR~(T2N) P means the permutation of order (2N)!, uk(r) is the Bloch function in the valence band and w?(r) that in the conduction band. For each spin state we have N’wave functions #(k-K, k) defined by vectors k-K, k representing respectively the Bloch function of the valence band from which an electron is missing and the Bloch function of the conduction band to which an electron has been raised. From these N2wave functions of the zeroth order we get, by a unitary transformation, N2 new wave functions :
1 y(KyR)=
--c dN
exp[-ikR]Y(k-K,
k).
k
Each of these represents a so-called excitation wave which corresponds to an electron-hole complex separated by R in space and moving with a wave vector K. Since we do not wish to consider phonons in this treatment we will take K = 0. (b) The excited electron returning to its ground state emits a photon. This corresponds to a coupling of the excitation wave with the radiation field. The emitted photon can be reabsorbed regenerating the excitation wave. This second step may seem similar to the mechanism evoked in the theory(s) of sensitized luminescence but actually it differs not only phenomenologically but also in theoretical treatment. The excitation we are considering is a property of the pure lattice and the transitions involve coupling between the excitation waves and the radiation field. The propagation of the excitation energy can be considered as exciton diffusion if the two following conditions are satisfied : (i) The excitation wave conserves the wave vector IKI over the whole process
EXCITONS
(ii) The multiple diffusion of the photon by the excitation waves is a coherent process. The coherence being defined as in the case of the multiple diffusion involved in optical resonance, which for a gas is as follows : If the state of an atom I after absorption of a photon is described by the wave function #1(t), this excitation can be re-emitted after a time 7 and reabsorbed by a second atom. The wave function @I(t) which will describe the second excited state after reabsorption can be considered in two different ways :(3) (1) If @I(t) is a proper atomic state, i.e., if between the coefficients of the expansion of the wave functions of the first atom Cf and the second atom C’d only an intensity relationship exists (in jC’i12 and IC’f12) then the diffusion is incoherent. The processes I and II are distinct and independent. We shall ignore such processes. (2) If @1(t) takes the form #11(t) = z: C,‘( t)@1 6 where the C’i are determined in phase and magnitude by the Ci of the first atom then the diffusion is coherent. In the case where Ci and C’a are equal in phase and magnitude one can say that the two atoms are exchanging their wave functions and in this case the notion of coherent multiple diffusion covers completely that of exciton propagation. A complete theoretical treatment of the multiple diffusion in the case of gases is carried out by J. P. Barrat which will be published soon in J. Phys. Radium. The method used in the theoretical treatment of multiple diffusion involves time dependent perturbation theory. The system first considered is set up from N atoms and the radiation field. The states taken into account are (1) the ground states plus a photon (2) one excited present.
state
(exciton)
and no photon
The system Hamiltonian is built up from a diagonal part 3?0 whose matrix elements are the energies cO and +, of the levels, and an off-diagonal part Zint which has non-zero matrix elements only between the ground state and the excited state. eF?Fint is the part of the radiation field responsible for the dipolar electronic transitions between the ground and excited states with absorption and
SESSION
emission of a photon. Only this part of the radiation field is quantised and all the possible states of the system for which the principal of energy conservation shall not be respected would not be accounted for. The exciton of energy Ek migrates through the crystal with a group velocity vk = l/A grad& _?3,+and as the matrix elements of 3?int depend on the position of the excited state, the matrix elements of Xrnt are consequently time dependent. We are considering in this treatment, in addition to the pure radiation field, the contribution of its interaction with particles. As we are considering the interaction as a perturbation, it is suitable to introduce the interaction representation which has the advantage of removing the trivial time dependence of the free fields and of introducing the time variation of the state vector due only to the interaction. Thus the computed radiative emission by recombination of an exciton is:
Ib,, km &d=q2_
1%121
181
G: EXCITONS
xQ*
For a small or negligible exciton-phonon interaction, i.e., when the effective mass of the exciton is not too large, the emitted optical line has the Lorentz form displaced by Doppler effect (connected with the term k*vk due to the velocity of propagation of the excitation wave). The breadth of the line is y. The exciton being propagated in a crystalline medium the right hand member is multiplied by the square of the ratio of the electrical field within the crystal and that of the exciton which is not homogeneous and depends on an effective dielectric constant, K = l/f(r), function of the radius of the electronic orbit T. For a strong phonon interaction the lattice vibrations which interfere with the radiation field have to be introduced as a probability function b,(t) of the states lp) which should be normalized on the atomic scale. The exciton effective mass will then be large and the absorbed and emitted line will have a Gaussian shape. The experimental study by optical and magnetic resonance will give information as to the degree of exciton-phonon interaction. This should also give information on
exciton propagation or degree of binding of the excited state. Considering two excitation waves one of which annihilates when the other is created by passage of the photon from one to the other one finds a probability of passage of the photon from exciton 1 to exciton 2 or l/~a~Rrs~. Extending this treatment to p passages of a photon after successive creation and annihilation of n excitons one finds an apparent lifetime T=--=
1
7
y(l-RX)
l-ax
>7
7 is the proper lifetime of the individual exciton. T is the macroscopic value measurable by magnetic resonance. On the other hand, the total probability of finding one exciton is Trace p(t) = e-y(l-z)t with X = 1 -&+“n. The time of trapping the radiation in the crystal is then
1
7
7’=-=__.
y(l--x)
l--x
This shows that in the solids where a reabsorption of the emitted photon is possible one should find the apparent lifetime of the exciton greater than that given by simple estimation on one excited state. Because this multiple process is possible it is also natural to find great diffusion length. This explains the known experimental results. Further experimentation on optical and magnetic resonance should give data on the velocity of exciton propagation and presumably on their effective mass as well. When it will become possible to resolve the lines from electron spin resonance one will become aware of the fine structure of the exciton. REFERENCES TAKEUTI Y., Prog. Theor. Phys. 18, 421 (1957). 2. DEXTER D. L., J. Chem. Phys. 21, 836 (1953). 3. GUIOCHON M. A., BLAMONT J. E. and BROSSEL J.,
1.
C.R. Acad. Sci., Paris. Phys. 18, 99 (1957).
243,
1859 (1956) and J.
G:
the photoconductivity peak at 500 rnp and the peak of the yellow band, we fmd that a monochromatic radiation of 575 rnp with the same intensity as that of the yellow luminescence will give the same photoconductivity. Indeed, 575 rnp is exactly the wavelength of the band maximum of the yellow luminescence. Measurements in the region of liquid-helium temperature have recently been performed. With the crystals investigated up to now no measurable diffusion of photoconductivity could be observed. This may be also an indirect argument for the light scattering theory, as in this case no excitation of photoconductivity with direct light in the long wavelength tail of the absorption band could be found, and therefore, neither the luminescent nor the scattered light could produce photoconductivity in the not directly illuminated part of the crystal. Summarizing the results of our measurements, we may state that there is no cogent reason to assume exciton diffusion in order to explain the propagation of photoconductivity in photo-
J. Phys.
Chem. Solids
Pergamon
ENERGY
G.4
conductors. On the other hand, it cannot be concluded from our investigations that exciton diffusion in the crystals does not occur at all, but if there is any energy transport by excitons, it will be masked by the much more effective mechanism of scattering and reabsorption of luminescent light. REFERENCES 1. BALKANSKI M. and BROSER I., paper given at International Conference on Semiconductors and Garmisch-Partenkirchen, Germany, Phosphors, August 2%September 1 (1956). 2. DIEMER G. and HOOCENSTRAATENW., J. Phys. Chem. Solids 2, 119 (1957). 3. BALKANSKIM. and BROSER I., 2. Elektrochem. 61, 715 (1957). 4. BROSER I. and BROSER-WARMINSKYR., J. Phys. Chem. Solids 6, 386 (1958). 5. BALKANSKIM. and WAL~RON R. D., Tech. Rep. 123, Laboratory for Insulation Research. Massachusetts Institute of Technology, Cambridge, Mass., U.S.A. (Nov. 1957). 6. BROSER I. and WARMINSKY R., X Naturf. 6a, 85 (1951). 7. DIEMER G., Private communication.
Press 1959. Vol. 8. pp. 179-181.
TRANSFER
179
EXCITONS
Printed in Great Britain
IN SEMICONDUCTORS
M. BALKANSKI Laboratoire
de Physique,
Ecole Norrnale SupCrieure, Paris, France
an exciton produced in a semiconductor annihilates after propagation by returning in its fundamental state emitting a photon, this photon can be reabsorbed creating a new exciton if its frequency corresponds to the resonance energy. The processes can repeat themselves as long as the photon emission takes place in the interior of the crystal at a few h beneath the surface. The energy propagates by alternative creation and annihilation of excitons coupled by the radiation field until the excited state reaches the surface where the photon is emitted out of the system and detected. This mechanism has as a model the multiple diffusion demonstrated for the first time in the case of gases WHEN
in the Laboratory of Prof. Kastler by J. E. Blamont. The idea for introducing multiple diffusion in the treatment of exciton diffusion was given by P. Aigrain and is based on two experimental facts observed in cadmium sulfide. (1) Migration of the excitation in the solid for a large distance from the point where it has been created. (2) The lifetime of the macroscopically observed excitation is much longer than the lifetime of a single excited state. The mechanism we are proposing to describe the energy transfer at great distances includes two successive steps repeated alternately.
180
SESSION
G:
(a) Absorption of light and creation of an electron-hole pair strongly interacting between them whose zeroth order excited wave function can be constructed as(l)
@c---K, k) = _Ic(-)pp’I’(k-K, v’(~N)!
k)Px
p
where @(k-K,
k) = ufi,(q) . . . z@(q) . . . u~,~(Q).
PCk,(TN+l)...U~_K(rN+Z)...UR~(T2N) P means the permutation of order (2N)!, uk(r) is the Bloch function in the valence band and w?(r) that in the conduction band. For each spin state we have N’wave functions #(k-K, k) defined by vectors k-K, k representing respectively the Bloch function of the valence band from which an electron is missing and the Bloch function of the conduction band to which an electron has been raised. From these N2wave functions of the zeroth order we get, by a unitary transformation, N2 new wave functions :
1 y(KyR)=
--c dN
exp[-ikR]Y(k-K,
k).
k
Each of these represents a so-called excitation wave which corresponds to an electron-hole complex separated by R in space and moving with a wave vector K. Since we do not wish to consider phonons in this treatment we will take K = 0. (b) The excited electron returning to its ground state emits a photon. This corresponds to a coupling of the excitation wave with the radiation field. The emitted photon can be reabsorbed regenerating the excitation wave. This second step may seem similar to the mechanism evoked in the theory(s) of sensitized luminescence but actually it differs not only phenomenologically but also in theoretical treatment. The excitation we are considering is a property of the pure lattice and the transitions involve coupling between the excitation waves and the radiation field. The propagation of the excitation energy can be considered as exciton diffusion if the two following conditions are satisfied : (i) The excitation wave conserves the wave vector IKI over the whole process
EXCITONS
(ii) The multiple diffusion of the photon by the excitation waves is a coherent process. The coherence being defined as in the case of the multiple diffusion involved in optical resonance, which for a gas is as follows : If the state of an atom I after absorption of a photon is described by the wave function #1(t), this excitation can be re-emitted after a time 7 and reabsorbed by a second atom. The wave function @I(t) which will describe the second excited state after reabsorption can be considered in two different ways :(3) (1) If @I(t) is a proper atomic state, i.e., if between the coefficients of the expansion of the wave functions of the first atom Cf and the second atom C’d only an intensity relationship exists (in jC’i12 and IC’f12) then the diffusion is incoherent. The processes I and II are distinct and independent. We shall ignore such processes. (2) If @1(t) takes the form #11(t) = z: C,‘( t)@1 6 where the C’i are determined in phase and magnitude by the Ci of the first atom then the diffusion is coherent. In the case where Ci and C’a are equal in phase and magnitude one can say that the two atoms are exchanging their wave functions and in this case the notion of coherent multiple diffusion covers completely that of exciton propagation. A complete theoretical treatment of the multiple diffusion in the case of gases is carried out by J. P. Barrat which will be published soon in J. Phys. Radium. The method used in the theoretical treatment of multiple diffusion involves time dependent perturbation theory. The system first considered is set up from N atoms and the radiation field. The states taken into account are (1) the ground states plus a photon (2) one excited present.
state
(exciton)
and no photon
The system Hamiltonian is built up from a diagonal part 3?0 whose matrix elements are the energies cO and +, of the levels, and an off-diagonal part Zint which has non-zero matrix elements only between the ground state and the excited state. eF?Fint is the part of the radiation field responsible for the dipolar electronic transitions between the ground and excited states with absorption and
SESSION
emission of a photon. Only this part of the radiation field is quantised and all the possible states of the system for which the principal of energy conservation shall not be respected would not be accounted for. The exciton of energy Ek migrates through the crystal with a group velocity vk = l/A grad& _?3,+and as the matrix elements of 3?int depend on the position of the excited state, the matrix elements of Xrnt are consequently time dependent. We are considering in this treatment, in addition to the pure radiation field, the contribution of its interaction with particles. As we are considering the interaction as a perturbation, it is suitable to introduce the interaction representation which has the advantage of removing the trivial time dependence of the free fields and of introducing the time variation of the state vector due only to the interaction. Thus the computed radiative emission by recombination of an exciton is:
Ib,, km &d=q2_
1%121
181
G: EXCITONS
xQ*
For a small or negligible exciton-phonon interaction, i.e., when the effective mass of the exciton is not too large, the emitted optical line has the Lorentz form displaced by Doppler effect (connected with the term k*vk due to the velocity of propagation of the excitation wave). The breadth of the line is y. The exciton being propagated in a crystalline medium the right hand member is multiplied by the square of the ratio of the electrical field within the crystal and that of the exciton which is not homogeneous and depends on an effective dielectric constant, K = l/f(r), function of the radius of the electronic orbit T. For a strong phonon interaction the lattice vibrations which interfere with the radiation field have to be introduced as a probability function b,(t) of the states lp) which should be normalized on the atomic scale. The exciton effective mass will then be large and the absorbed and emitted line will have a Gaussian shape. The experimental study by optical and magnetic resonance will give information as to the degree of exciton-phonon interaction. This should also give information on
exciton propagation or degree of binding of the excited state. Considering two excitation waves one of which annihilates when the other is created by passage of the photon from one to the other one finds a probability of passage of the photon from exciton 1 to exciton 2 or l/~a~Rrs~. Extending this treatment to p passages of a photon after successive creation and annihilation of n excitons one finds an apparent lifetime T=--=
1
7
y(l-RX)
l-ax
>7
7 is the proper lifetime of the individual exciton. T is the macroscopic value measurable by magnetic resonance. On the other hand, the total probability of finding one exciton is Trace p(t) = e-y(l-z)t with X = 1 -&+“n. The time of trapping the radiation in the crystal is then
1
7
7’=-=__.
y(l--x)
l--x
This shows that in the solids where a reabsorption of the emitted photon is possible one should find the apparent lifetime of the exciton greater than that given by simple estimation on one excited state. Because this multiple process is possible it is also natural to find great diffusion length. This explains the known experimental results. Further experimentation on optical and magnetic resonance should give data on the velocity of exciton propagation and presumably on their effective mass as well. When it will become possible to resolve the lines from electron spin resonance one will become aware of the fine structure of the exciton. REFERENCES TAKEUTI Y., Prog. Theor. Phys. 18, 421 (1957). 2. DEXTER D. L., J. Chem. Phys. 21, 836 (1953). 3. GUIOCHON M. A., BLAMONT J. E. and BROSSEL J.,
1.
C.R. Acad. Sci., Paris. Phys. 18, 99 (1957).
243,
1859 (1956) and J.