The problem of extracting the homogeneous contribution to an inhomogeneously broadened line is a classical problem of optical spectroscopy. Solving this problem allows one, in many cases, to get valuable information about the nature of broadening of spectral lines and characteristics of individual centers forming the inhomogeneously broadened line. This problem is encountered in atomic and molecular spectroscopy, in spectroscopic studies of surfaces, in spectroscopy of activated crystals and semiconductors, etc. At present, this problem has acquired a particular importance in studies of low-dimensional semiconductor structures (quantum wells and quantum dots), with the inhomogeneous broadening resulted from variance of the size quantization effects. As a rule, however, this problem cannot be solved within the framework of standard methods of linear spectroscopy. This is why, spectroscopic characteristics of individual emitters are determined, in practice, either using methods of nonlinear spectroscopy (spectroscopy of transient response, hole-burning spectroscopy, etc., see, e.g., [1
1. J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures, Springer series in Solid-State Sciences, (Springer-Verlag, Heidelberg, 1996).
]), or by realizing conditions (by means of near-field or far-field microscopy) when the number of emitters becomes sufficiently small and components of the inhomogeneously broadened line profile become, to some extent, resolved [2
2. A. Zrenner, L. V. Butov, M. Hagn, and G. Abstreiter,“Quantum dots formed by interface fluctuations in AlAs/GaAs coupled quantum well structures,”G. Böhm and G. Weimann. Phys. Rev. Lett. 72, 3382 (1994). [CrossRef] [PubMed]
3. K. Brunner, G. Abstreiter, G. Böhm, G. Tränkle, and G. Weimann, “Sharp-line photoluminescence and two photon absorption of zero-dimensional biexcitons in a GaAs/AlGaAs structure,” Phys. Rev. Lett. 73, 1138 (1994). [CrossRef] [PubMed]
4. H. F. Hess, E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W. West, Science264, 1740 (1994). [CrossRef] [PubMed]
5. D. Gammon, E. S. Snow, B. V. Shanobrook, D. S. Katzer, and D. Park, “Fine Structure Splitting in the Optical Spectra of Single GaAs Quantum Dots,” Phys. Rev. Lett. 76, 3005 (1996). [CrossRef] [PubMed]
6. Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, “Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well,” Phys. Rev. Lett. 83, 2652 (1999). [CrossRef]
]. Note that extracting the homogeneous linewidth from the luminescence decay measurements is justified only in unique situations when the line is not broadened by dephasing processes.
In this paper, we want to call attention to the fact that information about the homogeneous broadening of an inhomogeneously broadened line can be extracted from the profile of the latter provided that this profile is recorded with a sufficiently high accuracy. We show that the inhomogeneously broadened line profile, in view of statistical nature of distribution of oscillators’ frequencies, exhibits spectral fluctuations, with their correlation length (in spectral scale) being directly connected with the homogeneous linewidth and with their amplitude determined by the number of centers forming the profile in particular experimental conditions. We propose a method for measuring correlation characteristics of inhomogeneously broadened lines, estimate quantitatively the magnitude of the effect, and formulate requirements that determine the possibility of its practical observation and use.
2 Mathematical procedure
For definiteness, we will deal, in what follows, with spectra of photoluminescence (although the proposed approach can be applied to studies of other spectra of secondary emission or absorption). For this reason, individual centers forming the inhomogeneously broadened line (quantum dots, localized excitons, adsorbed atoms, impurity centers, etc.) will be further referred to as emitters.
As will be shown below, information about the homogeneous width of the spectrum can be obtained, in principle, from correlation analysis of a single realization. It is known, however, that reliability of information about a random process increases with the number of its realizations. In photoluminescence studies, the number of realization of the spectrum can be easily increased by shifting the spot of optical excitation over the sample. Each new position will give a new realization of the photoluminescence spectrum because the spectrum will be formed each time by a different combination of emitters. We assume, for simplicity, that the homogeneous spectral width does not depend in any regular way on position of the spot on the sample.
Let the inhomogeneously broadened line be formed by N emitters, with the i-th emitter being characterized by a homogeneously broadened line a(ω-ωi), where ωi (i=1, …,N) are mutually independent random quantities with the distribution function ρ(ω) (∫ ρ(ω)dω=1). We assume, for simplicity, that ∫ a(ω)dω=1. Then, each realization of the inhomogeneously broadened line will be described by the random function A(ω):
The homogeneous width of the spectral line is considered to be much smaller than the inhomogeneous width, i.e., the region where the function ρ(ω) is nonzero is much larger than that for the function a(ω). Then, for the inhomogeneously broadened line averaged over realizations <A(ω)>, we have:
Let us calculate the correlation function <A
)> (see, e.g., [7
7. A. van der Ziel, Noise in Measurements (John Wiley & Sons, New York, 1976).
Let the homogeneous lineshape be a Lorentzian:
Assuming that N≫1 and expressing ρ(ω) through the averaged profile of the inhomogeneously broadened line, we obtain:
Thus we see that auto-convolution of the homogeneous line can be directly expressed through characteristics of the experimentally measured functions A(ω) (left-hand side of the equation). Suppose we have a set of NR realizations of the inhomogeneously broadened line, i.e., NR functions Ar(ω), r=1, …, NR. Then, the expression for auto-convolution of the homogeneously broadened line can be written in the form:
Formulas (3) and (4) express in a quantitative form the fact that the inhomogeneously broadened line profile is, in essence, a random function which inevitably exhibits spectral fluctuations whose correlation properties contain information about its homogeneous broadening. The result thus obtained, evidently, does not depend qualitatively on the homogeneously broadened line shape and can be formulated as follows: The left-hand side of Eq.(4)
), measured experimentally, depends only on the frequency difference ω
and represents a function with a narrow peak (right-hand side), whose width is twice as large as that of the homogeneous line.
To demonstrate efficiency of the proposed correlation method, we have made a series of computer-simulation experiments. Each realization of the inhomogeneously broadened line formed by an ensemble of N
emitters was obtained using formula (1) by generating N
random numbers ωi
), with the Gaussian distribution function
. The homogeneous spectral line was taken in the form of a Lorentzian (2). By repeating this procedure NR
times, we obtained NR
independent realizations of the inhomogeneously broadened line Ar
. Then we calculated the correlation function of the inhomogeneously broadened line and checked formula (4). Note that the frequency increment used for numerical generation of the arrays Ar
) had to be essentially smaller than the homogeneous linewidth δ
. The results thus obtained for different numbers of realizations NR
, with the number of emitters N
and the inhomogeneous/homogeneous linewidth ratio α
=20, are shown in Fig.1
The dashed line shows auto-convolution of the homogeneously broadened line. As seen from Fig.1
, the results of computer simulation agree well with predictions of our analysis.
3 Magnitude of the effect
Let us estimate the amplitude of spectral fluctuations of the inhomogeneously broadened line. Variance of a random quantity, by definition, is given by the formula:
Whence it follows, as can be easily shown, that the relative fluctuation of the line profile
is given by the relationship:
If the homogeneous line is a Lorentzian (2), and the homogeneous linewidth δ is much smaller than the inhomogeneous linewidth, formula (5) can be approximately represented in the form
However, the magnitude proper of the spectral fluctuations d will evidently drop with increasing N and, for sufficiently large N, may ‘sink’ below the detected noise. It should be emphasized that the spectral noise of the inhomogeneously broadened line under study can be considered here, in the framework of this approach, as a statistical signal, which should be the same for identical realizations of the ensemble of emitters. The possibility of measuring the spectral fluctuations is determined by the ratio of their amplitude (amplitude of signal) to amplitude of real irreproducible noise of measurements. The latter can be associated not only with usual noise of detection (shot noise of the photocurrent, noise of electronics, etc.), but also with various kinds of spurious spectral modulation (caused, e.g., by interferometric effects on parasitic interferometers in the optical channel).
Let us make a realistic assumption that the main contribution to the noise of measurements is made by shot noise of the detector’s photocurrent. Then, the magnitude of the accumulated signal n of the spectral data array (expressed in the number of photoelectrons) needed to be able to neglect the noise of detection will be given by
is the variance of spectral fluctuations of the line profile, given by Eq.(6)
. In other words, the method under consideration can be used when the experimental numbers of the spectral data array n
essentially exceed (say by one or two orders of magnitude) the number of emitters falling into the interval equal to the homogeneous linewidth. In this case, the signal/noise ratio in the auto-convolution of the homogeneous line, obtained experimentally, will be determined, as before, by Eq. (7)
. Using this criterion, one can easily estimate that with the up-to-date methods of detection and accumulation of optical signals (with CCD detector arrays and automatic scanning of the optical excitation spot over the sample), this requirement can be fulfilled in a wide number of cases, which indicates feasibility of the proposed experimental approach. Note that to meet the requirement (8) one can increase the signal accumulation time and decrease the excitation spot size. As was already pointed out, spectral resolution for measurements of this kind should essentially exceed the measured homogeneous linewidth or, which is the same, the characteristic correlation interval of the spectral fluctuations.