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Optical line shapes of inhomogeneously broadened exciton transitions in the coherent potential approximation
Journal of Luminescence 48 & 49 (1991) 255-258 North-Holland
255
Optical line shapes of inhomogeneously broadened exciton transitions in the coherent potential approximation A. Boukahil and D.L. Huber Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
We have carried Out a series of calculations of the optical line shapes of inhomogeneously broadened exciton transitions within the framework of the coherent potential approximation (CPA). A Gaussian distribution of transition frequencies was assumed. In this paper, we present detailed results for Frenkel excitons in simple cubic systems with nearest-neighbor interactions. The theoretical predictions are compared with numerical simulations. Good agreement is obtained over a wide range of the ratio of the width of the distribution of transition frequencies to the exciton bandwidth. Previous limitations attributed to the CPA are shown to result from inaccuracies in the numerical evaluation of Green’s function or in the treatment of the Gaussian averaging.
1. Introduction Understanding the origin of the inhomogeneous broadening of optical transitions in solids continues to be a challenging problem [1]. Broadly speaking, one can divide sources of the line width into two categories depending on whether the broadening arises from macroscopic strains or has a microscopic origin. In our work, we have focussed on microscopic strains as a mechanism for the inhomogeneous line width in systems whose optical excitations are Frenkel excitons. We study a model system where the optically active ions occupy sites on a simple cubic lattice [2]. Nearestneighbor interactions are assumed. The effect of microscopic strain broadening is simulated by a Gaussian distribution of optical transition frequencies, with no correlation between the frequencies on different sites. In addition to being appropriate for strain broadening, the Hamiltonian has also been used as a model for the effect of the electron-phonon interaction on the optical spectra [3,4]. In this paper, we will compare the results of a theoretical calculation of the optical spectrum with the data from computer simulation studies [5,6]. The theoretical studies are based on the coherent poten~tialapproximation (CPA). In contrast to pre0022-2313/91/803.50
©
1991
—
vious CPA calculations [2,3], which have not had great success in accounting quantitatively for the effects of the disorder, we obtain good agreement with the simulations over a wide range of parameter values. Our success is attributed to an accurate treatment of the Gaussian averaging together with a closed-form approximation for the Green function appearing in the self-consistent equation for the coherent potential. As discussed in refs. [2,3], the use of the CPA involves the solution of a self-consistent equation. Appearing in this equation is a complex Green function, G. In ref. [2] this function was calculated by evaluating the integral representation. We have subsequently learned that we obtain better results using a simple analytic approximation introduced by Hubbard [7], which is appropriate for simple cubic arrays: 8 íE
where M is the exciton bandwidth. The Hubbard approximation was also used in ref. [3]. Unfortunately, the treatment of the Gaussian averaging given there does not give results that agree with numerical simulations in the weak disorder limit [5]. In contrast, we use an accurate numerical routine for treating the averaging which yields
Elsevier Science Publishers B.V. (North-Holland)
256
A. Boukahil, D.L. Huber
/ Optical
line shapes of inhomogeneouslv broadened exciton transitions
optical line shapes that are in good agreement with the simulations.
the equations of motion in ref. [6]. From the figure it is evident that there is very good agreement between the two sets of curves. In the analysis of ref. [5], a Gaussian distribution of transition frequencies was used to simulate the effects of thermal disorder. Thus the rms width of the Gaussian was expressed in terms of a ternperature according to a- = 0.0093 T1~2eV (T in K). In fig. 2, we display the line shifts versus ternperature calculated from the CPA along with the corresponding simulation data. Except at the highest temperature, where a- = 0.37, there is excellent agreement between the two sets of data points. Figure 3 displays the corresponding values of the line width (full width at half maximum). Although the agreement is not quite as good as with the line shift, the simulated and CPA line widths are nearly identical at the maximum and minimum ternperatures. We have also extended the calculations to smaller values of a- to ascertain the limiting behavior as a- approaches zero. We obtained the asymptotic form for the line shift and the half
2. Calculations We have carried out a series of calculations of the optical lineshape, F(E), in the CPA assuming a Gaussian distribution for the single-ion transition frequencies with rms width a-. Parameters were chosen so as to be able to make direct contact with the numerical simulations of refs. [5,6]. In fig. 1, we compare the CPA calculations with the curves generated by equation-of-motion techniques applied to arrays of 4096 ions occupying sites on a simple cubic lattice [6]. The energy scale is in units of the nearest-neighbor interaction. The curves shown are for a- = 4, 8, and 12, the latter value being equal to the exciton bandwidth. All curves have the same area. The CPA data were calculated with the energy having an imaginary part equal to 2. This is consistent with the exponential cut-off, exp(—2t), used in the integration of
(a)A ~
0.08
0,04
—
0.02
—
(b)
—
—
(c) 0.00
I
0
I
I
20
40
60
80
E Fig. 1. Optical line shape function, F(E) versus E. (a) u =4, (b) u = 8, (c) o = 12. Energy is measured in units of the nearest-neighbor interaction. The inset shows the simulation data from ref. [6] for the same parameters. All curves have the same area. The CPA results were obtained with the energy having an imaginary part equal to 2, corresponding to the exponential cut-off used in the simulation studies.
A. Boukahil, DL. Huber / Optical line shapes of inhomogeneously broadened exciton transitions 030
iii~uii,~ii,
025
—
020
—
~
257
I
—
0 —
0 >
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: 1..
0.15—
,~,
—
0
Co
~
0.10
—
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—
0 0
0.00
I
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I
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1000 TEMPERATURE (K)
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I
1500
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Fig. 2. Line shift versus temperature. The squares are the results from the CPA, and the diamonds are the simulation data from ref. 12 eV (T in K). [5]. The rms width of the Gaussian distribution of transition frequencies is equal to 0.0092T’
0 0
0
10—1
—
—
x
0
0
io2
—
50
‘c~
100
—
500 TEMPERATURE (K)
1000
Fig. 3. Line width (FWHM) versus temperature. The squares are the results from the CPA, and the diamonds are the simulation data from ref. [5]. The rms width of the Gaussian distribution of transition frequencies is equal to 0.0092 T~’2eV (T in K).
258
A. Boukahil, DL. Huber/ Optical line shapes of inhomogeneously broadened exciton transitions
width at half maximum on the high and low energy sides of the line [5]; i
2
line shift = ~aHWHM-HE=~a-3, HWHM-LE/HWHM-HE~1 in units of the nearest-neighbor interaction. The line shift is in agreement with lowest order perturbation theory, while the line width varies less rapidly than predicted by perturbation theory, where the HWHM-HE is proportional to a-4. It should be noted that had we used the exact Green function, the line shift coefficient would be 0.252 whereas the coefficient multiplying a-3 in HWHMHE would be 0.040.
3. Discussion The results reported here show that the CPA, when handled carefully, provides a quantitatively realistic treatment of the effects of Gaussian disorder on the exciton spectra of cubic systems with short-range interactions. Similar calculations have
also established that it works equally well in lower dimensions [8,9]. As such, it provides a model for the characterization of strain broadening in the extreme microscopic limit whereshifts there at is no correlation between the frequency different .
.
.
sites. In the small disorder limit, the line shift in three dimensions is accurately described by lowest order perturbation theory. The line width, however, is much more difficult to calculate. Even infinite order calculations give a-4 behavior [5]. Our CPA calculations are the first to reproduce the a-3 vanation. .
.
.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R.M. Macfarlane, J. Lumin. 45 (1990) 1. DL. Huber, Chem. Phys. 128 (1988) 1. H. Sumi, J. Phys. Soc. Jpn. 32 (1972) 616. J. Klafter and J. Jortner, J. Chem. Phys. 56 (1972) 5550. M. Schreiber and Y. Toyozawa, J. Phys. Soc. Jpn. 51(1982) DL. Huber and WY. Ching, Phys. Rev. B 39 (1989) 8652. J. Hubbard, Proc. Roy. Soc. A 281 (1964) 401. A. Boukahil and DL. Huber, J. Lumin. 45 (1990) 13. A. Boukahil, unpublished.
255
Optical line shapes of inhomogeneously broadened exciton transitions in the coherent potential approximation A. Boukahil and D.L. Huber Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA
We have carried Out a series of calculations of the optical line shapes of inhomogeneously broadened exciton transitions within the framework of the coherent potential approximation (CPA). A Gaussian distribution of transition frequencies was assumed. In this paper, we present detailed results for Frenkel excitons in simple cubic systems with nearest-neighbor interactions. The theoretical predictions are compared with numerical simulations. Good agreement is obtained over a wide range of the ratio of the width of the distribution of transition frequencies to the exciton bandwidth. Previous limitations attributed to the CPA are shown to result from inaccuracies in the numerical evaluation of Green’s function or in the treatment of the Gaussian averaging.
1. Introduction Understanding the origin of the inhomogeneous broadening of optical transitions in solids continues to be a challenging problem [1]. Broadly speaking, one can divide sources of the line width into two categories depending on whether the broadening arises from macroscopic strains or has a microscopic origin. In our work, we have focussed on microscopic strains as a mechanism for the inhomogeneous line width in systems whose optical excitations are Frenkel excitons. We study a model system where the optically active ions occupy sites on a simple cubic lattice [2]. Nearestneighbor interactions are assumed. The effect of microscopic strain broadening is simulated by a Gaussian distribution of optical transition frequencies, with no correlation between the frequencies on different sites. In addition to being appropriate for strain broadening, the Hamiltonian has also been used as a model for the effect of the electron-phonon interaction on the optical spectra [3,4]. In this paper, we will compare the results of a theoretical calculation of the optical spectrum with the data from computer simulation studies [5,6]. The theoretical studies are based on the coherent poten~tialapproximation (CPA). In contrast to pre0022-2313/91/803.50
©
1991
—
vious CPA calculations [2,3], which have not had great success in accounting quantitatively for the effects of the disorder, we obtain good agreement with the simulations over a wide range of parameter values. Our success is attributed to an accurate treatment of the Gaussian averaging together with a closed-form approximation for the Green function appearing in the self-consistent equation for the coherent potential. As discussed in refs. [2,3], the use of the CPA involves the solution of a self-consistent equation. Appearing in this equation is a complex Green function, G. In ref. [2] this function was calculated by evaluating the integral representation. We have subsequently learned that we obtain better results using a simple analytic approximation introduced by Hubbard [7], which is appropriate for simple cubic arrays: 8 íE
where M is the exciton bandwidth. The Hubbard approximation was also used in ref. [3]. Unfortunately, the treatment of the Gaussian averaging given there does not give results that agree with numerical simulations in the weak disorder limit [5]. In contrast, we use an accurate numerical routine for treating the averaging which yields
Elsevier Science Publishers B.V. (North-Holland)
256
A. Boukahil, D.L. Huber
/ Optical
line shapes of inhomogeneouslv broadened exciton transitions
optical line shapes that are in good agreement with the simulations.
the equations of motion in ref. [6]. From the figure it is evident that there is very good agreement between the two sets of curves. In the analysis of ref. [5], a Gaussian distribution of transition frequencies was used to simulate the effects of thermal disorder. Thus the rms width of the Gaussian was expressed in terms of a ternperature according to a- = 0.0093 T1~2eV (T in K). In fig. 2, we display the line shifts versus ternperature calculated from the CPA along with the corresponding simulation data. Except at the highest temperature, where a- = 0.37, there is excellent agreement between the two sets of data points. Figure 3 displays the corresponding values of the line width (full width at half maximum). Although the agreement is not quite as good as with the line shift, the simulated and CPA line widths are nearly identical at the maximum and minimum ternperatures. We have also extended the calculations to smaller values of a- to ascertain the limiting behavior as a- approaches zero. We obtained the asymptotic form for the line shift and the half
2. Calculations We have carried out a series of calculations of the optical lineshape, F(E), in the CPA assuming a Gaussian distribution for the single-ion transition frequencies with rms width a-. Parameters were chosen so as to be able to make direct contact with the numerical simulations of refs. [5,6]. In fig. 1, we compare the CPA calculations with the curves generated by equation-of-motion techniques applied to arrays of 4096 ions occupying sites on a simple cubic lattice [6]. The energy scale is in units of the nearest-neighbor interaction. The curves shown are for a- = 4, 8, and 12, the latter value being equal to the exciton bandwidth. All curves have the same area. The CPA data were calculated with the energy having an imaginary part equal to 2. This is consistent with the exponential cut-off, exp(—2t), used in the integration of
(a)A ~
0.08
0,04
—
0.02
—
(b)
—
—
(c) 0.00
I
0
I
I
20
40
60
80
E Fig. 1. Optical line shape function, F(E) versus E. (a) u =4, (b) u = 8, (c) o = 12. Energy is measured in units of the nearest-neighbor interaction. The inset shows the simulation data from ref. [6] for the same parameters. All curves have the same area. The CPA results were obtained with the energy having an imaginary part equal to 2, corresponding to the exponential cut-off used in the simulation studies.
A. Boukahil, DL. Huber / Optical line shapes of inhomogeneously broadened exciton transitions 030
iii~uii,~ii,
025
—
020
—
~
257
I
—
0 —
0 >
5)
: 1..
0.15—
,~,
—
0
Co
~
0.10
—
0
005—
—
0 0
0.00
I
0
I
I
500
I
I
I
1000 TEMPERATURE (K)
I
I
I
1500
I
Fig. 2. Line shift versus temperature. The squares are the results from the CPA, and the diamonds are the simulation data from ref. 12 eV (T in K). [5]. The rms width of the Gaussian distribution of transition frequencies is equal to 0.0092T’
0 0
0
10—1
—
—
x
0
0
io2
—
50
‘c~
100
—
500 TEMPERATURE (K)
1000
Fig. 3. Line width (FWHM) versus temperature. The squares are the results from the CPA, and the diamonds are the simulation data from ref. [5]. The rms width of the Gaussian distribution of transition frequencies is equal to 0.0092 T~’2eV (T in K).
258
A. Boukahil, DL. Huber/ Optical line shapes of inhomogeneously broadened exciton transitions
width at half maximum on the high and low energy sides of the line [5]; i
2
line shift = ~aHWHM-HE=~a-3, HWHM-LE/HWHM-HE~1 in units of the nearest-neighbor interaction. The line shift is in agreement with lowest order perturbation theory, while the line width varies less rapidly than predicted by perturbation theory, where the HWHM-HE is proportional to a-4. It should be noted that had we used the exact Green function, the line shift coefficient would be 0.252 whereas the coefficient multiplying a-3 in HWHMHE would be 0.040.
3. Discussion The results reported here show that the CPA, when handled carefully, provides a quantitatively realistic treatment of the effects of Gaussian disorder on the exciton spectra of cubic systems with short-range interactions. Similar calculations have
also established that it works equally well in lower dimensions [8,9]. As such, it provides a model for the characterization of strain broadening in the extreme microscopic limit whereshifts there at is no correlation between the frequency different .
.
.
sites. In the small disorder limit, the line shift in three dimensions is accurately described by lowest order perturbation theory. The line width, however, is much more difficult to calculate. Even infinite order calculations give a-4 behavior [5]. Our CPA calculations are the first to reproduce the a-3 vanation. .
.
.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R.M. Macfarlane, J. Lumin. 45 (1990) 1. DL. Huber, Chem. Phys. 128 (1988) 1. H. Sumi, J. Phys. Soc. Jpn. 32 (1972) 616. J. Klafter and J. Jortner, J. Chem. Phys. 56 (1972) 5550. M. Schreiber and Y. Toyozawa, J. Phys. Soc. Jpn. 51(1982) DL. Huber and WY. Ching, Phys. Rev. B 39 (1989) 8652. J. Hubbard, Proc. Roy. Soc. A 281 (1964) 401. A. Boukahil and DL. Huber, J. Lumin. 45 (1990) 13. A. Boukahil, unpublished.