Microscopic quantum theory of spatially resolved photoluminescence in semiconductor quantum structures
G. Pistone,a)S. Savasta, O. Di Stefano, and R. Girlanda
Istituto Nazionale per la Fisica della Materia INFM and Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, Universita` di Messina, Salita Sperone 31, I-98166 Messina, Italy
We present a microscopic analysis of spatially resolved photoluminescence and photoluminescence excitation spectroscopy in semiconductor quantum structures. Such theoretical and numerical framework provides a general basis for the description of spectroscopic imaging in which the excitation and detection energies and spatial positions can all independently be scanned. The numerical results clarify the impact of the near-field optical setup on the obtained images and resolutions. 关DOI: 10.1063/1.1711184兴
In recent years, measurements based on spatially re- solved photoluminescence 共PL兲 provided direct information on the spatial and energy distribution of light emitting nano- metric centers of semiconductor quantum structures, thus opening a rich area of physics involving spatially resolved quantum systems in a complex solid state environment.1– 6In particular, near-field optical microscopy and spectroscopy can detect the optical characteristics of individual quantum dots共QDs兲 among, e.g., a high-density ensemble of naturally formed QDs,7 whereas usual far-field methods provide only ensemble-averaged properties. Detailed simulations of Zim- mermann, Runge, and Savona8,9have clarified many aspects of the intrigued nonequilibrium dynamics giving rise to pho- toluminescence spectra in these quantum structures. How- ever, theoretical simulations of near-field imaging spectros- copy of semiconductor quantum structures focus on calcu- lations of local absorption.10–13 In contrast, as a matter of fact, almost all experimental images are obtained from PL measurements. Here we present a microscopic theory of spatially resolved photoluminescence in quantum structures that includes both light quantization 共essential to describe radiative recombination兲 and phonon scattering. The theory also includes the description of spatially confined excitation 共illumination mode兲 and/or detection 共collection mode兲. It is worth noting that we are faced with a strictly nonequilibrium problem. Nonequilibrium here arising from both radiative recombination 共preventing full thermalization兲 and from the local nature of the excitation source 共in the illumination- mode setup兲. We are also faced with the problem of the dif- ferences between illumination (I) and collection (C) scan- ning near-field optical microscopy共SNOM兲 instruments. The equivalence between these two working modes has been established on the basis of the reciprocity theorem for elec- tromagnetic fields.14 However, this theorem holds for linear and passive media. While semiconductor structures, at low excitation densities, show a linear behavior, phonon scatter- ing and radiative recombination prevent them from being passive.
The positive frequency components of the operator de-
scribing the signal that can be detected by a general near- field setup can be expressed as15Sˆt⫹⫽Aˆbg⫹⫹Sˆ⫹, where Aˆ
bg⫹ is the elastic background signal 共largely uniform along the x – y plane兲 proportional to the input electric-field operator, and Sˆ⫹is related to the sample polarization density operator Pˆ⫹(r): Sˆ⫹⫽A兰drPˆ⫹(r)•EC(r), whereA is a complex con- stant and EC(r) is the signal mode delivered by the tip. Pho- toluminescence can be defined as the incoherent part of the emitted light intensity.16 The PL that can be measured by a photodetector after the collection setup 共broadband detec- tion兲 is proportional to IPL⫽I⫺Icoh⫽具Sˆ⫺()Sˆ⫹()典
⫺兩具Sˆ⫹()典兩2. Analogously, the steady-state spectrum of in- coherent light emitted by the semiconductor quantum struc- ture and detected by the SNOM setup can be expressed as
I共C兲⫽1
冕
0⬁
d具Sˆ⫺共0兲Sˆ⫹共兲典eiC. 共1兲
The interband polarization density operator can be expressed in terms of exciton operators as Pˆ⫹(r)⫽ehf (z)⌿␣eh(
⫽0,R)Bˆ␣, where e and h are appropriate sets of quantum numbers which label the carrier states involved in the optical transition. The operator Bˆ
␣† creates an exciton state 共one electron-hole pair兲 Bˆ␣†兩0典⬅兩E1,␣典 with energy 1,␣.
⌿␣eh(,R) is the exciton wave function withindicating the relative in-plane electron-hole eh coordinate and R describes the center of mass motion, while f (z)⫽ue(z)uh(z) is the product of the electron and hole envelope functions along the confinement direction. If the disorder induced broadening is small compared to the exciton binding energy, only the low- est bound state 1s at the fundamental sublevel transition has to be considered and the exciton wave function can be fac- torized as ⌿␣eh(,R)⫽1s()␣(R).9 In the low-excitation regime the relation Bˆ␣⯝兩0典具E1,␣兩 can be assumed and it can be shown that the operators Bˆ␣behave as Boson operators.17 Within these assumptions, the Hamiltonian determining the dynamics of the semiconductor system is given by the three contributions: the bare excitonic Hamiltonian of the semi- conductor system, the interaction Hamiltonian of the semi- conductor with the quantum共possibly inhomogeneous兲 light field and the interaction Hamiltonian of excitons with the phonon bath determining inelastic scattering and dephasing.9 The kinetic equation for diagonal terms of the exciton den-
a兲Author to whom correspondence should be addressed; electronic mail:
pistone@unime.it
sity matrix N␣⫽具Bˆ␣†Bˆ␣典 can be derived starting from the Heisenberg equation of motion for the exciton operators un- der the influence of H. The main approximations are the neglect of possible coherent phonon states and of memory effects induced by the photon and phonon fields. The result is
tN␣⫽G␣共I兲⫹
兺
 ␥␣←N⫺2⌫␣N␣, 共2兲where the resulting phonon-assisted scattering rates ␥←␣
are given by ␥←␣⫽(2/ប2)兺q兩t␣,q 兩2关n(q)␦␣,,q⫹ ⫹(1
⫹n(q))␦␣,,q⫺ 兴, n(q) being the thermal phonon occupa- tion functions and ␦␣,,q⫾ ⫽␦(␣⫺⫾q). The scattering matrix elements t␣,q depend on the lattice deformation po- tentials and the overlap between the exciton states.9 2⌫␣
⫽r␣⫹兺␥←␣ is the total out-scattering rate, r␣ being the rate for radiative recombination proportional to the exciton oscillator strength: r␣⫽r0兩兰d2R␣(R)兩2 共we assume that the tip-sample interaction does not alter the radiative decay rates兲. In the above equation the generation term depends on the spatial overlap between the illuminating field EI(r) and the exciton wave functions corresponding to exciton levels resonant with the input light: G␣(RI)⫽r0兩o␣I(RI)兩2L␣(I) with L␣()⫽⌫/关(⫺␣)2⫹⌫2兴 and o␣I(RI)
⫽兰d2RE˜I(R)␣(R) where E˜I(R)⫽兰EI(r) f (z)dz. In the subsequent numerical calculations concerning the illumina- tion mode we will assume an input light field with a given Gaussian profile centered around the tip position RI: E˜I(R)⫽EI
0g(R⫺RI). In this case the generation term be- comes function of the beam position and shape共spatial reso- lution兲.
Once the exciton densities have been derived, the fre- quency integrated PL can be readily obtained. It results in I(RC)⫽r0兺␣兩o␣C(RC)兩2N␣, where o␣C, analogously to o␣I , contains the overlap of the exciton wave functions with the signal mode E˜C(R) delivered by the tip 共collection mode兲.
According to the quantum regression theorem, 具Sˆ⫺(0)Sˆ⫹()典 has the same dynamics of 具Sˆ⫹()典 共propor- tional to the exciton operator兲, but with 具Sˆ⫺(0)Sˆ⫹(0)典 as initial condition. Following this procedure we obtain I(RC,C)⫽r0兺␣兩o␣C(RC)兩2L␣(C) N␣.
We apply the above developed theoretical scheme to analyze the spatially resolved light emission of a QD system arising from interface fluctuations of GaAs QWs. The effec- tive disordered potential felt by 1s excitons used in our simulations is obtained summing up two different contribu- tions, both modeled as a zero mean, Gauss distributed and spatially correlated process. We used⫽16 nm 共the correla- tion length兲, v0⫽1.5 meV 共the width of the energy distribu- tion兲; and⫽8 nm, v0⫽0.5 meV, respectively.
Figure 1共a兲 displays the specific realization of the effec- tive disordered potential used for all the calculations. Figures 1共b兲–1共d兲 show energy-integrated PL images obtained after uniform illumination of the sample at energy I⫽1 meV 共the zero of energy is fixed at the energy of the 1s exciton in absence of disorder兲 and collecting locally the emitted light 关C mode with spatial resolution full width at half maximum 共FWHM兲⫽47 nm兴. It is worth noting that the energy- integrated excitonic local density of states does not depend on position. So the observed structures are a direct conse- quence of the increasing ratio between radiative and nonra-
diative scattering rates for exciton states localized at the po- tential minima关compare, e.g., images 1共a兲 and 1共c兲兴. As the temperature of the structure is lowered, a transition from a broad and fairly continuous PL to an intense set of few spa- tially localized luminescence centers can be observed. An- other interesting feature is the nonmonotonous brightness of some luminescence centers when the temperature is in- creased 关see, e.g., the location indicated by an arrow in im- age 1共b兲兴, due to excitons which can overcome shallow local minima by thermal activation and fall down in still deeper states.8
In Fig. 2 we report the emission intensity as a function of both the illumination and collection energy for the three main working modes of the SNOM: I 共local illumination, far-field collection兲, C 共far-field illumination, local collec- tion兲 and I-C mode 共both local illumination and collection兲.
The tip 共a spatial resolution of FWHM⫽94 nm is assumed兲 for local illumination and/or detection is centered at the po-
FIG. 1. 共a兲. Specific realization of disorder potential; 共b兲–共d兲 energy- integrated PL images obtained after uniform illumination of the sample at energyI⫽1 meV and collecting locally the emitted light.
FIG. 2.共Color兲 Emission intensity as a function of both the illumination and collection energy for the three main working modes of the SNOM. A small region along the diagonal (C⫽I) containing strong elastic contributions is canceled.
sition indicated by a circle in Fig. 1共a兲. Figure 2 clearly shows the impact that the specific working mode of the in- strument can have on the obtained results. The I-C images are the most selective and show a series of discrete peaks that merge into quasicontinua 共corresponding to less local- ized exciton states兲 at higher energy, whereas the I mode is selective only for photoluminescence excitation 共PLE兲 mea- surements (Cfixed,Ituned兲. On the contrary, the C mode shows good selectivity for PL measurements (C tuned,I
fixed兲, being of poor selectivity for PLE. This picture can be better pointed out by looking at the sections of these gray- scale plots shown in Fig. 3. Panel 3共a兲 displays for reference the far-field and local absorption spectra. Figure 3共b兲 shows that the C and I-C modes give the same results when the sample is excited at quite a high energy and C is tuned.
Also, the I and I-C modes almost coincide 关see Fig. 3共c兲兴 whenCis fixed at a low energy absorption peak. However, Fig. 3共d兲 shows that in other circumstances the three working modes produce different results. It is also interesting to no- tice the differences in the peaks height between PLE and local absorption spectra. The strong nonequivalence between different SNOM kinds, here observed, is consequence共and at the same time a direct proof兲 of the nonlocal nature of the incoherent energy transfer between different regions of the sample involving length scales significantly larger than
SNOM spatial resolutions 共even in presence of well defined dots兲. Further evidence is given by the strong differences between the calculated local absorption and emission spectra in some working mode. Finally, we note the thermal activa- tion of light up-conversion due to phonon-assisted exciton scattering clearly visible in the lower panels of Fig. 2.
In summary we have presented a microscopic quantum theory of spatially resolved photoluminescence in quantum wells with interface fluctuations. The theory establishes a clear functionality of the different SNOM kinds when ap- plied to the analysis of semiconductor quantum structures.
Moreover, this formulation permits one to model PL excita- tion spectroscopy in which the excitation and detection en- ergies and spatial positions can all independently be scanned.
The numerical results here presented constitute an intriguing example of the impact of sample temperature, exciton local- ization, and microscope setup in the formation of subwavelength-resolution images.
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FIG. 3. 共a兲 Far-field spectrum and local absorption spectrum calculated centering the tip at the position indicated by a circle in Fig. 1共a兲. 共b兲,共c兲,共d兲 Line scans共PL and PLE spectra兲 indicated by the letters 共b,c,d兲 in Fig. 2 for the three different working modes (T⫽100K).
