Nonlinear two-dimensional Fourier transform (2DFT) and linear absorption spectroscopy are used to study the electronic structure and optical properties of excitons in the layered semiconductor GaSe. At the 1s exciton resonance, two peaks are identified in the absorption spectra, which are assigned to splitting of the exciton ground state into the triplet and singlet states. 2DFT spectra acquired for co-linear polarization of the excitation pulses feature an additional peak originating from coherent energy transfer between the singlet and triplet. At cross-linear polarization of the excitation pulses, the 2DFT spectra expose a new peak likely originating from bound biexcitons. The polarization dependent 2DFT spectra are well reproduced by simulations using the optical Bloch equations for a four level system, where many-body effects are included phenomenologically. Although biexciton effects are thought to be strong in this material, only moderate contributions from bound biexciton creation can be observed. The biexciton binding energy of ∼2 meV was estimated from the separation of the peaks in the 2DFT spectra. Temperature dependent absorption and 2DFT measurements, combined with “ab initio” theoretical calculations of the phonon spectra, indicate strong interaction with the A 1 phonon mode. Excitation density dependent 2DFT measurements reveal excitation induced dephasing and provide a lower limit for the homogeneous linewidth of the excitons in the present GaSe crystal.

Layered semiconductor materials have generated a wealth of interest because of their unusual electronic properties and the potential for optoelectronic and photonic applications. Transition metal dichalcogenides and other layered materials have recently been suggested as excellent candidates for numerous applications due to their sizable bang gap, their coupled spin and valley degrees of freedom, strong optical absorption, and controllable doping via electrodes.1,2 When reduced to monoatomic layers, these materials exhibit drastic changes in the electronic properties. The transition from two atomic layers to a single atomic layer leads to a transformation from an indirect to a direct band gap semiconductor in transition metal dichalcogenides.3 Furthermore, single layer transition metal dichalcogenides have intrinsic inversion symmetry breaking leading to coupled spin and valley degrees of freedom resulting in circularly polarized optical selection rules.4–10 However, it is important to explore new layered materials that could exhibit higher carrier mobilities as well as different band gap energies. Finally, group III-VI chalcogenides have exhibited interesting many-body physics, including non-Markovian memory effects observed in GaSe, resulting from exciton-phonon interactions.11 Recently, GaSe was also predicted to belong to a new class of topological insulator.12 The unusual electronic properties make GaSe interesting both for theoretical investigations and potential applications.

Bulk GaSe is one of the most studied layered semiconductors of the III-VI family. In layered materials, the crystal structure is highly anisotropic and consists of covalently bonded atomic layers that are bound by weak van der Waals interactions. The two-dimensional layers are composed of gallium and selenium atoms at opposite corners of the hexagons. In the most common ε− polytype, there are two layers and eight atoms in the unit cell with the resulting crystal belonging to the D 3 h 1 space group.13 Early experiments together with band-structure calculations concluded that the direct and indirect gap energies are very close at ambient pressure and at room temperature.14 

Furthermore, time-resolved spectroscopy has provided important insights into the carrier dynamics in GaSe. The relaxation of high energy carriers and the formation of exciton in GaSe were studied by time-resolved photoluminescence techniques.15 These studies observed a faster relaxation in the sub-picosecond time scale due to Fröhlich interaction of carriers with the longitudinal optical E′ phonons and a slower component that is caused by deformation potential interaction with the nonpolar optical A 1 phonons. Furthermore, theoretical studies, Hall mobility, and absorption measurements have suggested that the electron-phonon interactions in GaSe are primarily caused by the deformation potential coupling to the A 1 phonon mode of ∼16.7 meV.16–20 

More recently, bulk GaSe has been studied using time-integrated, time-resolved, and spectrally resolved four wave mixing (FWM) experiments. The dephasing of excitons was measured using time-integrated FWM and the homogeneous linewidth deduced, assuming an echo behavior. The echo behavior has been observed using time-resolved FWM and occurs as a result of the large inhomogeneous broadening due to stacking disorders as a result of the mixture of the two different γ − and ε− polytypes. The inhomogeneous excitonic linewidths observed were on the order of ∼5 meV indicating substantial inhomogeneous broadening as compared to the estimated homogeneous linewidth ∼0.4 meV.22 Combined studies of time-resolved and spectrally resolved FWM have observed non-Markovian dynamics resulting from electron-phonon interactions.11,23,24

In the present study, two-dimensional Fourier transform (2DFT) measurements reveal coherent energy transfer between the two substates of the excitonic resonance. The doublet structure observed is assigned to the splitting of the excitonic ground spin states due to exchange interactions, after careful consideration of other possibilities. Polarization dependent 2DFT spectra reveal a new peak originating from the bound biexciton. The relative strength of the biexciton peak suggests only moderate biexciton creation. From the 2DFT spectra, a biexciton binding energy of ∼2 meV is estimated. Simulation of the 2DFT spectra using the optical Bloch equations for a four-level system reproduces the experimental data well and confirms coherent energy transfer between the two excitonic fine structure components induced by many-body interactions. Furthermore, detailed temperature dependent absorption and 2DFT measurements, combined with “ab initio” phonon calculations, provide important insights into the electron-phonon interactions. In particular, we identify the A 1 mode as the dominant phonon responsible for excitonic dephasing in this material. The energy shift of the band gap with temperature provides a close estimate of the “zero-Kelvin” band gap renormalization. Excitation density dependent measurements reveal excitation induced dephasing (EID) likely due to exciton-exciton scattering. The residual homogeneous linewidth of the excitonic resonance in the limit of low excitation density and low temperature is measured be ∼0.61 meV.

The coherent response of semiconductors is directly related to both the electronic structure of the excitonic ground state and the many-body interactions taking place. In three pulse FWM, three pulses are incident on the sample with wavevectors ka, kb, and kc. The nonlinear interaction gives rise to a signal in the direction −ka + kb + kc. The phase conjugate pulse −ka and the second pulse kb are separated by the time delay τ, whereas pulse kb and the third pulse kc are separated by the population time T (Fig. 1(b)). By varying the time delay τ and monitoring the time-integrated FWM intensity, the dephasing time of excitons can be measured. By varying the time delay T, the population relaxation or excitonic lifetime can be measured. In order to monitor the third time evolution, t additional time resolved experiments need to be performed.

FIG. 1.

(a) The four phase-stabilized linearly polarized beams obtained from the multi-dimensional optical nonlinear spectrometer (MONSTR) instrument described in Ref. 21 are focused on the sample, which is held in the cryostat at 5 K. A portion of the laser pulse has been split off and co-linearly recombined with the FWM signal for heterodyne detection. The combined beams are dispersed in the spectrometer resulting in the spectral interferogram. (b) The sequence of the laser pulses used in the experiments, where A corresponds to the phase conjugated pulse. The time delay τ corresponds to the time between the pulses A and B, T is the time delay between the pulses B and C, and t is the evolution of the echo in “real time.” (c) Crystal structure of the ε− GaSe, where the unit cell extends over two layers bound through van der Waals interaction along the c-axis. The laser excitation shown is perpendicular to the covalently bound layers or parallel to the c-axis.

FIG. 1.

(a) The four phase-stabilized linearly polarized beams obtained from the multi-dimensional optical nonlinear spectrometer (MONSTR) instrument described in Ref. 21 are focused on the sample, which is held in the cryostat at 5 K. A portion of the laser pulse has been split off and co-linearly recombined with the FWM signal for heterodyne detection. The combined beams are dispersed in the spectrometer resulting in the spectral interferogram. (b) The sequence of the laser pulses used in the experiments, where A corresponds to the phase conjugated pulse. The time delay τ corresponds to the time between the pulses A and B, T is the time delay between the pulses B and C, and t is the evolution of the echo in “real time.” (c) Crystal structure of the ε− GaSe, where the unit cell extends over two layers bound through van der Waals interaction along the c-axis. The laser excitation shown is perpendicular to the covalently bound layers or parallel to the c-axis.

Close modal

Enhanced understanding of the coherent dynamics of semiconductors and semiconductor nanomaterials can be obtained by using multidimensional Fourier-transform spectroscopy. In the present 2DFT experiments, the FWM signal field is recorded during the time delays τ and t simultaneously, and the phase information is preserved. The Fourier transform with respect to these two time delays leads to a two-dimensional map in frequency domain where the ωτ and ωt axes are correlated. It is the phase preservation throughout the experiment that makes these measurements challenging. In order to overcome this challenge, new instrumentation has recently been developed.21,25 The advantages of multidimensional spectroscopy over one dimensional techniques are well known and thoroughly described in the literature.26–32 In semiconductor nanomaterials, 2DFT spectroscopy has allowed accurate measurements of the homogeneous linewidth and provided insights into the microscopic details of the many-body interactions.33–37 

In the present time resolved study, ∼130 fs laser pulses at 76 MHz repetition rate were provided by an oscillator. The light generated by the oscillator is used to pump an optical parametric oscillator (OPO), which splits the frequency of the pumping oscillator into signal and idler. The intracavity second harmonic of the signal provided the excitation source at 587.6 nm for the experiments. The cavity length of the OPO is actively stabilized in order to provide a stable laser power. The 2DFT setup that is used to perform measurements at these wavelengths was recently developed and a detailed description can be found in Ref. 21. High quality undoped ε− GaSe crystals were grown using the Bridgman method. The primitive unit cell of ε− GaSe contains two layers, each consisting of four closely packed, covalently bound, monoatomic sheets in the sequence Se–Ga–Ga–Se. Within each plane, atoms form hexagons interacting by weak van der Waals forces between the layers (Fig. 1(c)). The ε− GaSe crystals exhibited p-type conductivity with a majority carrier concentration in the (1 − 5) × 1013 cm−3 range. The sample used in the present studies was exfoliated down to sub-micron thickness. The data were collected from a ∼0.9 μm thick region shown in Fig. 1(a). Absorption spectra were collected using a FTIR spectrometer.

We start by discussing the absorption measurements on undoped high-quality GaSe crystals of Figs. 2(a) and 2(b), which show a doublet structure in the excitonic ground state. Several potential mechanisms could be responsible for the observed doublet fine structure. (1) In bulk GaSe, the direct band gap and an indirect band gap are very close in energy at ambient pressure.38 Therefore, the doublet structure could originate from direct and indirect excitonic transitions. (2) Furthermore, two polytypes, namely, ε- and γ-polytypes appear to coexist in crystals of the present study. Excitonic transitions are slightly shifted in energy for the two polytypes.16 (3) Finally, exchange interaction leads to an energy splitting into a singlet and triplet state, assuming that the band mixing induced by the spin-orbit coupling is relatively weak.16,38

FIG. 2.

(a) Absorption spectra at 5 K of the 1s exciton from a large area of the sample. The overall linewidth is comparable to the values reported in the literature. The shoulder at lower energy indicates the presence of two transitions. (b) Absorption spectra at 5 K of the 1s exciton from a region of the sample with 0.9 μm uniform thickness. The same area was used in the 2DFT measurements. Two separate peaks are observed at 2.110 67 eV and 2.114 35 eV, with equal FWHM of 2.7 meV. Two Gaussian lines were used to fit the individual components (blue), whereas the red line is the addition of the two Gaussian lines and the black line is the experimental data.

FIG. 2.

(a) Absorption spectra at 5 K of the 1s exciton from a large area of the sample. The overall linewidth is comparable to the values reported in the literature. The shoulder at lower energy indicates the presence of two transitions. (b) Absorption spectra at 5 K of the 1s exciton from a region of the sample with 0.9 μm uniform thickness. The same area was used in the 2DFT measurements. Two separate peaks are observed at 2.110 67 eV and 2.114 35 eV, with equal FWHM of 2.7 meV. Two Gaussian lines were used to fit the individual components (blue), whereas the red line is the addition of the two Gaussian lines and the black line is the experimental data.

Close modal

The exciton absorption spectra collected from a large area of the sample using the FTIR spectrometer are shown in Fig. 2(a). A doublet is marginally resolved in the spectra. The line width is dominated by inhomogeneous broadening likely due to the varying thickness of the sample, stacking disorder, and the occurrence of the γ− GaSe polytype. The mixing of the two ε − and γ− polytypes is thought to lead to an energy shift of the exciton smaller than ∼0.2 meV.39 The inhomogeneous linewidth of ∼6.5 meV is comparable with what has been reported in the literature.22 The doublet structure of the excitonic peak is well resolved in the data shown in Fig. 2(b), obtained instead using the excitation laser pulse from a small area of the sample with uniform thickness. The thickness was estimated using the absorption coefficient measured in Ref. 40 to be ∼0.9 μm. The doublet is well resolved with a 3.7 meV energy separation between the two peaks. The energy splitting of the fine structure is too large to originate from excitons residing in the two different polytypes. The linewidths are ∼2.7 meV and are significantly narrower than when averaged over a large area of the sample.

To further investigate the origin of the excitonic structure, we performed 2DFT measurements using different polarizations of the excitation laser pulses. 2DFT measurements can provide useful information about the origin of the electronic transitions. The 2DFT spectra at 5 K using co-linearly polarized excitations are shown in Fig. 3(b). The linear polarizations (XXXX) correspond to pulses A, B, C, and detection, respectively. Two main peaks labeled as A and B are observed along the diagonal dashed line. The absorption spectra are shown in Fig. 3(a) at the same energy scale. Peak A energetically coincides with the dominant high energy peak observed in the absorption spectra, whereas B energetically corresponds to the weaker low energy peak. The appearance of peak B along the diagonal in the 2DFT spectra excludes biexciton formation as the origin of this resonance, since the biexciton peak is expected to appear off-diagonal.41 Biexciton effects are predicted to be strong in this material, and the energy of the separation of peak B from the resonance A scales reasonably well with the exciton binding energy.42 Thus, 2DFT spectroscopy is advantageous compared to other linear and nonlinear techniques for discerning between biexciton features from a single resonance and two individual resonances.

FIG. 3.

(a) Red line is the absorption spectrum of the 1s exciton from the 0.9 μm uniform region of the sample. Black line is the excitation laser spectrum. (b) Magnitude and (c) real part of the 2DFT spectra of the GaSe 1s exciton using co-linearly (XXXX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively. Diagonal peaks are labeled as A and B, where A corresponds to the main 1s exciton peak, whereas B corresponds to the weaker low energy peak observed in the absorption spectra. The weak cross-peak in the 2DFT spectra above the diagonal is labeled as C. (d) Schematic of the transitions taking place. (e) Magnitude and (f) real part of the 2DFT spectra of the GaSe 1s exciton using cross-linearly (XYYX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively. Using the cross-polarized sequence the diagonal peaks are suppressed, revealing the existence of an off-diagonal peak, labeled as D.

FIG. 3.

(a) Red line is the absorption spectrum of the 1s exciton from the 0.9 μm uniform region of the sample. Black line is the excitation laser spectrum. (b) Magnitude and (c) real part of the 2DFT spectra of the GaSe 1s exciton using co-linearly (XXXX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively. Diagonal peaks are labeled as A and B, where A corresponds to the main 1s exciton peak, whereas B corresponds to the weaker low energy peak observed in the absorption spectra. The weak cross-peak in the 2DFT spectra above the diagonal is labeled as C. (d) Schematic of the transitions taking place. (e) Magnitude and (f) real part of the 2DFT spectra of the GaSe 1s exciton using cross-linearly (XYYX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively. Using the cross-polarized sequence the diagonal peaks are suppressed, revealing the existence of an off-diagonal peak, labeled as D.

Close modal

Furthermore, a third weak cross-peak is observed labeled as C. The fact that a cross peak appears between the two resonances A and B suggests that they are correlated and likely belong to the same center. Therefore, it is unlikely that the A and B resonances originate from excitons in two different ε and γ polytypes at different locations in the sample. The appearance of peak C suggests the presence of coherent energy transfer between the A and B excitons, mediated by many-body interactions. The fact that the lower energy peak B is weaker than the higher energy peak A indicates a weaker transition dipole moment for peak B compared to peak A.

In order to retrieve the real part of the complex FWM signal field amplitude, two methods were used to obtain the absolute phase. The absolute phase of the signal was obtained using the method described in Ref. 43. Furthermore, the projection of the real part of the 2DFT spectra on the ωt axis was also compared with differential absorption data to probe the accuracy of the retrieved phase.26 The real parts are shown in Figs. 3(c) and 3(f) and show dispersive line shapes.

Previous studies have shown that using a cross-polarized excitation scheme (XYYX) isolates many-body effects, specifically in this case, bound biexcitons and unbound two-excitons contributions. As a result, the biexciton peak can be enhanced with respect to the exciton resonance. Again, the polarizations correspond to A, B, C, and detection, respectively. The 2DFT spectra using cross-polarized excitations are shown in Fig. 3(e). Peak B is too weak to be observed, whereas the stronger peak A is significantly weaker. As a result, the cross peak C also disappears. However, a new peak appears shifted off the diagonal along ωt. The position of the peak D in the 2DFT spectra and the energy separation toward lower energies along ħωt from the main exciton peak A by ∼2 meV suggest that peak D is due the creation of biexcitons. Although biexcitonic effects are thought to dominate in this material, the strength of peak D indicates only moderate biexciton creation with a binding energy of ∼2 meV.

In order to probe the relaxation pathway between resonances A and B, 2DFT spectra at three different temperatures were collected. The temperature dependent 2DFT spectra are shown in Fig. 4 at 5, 20, and 40 K. Furthermore, 2DFT spectroscopy enables the accurate measurement of the homogeneous broadening of excitonic resonances. The lineshape along the diagonal of the 2DFT spectra, marked with a dashed line in Figs. 4(a)-4(f), reflects the inhomogeneous width of the excitonic ensemble, whereas the cross diagonal profile provides the homogeneous linewidth.44 The homogeneous linewidth increases with temperature as a result of electron-phonon interactions. In addition, the resonances shift toward lower energies with increasing temperature as a result of the temperature shift of the band gap.

FIG. 4.

(a)-(c) Magnitude and (d)-(f) real part of the 2DFT spectra of the GaSe 1s exciton using co-linearly (XXXX) polarized excitation at different temperatures, (a) and (d) at 5 K, (b) and (e) at 20 K, and (c) and (f) at 40 K. Peaks B and C become weaker with increasing temperature. All peaks shift towards lower energy with increasing temperature as a result of the band gap renormalization.

FIG. 4.

(a)-(c) Magnitude and (d)-(f) real part of the 2DFT spectra of the GaSe 1s exciton using co-linearly (XXXX) polarized excitation at different temperatures, (a) and (d) at 5 K, (b) and (e) at 20 K, and (c) and (f) at 40 K. Peaks B and C become weaker with increasing temperature. All peaks shift towards lower energy with increasing temperature as a result of the band gap renormalization.

Close modal

With increasing temperature, the intensity ratio between peaks A and B changes. At 20 K (Fig. 4(b)), all three peaks are still visible; however, peak B has become weak with respect to peak A. The 2DFT spectra at 40 K are shown in Fig. 4(c). Peaks B and C are completely absent, whereas peak A is the only dominant resonance peak in the 2DFT spectra. The temperature dependence of the 2DFT spectra shows a change in the population ratio between peaks A and B, which could be caused by phonon scattering of excitons at a higher rate for resonance peak B.

The weaker peak B at lower energy could suggest that it originates from the indirect band gap. However, the transitions from the indirect to the direct must involve excitations with momentum conserving phonons. Such transitions are expected to have a low probability and would have very weak absorption spectra. In the absorption spectra of Fig. 2, peak B has a strength of at least 25% of peak A, which suggest a direct transition. Furthermore, the direct and indirect transitions in γ− GaSe have been carefully studied using absorption and photoluminescence spectroscopy.38 The energy separation between the lowest direct excitonic transition and the indirect band gap is thought to be at least 6 meV. The energy separation between the lowest direct excitonic transition and impurity bound excitons at the indirect gap is even larger ∼20 meV. Recent studies of the electronic structure of GaSe at high pressure indicate a much larger energy difference between the direct and indirect band gaps.14,45 Therefore, an indirect to direct band gap relaxation is rather unlikely.

Finally, the exchange interaction can give rise to an additional fine structure of the exciton states. The band mixing induced by the spin orbit coupling is relatively weak, therefore the multiplet character of the exciton states should be well preserved.39 The exchange interaction leads to two components, one singlet and one triplet. Substructure of the excitonic resonance has been observed in absorption spectra, both in predominantly ε − and γ− polytypes.38,39 The splitting between the two components was 1.9 ± 0.1 meV, somewhat smaller than the splitting observed in the present study. Using the assignment of Refs. 39 and 38, we associate the singlet state E(1s)4) with the higher energy peak A and the triplet state E(1s)6) with the lower energy peak B. The singlet component E(1s)4) is highly dependent on the incoming polarization with respect to the plane of incidence of the exciting radiation.39 The size of the energy splitting between the peaks A and B, although somewhat larger, is comparable to the excitonic fine structure observed earlier. The size of the singlet-triplet splitting depends on the strength of the spin-orbit coupling which appears to be affected by the mixture of the ε − and γ− polytypes in the GaSe crystal. Therefore, the sample exhibits large inhomogeneous broadening and sample dependent energy splitting of the excitonic ground state. Furthermore, the assignment of peaks A and B to the singlet and triplet states is in agreement with both transitions originating at the direct band gap and belonging to the same center as previously revealed by the coupling peak C. The temperature and polarization dependent 2DFT measurements can be understood in terms of the exchange splitting of the excitonic ground state. Therefore, we assign peak A to the singlet state E(1s)4) and peak B to the triplet state E(1s)6) of the direct exciton.

In order to understand the mechanism giving rise to peak C, we use a phenomenological model to simulate coupling between the singlet and triplet exciton states. Without making any assumptions regarding whether these exciton resonances are coupled, one can map two independent two-level systems into a four-level system through a Hilbert space transformation, as illustrated in Fig. 5(a). State | g ¯ refers to the crystal ground state prior to optical excitation; states | X B ¯ and | X A ¯ represent the triplet and singlet exciton states, respectively; and state | X A B ¯ represents simultaneous excitation of the singlet and triplet. The system dynamics, including incoherent population relaxation and dephasing processes, can be described using the density matrix whose elements are connected to the coherent nonlinear signal via the optical Bloch equations (OBE).

FIG. 5.

(a) Level scheme describing the excitonic system: two independent two-level systems are coupled to a four-level system through a Hilbert space transformation. Simulations using the optical Bloch equations, where many-body effects such EID and EIS are included phenomenologically. (b) Magnitude and (c) real part of the 2DFT spectra of the GaSe 1s exciton using co-linearly (XXXX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively. (d) Magnitude and (e) real part of the 2DFT spectra of the GaSe 1s exciton using cross-linearly (XYYX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively.

FIG. 5.

(a) Level scheme describing the excitonic system: two independent two-level systems are coupled to a four-level system through a Hilbert space transformation. Simulations using the optical Bloch equations, where many-body effects such EID and EIS are included phenomenologically. (b) Magnitude and (c) real part of the 2DFT spectra of the GaSe 1s exciton using co-linearly (XXXX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively. (d) Magnitude and (e) real part of the 2DFT spectra of the GaSe 1s exciton using cross-linearly (XYYX) polarized excitation. The polarizations correspond to A, B, C, and detection, respectively.

Close modal

We perturbatively solve the optical Bloch equations up to third order in the excitation field to simulate the two-dimensional spectra in Figs. 3(b)-3(f). The four-level energy scheme allows for the inclusion of exciton many-body effects phenomenologically. The lower transitions ( | g ¯ | X A ¯ ) and ( | g ¯ | X B ¯ ) are excited to first order in perturbation theory, whereas the upper transitions ( | X A ¯ | X A B ¯ ) and ( | X B ¯ | X A B ¯ ) can contribute to third order provided the lower transitions have been excited; therefore, the electronic and optical properties of the upper transitions dictate the type and strength of the interactions.46 Here, we consider the exciton renormalization energy for which the real and imaginary parts correspond to excitation-induced energy shift (EIS) and EID effects. EIS effects are modeled by breaking the energy equivalence of the upper and lower transitions through a shift ΔAB of state | X A B ¯ . EID effects are introduced by increasing the dephasing rate of the upper state transition with respect to its equivalent lower transition by an amount γ′.

This phenomenological technique has been applied to describe coupling between semiconductor double quantum wells,47 coherent coupling between neutral and charged excitons in modulation doped quantum wells48 and atomically thin transition metal dichalcogenides,49 and biexciton effects in quantum dots.50,51 Here, we show that the singlet and triplet exciton states in layered GaSe are coherently coupled through both EIS and EID effects. Simulated absolute value and real 2DFT spectra are shown in Figs. 5(b) and 5(c) for the co-linearly polarized case. The measured homogeneous and inhomogeneous linewidths and relative magnitudes from the absolute value spectrum and the positive/negative amplitudes from the real spectrum are used as input parameters for the simulation. We note that the inclusion of additional doubly excited states representing interactions between two singlet excitons or two triplet excitons is necessary in order to reproduce the dispersive lineshape of the A and B peaks in the real part of the 2D spectra in Figs. 3(c) and 3(f). We refer the reader to Ref. 48 and references therein for details of the calculations. The simulation qualitatively reproduces the essential features of Figs. 3(b) and 3(c) when including ΔAB ≅ 0.1 meV (EIS) and γAB ≅ 2γB (EID), where γAB and γB are the dephasing rates of transitions ( | X B ¯ | X A B ¯ ) and ( | g ¯ | X B ¯ ) , respectively.

The inclusion of EIS effects gives rise to off-diagonal cross peaks both above and below the diagonal dashed line. In order to model the asymmetry between the peak amplitudes, it is necessary to include EID effects as previously described. The presence of an upper cross peak (peak C) and the apparent absence of a cross-peak below the diagonal indicate that excitation of lower-energy triplet excitons causes additional pure dephasing of singlet excitons through triplet-singlet many-body interactions. We rule out incoherent energy transfer from the triplet to the singlet state as the origin of the coupling peak C due the zero delay T = 0 fs used for the experiments; this result implies that incoherent uphill energy transfer would have to occur within the pulse duration, which is unlikely. This conclusion is further supported by the absence of any significantly populated phonon modes at the singlet-triplet energy separation of ∼3.7 meV at 5 K. Therefore, the presence of the coupling peaks indicates that energy transfer between the singlet and triplet states can occur coherently since these states are coherently coupled. Future two-quantum experiments investigating such coupling can be performed by scanning the delay T while holding the delay τ fixed, which could isolate coherent effects from Pauli blocking nonlinearities.34 

We further examine interactions between two singlet excitons by simulating the measured spectra in Figs. 3(e) and 3(f) for cross-linearly polarized excitation. The absolute value and real parts of the simulated spectrum are shown in Figs. 5(d) and 5(e). The spectra are obtained by generating expressions using the energy level diagram in Fig. 5(a) for linearly polarized selection rules, which is also equivalent to the uncoupled two-level systems as long as equivalence of the upper and lower transitions is maintained. Through the linearly polarized dipole selection rules, the energy scheme permits the inclusion of two independent many-body contributions to the nonlinear spectra. The first arises from the bound biexciton (peak D) red-shifted along the ħωt axis by the biexciton binding energy of ∼2 meV. The second term reproduces peak A on the diagonal, which arises from the scattering state of two interacting singlet excitons.

The energy shift of the electronic band gap with temperature can provide important details on the electron-phonon interactions taking place. Temperature dependent absorption spectra of the exciton provide the energy shift of the band gap with temperature and the broadening of the excitonic resonance. Both effects are in first approximation, a result of electron-phonon interactions.52 The renormalization of the electron with energy E k o and wave vector k caused by the electron-phonon interactions can be written as

E k = E k o + Δ k DW + Δ k SE + i γ k ,
(1)

where Δ k DW is the energy shift induced by the Debye-Waller term. The second term, the complex self-energy has a real part Δ k SE , which leads to an energy shift of the electronic states, and an imaginary part γk, which causes a lifetime broadening of the electronic states.53–56 The evaluation of both the self-energy and Debye-Waller contributions is quite challenging. Therefore, approximate models have been often used in the evaluation of the temperature dependence of the band gap of solids.56 

The energy shift of the exciton resonance with temperature obtained from the absorption spectra is shown in Fig. 6(a), where the black squares are the experimental data and the red line is the theoretical fit. The fitting procedure was performed using the approximate model introduced in Ref. 56,

E g = E 0 E 1 2 ( exp Θ k T 1 ) 1 + 1 ,
(2)

where E0 and E1 are fitting parameters, whereas Θ is an average phonon energy. The exact determination of the average phonon energy is difficult, since it requires complicated integration over the entire Brillouin zone. When the average phonon energy is exactly known, the two coefficients E0 and E1 correspond to the unrenormalized “bare” band gap and the “zero-Kelvin” renormalization energy, respectively.57 However, it is common to use the energy of a dominant phonon for Θ. Excellent fitting of the experimental data was achieved using Θ = 16.6 meV corresponding to the energy of the A 1 phonon, E0 = 2.1596 eV and E1 = 47.2 meV.

FIG. 6.

(a) Energy position of the exciton as a function of temperature obtained from absorption spectra reflecting the temperature dependence of the band gap. The black squares are the experimentally measured energy positions, whereas the red line is the fitting using Eq. (2). (b) The linewidth of the exciton obtained from absorption spectra as a function of temperature. The black squares are experimentally measured widths whereas the red line is the fit using Eq. (3). (c) Purely homogeneous linewidths obtained from the cross-diagonal profiles of the 2DFT spectra as a function of temperature. Black squares are the experimental data, whereas the red line is a linear fit which takes into account scattering with acoustic phonons. (d) Partial phonon density of states calculated “ab initio.” The blue line corresponds to the Ga atom contribution to the phonon DOS, whereas the red is the Se atom contribution. (e) The calculated out-of-plane A 1 phonon mode.

FIG. 6.

(a) Energy position of the exciton as a function of temperature obtained from absorption spectra reflecting the temperature dependence of the band gap. The black squares are the experimentally measured energy positions, whereas the red line is the fitting using Eq. (2). (b) The linewidth of the exciton obtained from absorption spectra as a function of temperature. The black squares are experimentally measured widths whereas the red line is the fit using Eq. (3). (c) Purely homogeneous linewidths obtained from the cross-diagonal profiles of the 2DFT spectra as a function of temperature. Black squares are the experimental data, whereas the red line is a linear fit which takes into account scattering with acoustic phonons. (d) Partial phonon density of states calculated “ab initio.” The blue line corresponds to the Ga atom contribution to the phonon DOS, whereas the red is the Se atom contribution. (e) The calculated out-of-plane A 1 phonon mode.

Close modal

The absorption spectra of the exciton at low temperature are mostly inhomogeneously broadened. However, as the temperature rises, the phonon scattering leads to observable broadening since the lineshapes are a convolution of homogeneous and inhomogeneous contributions. The linewidths of the exciton absorption spectra as a function of temperature are plotted in Fig. 6(b), where the black squares are the experimental data, whereas the red line is the theoretical fit. At low temperature, the acoustic phonon scattering is the dominant line broadening mechanism, whereas at higher temperatures, optical phonons become more important. The theoretical fit was performed according to

γ = γ I + a T + b exp ( Θ / k T ) 1 ,
(3)

where γI is the temperature independent inhomogeneous broadening. The homogeneous “zero-Kelvin” broadening γ is much smaller and can be neglected in this case. The linear term is due to the line broadening as a result of acoustic phonon scattering, whereas the phonon population term takes into account optical phonon broadening. Since the inhomogeneous broadening dominates at low temperature, the linear term has to also be neglected. Taking into account the temperature independent inhomogeneous contribution to the linewidth γI ∼ 6.5 meV, the fitting leads to an optical phonon scattering parameter of b = 8.92 meV. The phonon energy Θ = 16.6 meV used in the fitting corresponds to the energy of the A 1 phonon.

At low temperature, the inhomogeneous broadening dominates the exciton linewidth, therefore it is difficult to estimate the scattering by acoustic phonons. However, using 2DFT spectroscopy, the homogeneous linewidth can be measured directly. Exciton-phonon interaction can lead to efficient excitonic dephasing, which reflects itself in changes of the cross-diagonal profile of temperature dependent 2DFT measurements. In particular, due to the anisotropic nature of layered crystals, the carrier-phonon interactions can lead to peculiar effects. In order to further investigate the effect of acoustic phonon scattering, 2DFT spectra were collected at different temperatures. The homogeneous widths obtained from the cross diagonal profile of the 2DFT spectra are plotted as a function of temperature in Fig. 6(c). The elastic acoustic phonon scattering results in a linear dependence with temperature according to γ = γ + aT. The temperature independent homogeneous linewidth γ reflects the dephasing caused by the remaining temperature independent contributions. In order to estimate the effect of acoustic phonon scattering, a linear fit was used in Fig. 6(c) (red line). The fitting procedure leads to a temperature independent homogeneous linewidth γ = 0.74 meV and an acoustic phonon scattering coefficient a ∼ 20 μeV/K.

In order to further investigate the phonon modes thought be the predominant vibrations leading to excitonic dephasing, total energy calculations were performed within the framework of the density functional theory and the projector-augmented wave method, as implemented in the Vienna “ab initio” simulation package (VASP).58–63 We used a plane-wave energy cutoff of 620 eV in order to ensure high precision in our calculations. The exchange and correlation energy was described within the generalized gradient approximation in the Perdew-Burke-Ernzerhof functional.63 The Monkhorst-Pack scheme was employed for the Brillouin-zone integrations, with a mesh of 12 × 12 × 3 for the hexagonal setup with 8 atoms per unit cell.64 In the configuration study, GaSe is composed of a series of layers with strong van der Waals interactions along the c axis. In order to account for these forces, we have used the proposed van der Waals functional as described in Ref. 65 and implemented by VASP in Ref. 66. The optimized ground state corresponds to a crystal structure with P 6 ̄ m 2 symmetry (space group 187) and with cell parameters of a = 3.72 Å and c = 15.13 Å (the primitive cell has only 4 atoms per unit cell).

In the equilibrium configuration, the forces are less than 3 meV/Å per atom along each of the cell directions. The calculation of the dynamical matrix using the direct force constant approach (or supercell method), as implemented in the Phonopy code with a 3 × 3 × 3 supercell, requires highly converging results on forces.67 The construction of the dynamical matrix at the Γ point is performed by using separate calculations of the forces, in which a fixed displacement from the equilibrium position of the atoms within the unit cell is considered. The symmetry reduces the number of such independent distortions to 8 independent displacements in the 8 atom unit cell. The calculated partial phonon density of states (DOS) is shown in Fig. 6(d). A close examination of the phonon energies at the Γ point reveals a mode at 16.6 meV with a A 1 irreducible representation, which corresponds to the vibration of the Ga and Se layers moving against each other as shown in Fig. 6(e). At the Γ point, no other phonon mode can be found in this energy range, which indicates that this is the only out-of-plane vibration of the Ga and Se atoms that dominates the exciton-phonon interactions. The amplitude of the A 1 vibration is almost the same for Ga and Se atoms, which implies that the layers are moving almost rigidly with respect to each other.

The homogeneous broadening at low temperature is also affected by excitation induced effects, such as exciton-exciton and exciton-carrier scattering.68 In order to estimate the limiting homogeneous linewidth in our sample and investigate the role of excitation induced effects, we performed 2DFT experiments at different excitation powers. Excitation powers from 2.1 μJ/cm2 and all the way to 0.58 μJ/cm2 per beam were used to excite the sample. Some reduction in homogeneous linewidth was observed over this excitation range, indicating excitation induced broadening effects. It was shown early on that excitation induced dephasing is enhanced as a result of reducing the dimensions in GaAs quantum wells as compared to the bulk crystals.69 Strong exciton-exciton scattering induced dephasing is common in nanomaterials as a result of the reduced dimensionality and has also been observed in single-walled semiconducting carbon nanotubes.70 

The homogeneous linewidth is measured using the cross-diagonal linewidth of the 2DFT spectra and the obtained linewidths are plotted as a function of excitation density in Fig. 7. The data are fitted using a known linear dependence γ = γ + βnx, where γ is the density independent homogeneous width, β is the exciton-exciton and exciton-carrier scattering coefficient, and nx the excitation density.69–71 As a result of the fitting procedure, we obtain a residual excitation density independent homogeneous linewidth γ of ∼0.61 meV.

FIG. 7.

Homogeneous linewidths of the exciton obtained from cross-diagonal fits of the 2DFT spectra as a function of excitation density. Black squares are the experimental values, whereas the red line is a linear fit.

FIG. 7.

Homogeneous linewidths of the exciton obtained from cross-diagonal fits of the 2DFT spectra as a function of excitation density. Black squares are the experimental values, whereas the red line is a linear fit.

Close modal

We performed linear absorption and 2DFT spectroscopy combined to study the direct 1s excitons in GaSe. Several possibilities were considered for the origin of the excitonic ground state splitting. The doublet was assigned to a splitting of the excitonic ground state into the E(1s)4) singlet (peak A) and E(1s)6) triplet (peak B). The 2DFT spectra reveal a cross-peak between the two main resonances A and B, revealing coherent energy transfer from the triplet to the singlet state. Polarization dependent 2DFT spectroscopy reveals a new peak likely originating from biexcitons. Simulations using OBE where many-body effects are included phenomenologically reproduced the experimental 2DFT spectra. Although biexciton effects are thought to be strong in this material, only moderate contributions from biexciton creation could be observed. The biexciton binding energy was measured from the experimental data to be ∼2 meV. Temperature dependent absorption and 2DFT measurements, combined with “ab initio” theoretical calculations of the phonon spectra, indicate strong interaction with the A 1 phonon mode. Finally, excitation density dependent 2DFT measurements reveal excitation induced dephasing and reveal the lower limit for the homogeneous linewidth of the excitons in the present GaSe crystal to be ∼0.61 meV.

The research at USF is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-SC0012635. A.H.R. recognizes the support of the Marie Curie Actions from the European Union in the international incoming fellowships (Grant No. PIIFR-GA-2011-911070) and the computer resources provided by RES (Red Espãnola de Supercomputacíon), the TACC-Texas Supercomputer Center, and MALTA-Cluster. A.C. acknowledges support from project MAT2012-33483 of the Spanish Agency for Research.

1.
H. S.
Lee
,
S.-W.
Min
,
Y.-G.
Chang
,
M. K.
Park
,
T.
Nam
,
H.
Kim
,
J. H.
Kim
,
S.
Ryu
, and
S.
Im
,
Nano Lett.
12
,
3695
(
2012
).
2.
T.
Georgiou
,
R.
Jalil
,
B. D.
Belle
,
L.
Britnell
,
R. V.
Gorbachev
,
S. V.
Morozov
,
Y.-J.
Kim
,
A.
Gholinia
,
S. J.
Haigh
,
O.
Makarovsky
,
L.
Eaves
,
L. A.
Ponomarenko
,
A. K.
Geim
,
K. S.
Novoselov
, and
A.
Mishchenko
,
Nat. Nanotechnol.
8
,
100
(
2013
).
3.
K. F.
Mak
,
C.
Lee
,
J.
Hone
,
J.
Shan
, and
T. F.
Heinz
,
Phys. Rev. Lett.
105
,
136805
(
2010
).
4.
H.
Zeng
,
J.
Dai
,
W.
Yao
,
D.
Xiao
, and
X.
Cui
,
Nat. Nanotechnol.
7
,
490
(
2012
).
5.
K. F.
Mak
,
K.
He
,
J.
Shan
, and
T. F.
Heinz
,
Nat. Nanotechnol.
7
,
494
(
2012
).
6.
T.
Cao
,
G.
Wang
,
W.
Han
,
H.
Ye
,
C.
Zhu
,
J.
Shi
,
Q.
Niu
,
P.
Tan
,
E.
Wang
,
B.
Liu
, and
J.
Feng
,
Nat. Commun.
3
,
887
(
2012
).
7.
S.
Wu
,
J. S.
Ross
,
G.-B.
Liu
,
G.
Aivazian
,
A.
Jones
,
Z.
Fei
,
W.
Zhu
,
D.
Xiao
,
W.
Yao
,
D.
Cobden
, and
X.
Xu
,
Nat. Phys.
9
,
149
(
2013
).
8.
A. M.
Jones
,
H.
Yu
,
N. J.
Ghimire
,
S.
Wu
,
G.
Aivazian
,
J. S.
Ross
,
B.
Zhao
,
J.
Yan
,
D. G.
Mandrus
,
D.
Xiao
,
W.
Yao
, and
X.
Xu
,
Nat. Nanotechnol.
8
,
634
(
2013
).
9.
K. F.
Mak
,
K.
He
,
C.
Lee
,
G.
Hyoung
,
J.
Hone
,
T. F.
Heinz
, and
J.
Shan
,
Nat. Mater.
12
,
207
(
2012
).
10.
J. A.
Schuller
,
S.
Karaveli
,
T.
Schiros
,
K.
He
,
S.
Yang
,
I.
Kymissis
,
J.
Shan
, and
R.
Zia
,
Nat. Nanotechnol.
8
,
271
(
2013
).
11.
H.
Tahara
,
Y.
Ogawa
, and
F.
Minami
,
Phys. Rev. Lett.
107
,
037402
(
2011
).
12.
Z.
Zhu
,
Y.
Cheng
, and
U.
Schwingenschlögl
,
Phys. Rev. Lett.
108
,
266805
(
2012
).
13.
M.
Gauthier
,
A.
Polian
,
J. M.
Besson
, and
A.
Chevy
,
Phys. Rev. B
40
,
3837
(
1989
).
14.
D.
Olguin
,
A.
Rubio-Ponce
, and
A.
Cantarero
,
Eur. Phys. J. B
86
,
350
(
2013
).
15.
S.
Nüsse
,
P. H.
Bolivar
,
H.
Kurz
,
V.
Klimov
, and
F.
Levy
,
Phys. Rev. B
56
,
4578
(
1997
).
16.
M.
Schlüter
,
IL Nuovo Cimento B
13
,
313
(
1973
).
17.
V.
Augelli
,
C.
Manfredotti
,
R.
Murri
, and
L.
Vasanelli
,
Phys. Rev. B
17
,
3221
(
1978
).
18.
P.
Schmid
and
J.
Voitchovsky
,
Phys. Status Solidi B
65
,
249
(
1974
).
19.
G.
Antonioli
,
D.
Bianchi
,
U.
Emiliani
,
P.
Podini
, and
P.
Franzosi
,
II Nuovo Cimento B
54
,
211
(
1979
).
20.
N.
Piccioli
and
R. L.
Toullec
,
J. Phys. (France)
50
,
3395
(
1989
).
21.
P.
Dey
,
J.
Paul
,
J.
Bylsma
,
S.
Deminico
, and
D.
Karaiskaj
,
Rev. Sci. Instrum.
84
,
023107
(
2013
).
22.
F.
Minami
,
A.
Hasegawa
,
T.
Kuroda
, and
K.
Inoue
,
J. Lumin.
53
,
371
(
1992
).
23.
H.
Tahara
,
Y.
Ogawa
, and
F.
Minami
,
Phys. Rev. B
82
,
113201
(
2010
).
24.
T.
Kishimoto
,
A.
Hasegawa
,
Y.
Mitsumori
,
J.
Ishi-Hayase
,
M.
Sasaki
, and
F.
Minami
,
Phys. Rev. B
74
,
073202
(
2006
).
25.
A. D.
Bristow
,
D.
Karaiskaj
,
X.
Dai
,
T.
Zhang
,
C.
Carlsson
,
K. R.
Hagen
,
R.
Jimenez
, and
S. T.
Cundiff
,
Rev. Sci. Instrum.
80
,
073108
(
2009
).
26.
D. M.
Jonas
,
Annu. Rev. Phys. Chem.
54
,
425
(
2003
).
27.
S. T.
Cundiff
,
Opt. Express
16
,
4639
(
2008
).
28.
J.
Kim
,
S.
Mukamel
, and
G. D.
Scholes
,
Acc. Chem. Res
42
,
1375
(
2009
).
29.
S.
Mukamel
,
D.
Abramavicius
,
L.
Yang
,
W.
Zhuang
,
I. V.
Schweigert
, and
D. V.
Voronine
,
Acc. Chem. Res
42
,
553
(
2009
).
30.
S.
Mukamel
,
Y.
Tanimura
, and
P.
Hamm
,
Acc. Chem. Res
42
,
1207
(
2009
).
31.
M.
Cho
,
Two-Dimensional Optical Spectroscopy
(
CRC Press
,
2010
), p.
378
.
32.
P.
Hamm
and
M.
Zanni
,
Concepts and Methods of 2D Infrared Spectroscopy
(
Cambridge University Press
,
2011
), p.
286
.
33.
D.
Karaiskaj
,
A. D.
Bristow
,
L.
Yang
,
X.
Dai
,
R. P.
Mirin
,
S.
Mukamel
, and
S. T.
Cundiff
,
Phys. Rev. Lett.
104
,
117401
(
2010
).
34.
K. W.
Stone
,
K.
Gundogdu
,
D. B.
Turner
,
X.
Li
,
S. T.
Cundiff
, and
K. A.
Nelson
,
Science
324
,
1169
(
2009
).
35.
D.
Turner
and
K.
Nelson
,
Nature
466
,
1089
(
2010
).
36.
J.
Bylsma
,
P.
Dey
,
J.
Paul
,
S.
Hoogland
,
E. H.
Sargent
,
J. M.
Luther
,
M. C.
Beard
, and
D.
Karaiskaj
,
Phys. Rev. B
86
,
125322
(
2012
).
37.
P.
Dey
,
J.
Paul
,
N.
Glikin
,
Z. D.
Kovalyuk
,
Z. R.
Kudrynskyi
,
A. H.
Romero
, and
D.
Karaiskaj
,
Phys. Rev. B
89
,
125128
(
2014
).
38.
A.
Mercier
,
E.
Mooser
, and
J. P.
Voitchovsky
,
Phys. Rev. B
12
,
4307
(
1975
).
39.
E.
Mooser
and
M.
Schluter
,
II Nuovo Cimento
18
,
164
(
1973
).
40.
R. L.
Toullec
,
N.
Piccioli
, and
J. C.
Chervin
,
Phys. Rev. B
22
,
6162
(
1980
).
41.
A. D.
Bristow
,
D.
Karaiskaj
,
X.
Dai
,
R. P.
Mirin
, and
S. T.
Cundiff
,
Phys. Rev. B
79
,
161305
(
2009
).
42.
A.
Quattropani
,
J. J.
Forney
, and
F.
Bassani
,
Phys. Status Solidi B
70
,
497
(
1975
).
43.
A. D.
Bristow
,
D.
Karaiskaj
,
X.
Dai
, and
S. T.
Cundiff
,
Opt. Express
16
,
18017
(
2008
).
44.
M. E.
Siemens
,
G.
Moody
,
H.
Li
,
A. D.
Bristow
, and
S. T.
Cundiff
,
Opt. Express
18
,
17699
(
2010
).
45.
U.
Schwarz
,
D.
Olguin
,
A.
Cantarero
,
M.
Hanfland
, and
K.
Syassen
,
Phys. Status Solidi B
244
,
244
(
2007
).
46.
K.
Bott
,
O.
Heller
,
D.
Bennhardt
,
S. T.
Cundiff
,
P.
Thomas
,
E. J.
Mayer
,
G. O.
Smith
,
R.
Eccleston
,
J.
Kuhl
, and
K.
Ploog
,
Phys. Rev. B
48
,
17418
(
1993
).
47.
G.
Nardin
,
G.
Moody
,
R.
Singh
,
T. M.
Autry
,
H.
Li
,
F.
Morier-Genoud
, and
S. T.
Cundiff
,
Phys. Rev. Lett.
112
,
046402
(
2014
).
48.
G.
Moody
,
I. A.
Akimov
,
H.
Li
,
R.
Singh
,
D. R.
Yakovlev
,
G.
Karczewski
,
M.
Wiater
,
T.
Wojtowicz
,
M.
Bayer
, and
S. T.
Cundiff
,
Phys. Rev. Lett.
112
,
097401
(
2014
).
49.
A.
Singh
,
G.
Moody
,
S.
Wu
,
Y.
Wu
,
N. J.
Ghimire
,
J.
Yan
,
D. G.
Mandrus
,
X.
Xu
, and
X.
Li
,
Phys. Rev. Lett.
112
,
216804
(
2014
).
50.
J.
Kasprzak
,
B.
Patton
,
V.
Savona
, and
W.
Langbein
,
Nat. Photonics
5
,
57
(
2011
).
51.
G.
Moody
,
R.
Singh
,
H.
Li
,
I. A.
Akimov
,
M.
Bayer
,
D.
Reuter
,
A. D.
Wieck
,
A. S.
Bracker
,
D.
Gammon
, and
S. T.
Cundiff
,
Phys. Rev. B
87
,
041304
(
2013
).
52.
P. B.
Allen
and
V.
Heine
,
J. Phys. C: Solid State Phys.
9
,
2305
(
1976
).
53.
D.
Olguin
,
M.
Cardona
, and
A.
Cantarero
,
Solid State Commun.
122
,
575
(
2002
).
54.
S.
Gopalan
,
P.
Lautenschlager
, and
M.
Cardona
,
Phys. Rev. B
35
,
5577
(
1987
).
55.
P.
Lautenschlager
,
P. B.
Allen
, and
M.
Cardona
,
Phys. Rev. B
33
,
5501
(
1986
).
56.
L.
Viña
,
S.
Logothetidis
, and
M.
Cardona
,
Phys. Rev. B
30
,
1979
(
1984
).
57.
M.
Cardona
and
M. L. W.
Thewalt
,
Rev. Mod. Phys.
77
,
1173
(
2005
).
58.
P. E.
Blöchl
,
Phys. Rev. B
50
,
17953
(
1994
).
59.
G.
Kresse
and
D.
Joubert
,
Phys. Rev. B
59
,
1758
(
1999
).
60.
G.
Kresse
and
J.
Hafner
,
Phys. Rev. B
47
,
558
(
1993
).
61.
G.
Kresse
and
J.
Hafner
,
Phys. Rev. B
49
,
14251
(
1994
).
62.
G.
Kresse
and
J.
Furthmüller
,
Phys. Rev. B
54
,
11169
(
1996
).
63.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
64.
H. J.
Monkhorst
and
J. D.
Pack
,
Phys. Rev. B
13
,
5188
(
1976
).
65.
M.
Dion
,
H.
Rydberg
,
E.
Schröder
,
D. C.
Langreth
, and
B. I.
Lundqvist
,
Phys. Rev. Lett.
92
,
246401
(
2004
).
66.
J.
Klimeš
,
D. R.
Bowler
, and
A.
Michaelides
,
Phys. Rev. B
83
,
195131
(
2011
).
67.
A.
Togo
,
F.
Oba
, and
I.
Tanaka
,
Phys. Rev. B
83
,
134106
(
2008
).
68.
L.
Schultheis
,
J.
Kuhl
,
A.
Honold
, and
C. W.
Tu
,
Phys. Rev. Lett.
57
,
1635
(
1986
).
69.
A.
Honold
,
L.
Schultheis
,
J.
Kuhl
, and
C. W.
Tu
,
Phys. Rev. B
40
,
6442
(
1989
).
70.
K.
Leo
,
E. O.
Göbel
,
T. C.
Damen
,
J.
Shah
,
S.
Schmitt-Rink
,
W.
Schäfer
,
J. F.
Müller
,
K.
Köhler
, and
P.
Ganser
,
Phys. Rev. B
44
,
5726
(
1991
).
71.
J.
Shah
,
Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures
(
Springer-Verlag
,
1999
).