Fig. 2(a) and (b) illustrates the polarization dependence of the PzR spectra of Cu2ZnSiS4in the vicinity of the direct band-edge at 10 and 300 K, respectively. The PzR spectra show redshifts of the transition energies and lineshape broadening characteristics as the temperature increases from 10 to 300 K. The results indicate that featureEex⊥ is observed in E⊥c polarization (open-circles) while fea-tureE||exonly appears in E||c polarization (open-triangles). In order to determine the transition energies accurately, we have performed a theoretical lineshape fit. The functional form used in the fitting procedure corresponds to a first derivative Lorentzian line shape function of the form[10–13]:
R
R = Re
i=1
Aiej˚i(E − Ei+ ji)−n (1)
Fig. 2. The unpolarized (solid-squares), E⊥c polarization (open-circles) and E||c polarization (open-triangles) PzR spectra of Cu2ZnSiS4at (a) 10 K and (b) 300 K. The solid lines are the least-squares fits of experimental data to Eq.(1). The obtained values of the transition energies denoted asEex⊥ andE||exare indicated by the arrows.
where i =⊥ or ||, Aiand ˚iare the amplitude and phase of the lineshape, Eiandiare, respectively, the energy and broadening parameter of the interband transitions. For the first derivative func-tional form, n = 2 is appropriate for the excitonic transitions. For M0
type three-dimensional critical point interband transitions, n = 0.5 is appropriate[12,13]. Our experimental signatures are more con-sistent with excitonic lineshape. From the spectral characteristics of the PzR spectra, the features are most probably originating from the interband excitonic transitions. Shown by the solid curves in Fig. 2are the least-squares fits to Eq.(1). Arrows under the curves in Fig. 2(a) and (b) show the positions of the two interband excitonic features,Eex⊥ andE||exat 10 and 300 K, respectively. The obtained values ofE⊥exandE||exat 10/300 K are 3.389± 0.003/3.323 ± 0.005 and 3.482± 0.003/3.413 ± 0.005 eV, respectively, and are listed in Table 1. The fitted values of the broadening parameters are also listed inTable 1. The unpolarized spectrum (solid-squares) can be regarded as a random superposition of the spectra with E⊥c and E||c polarizations. The energy difference between the low- and high-energy transitions represents the crystal-field splitting between the two levels of the valence band and will be discussed later.
At 10 K the broadening parameters for both excitonic features are∼40 meV and increase to ∼60 meV at 300 K. The results imply that the samples probably have many scattering centers which lead to the large values of the broadening parameters at low tempera-ture and thereby dominate the natural intrinsic exciton linewidth which probably is much narrower. The observation of excitons at room temperature implies that the exciton binding energy is larger than 25 meV. When the temperature increases from 10 to 300 K, the broadening parameters increase by∼kT, where T is the temper-ature difference. This suggests that the exciton binding energy is of order of several kT; otherwise, the excitons would become too broad to be observed. Furthermore, the similar broadening param-eter of E⊥c and E||c excitonic features which is ∼90 meV apart also suggests a fairly large binding energy of Cu2ZnSiS4. An esti-mate of the exciton binding energy may be obtained using the
Table 1
48 S. Levcenco et al. / Journal of Alloys and Compounds 506 (2010) 46–50
Fig. 3. A schematic representation of the plausible assignments for the observed optical transitions near direct band-edge for Cu2ZnSiS4.
expression for the binding energy of the ground-state exciton[14], Eb=− (re4)/2¯h2ε2, where e is the electron charge andε is the dielectric constant. The index of refraction n may be deduced from an empirical relation called Moss rule where n4Eg= 77[15]. Take the value of Eg= 3.05 eV[1], we may then evaluateε ≈ n2= 5.02. For a wide band gap semiconductor, a larger effective mass of electron and hole is expected[16]. Therefore, by assuming that the exciton reduced massr= 0.2me, the exciton binding energy is estimated to be∼110 meV. This result agrees well with the previous report on the orthorhombic Cu2Zn1−xMnxGeS4[17]. Shih et al. reported a magnetoreflectance study of orthorhombic Cu2Zn1−xMnxGeS4, where the results suggest a large exciton binding energy of the order 100 meV[17]. The experimental evidences thus far point to a general characteristic of the Cu2–II–IV–VI4compounds with orthorhombic structure of having a rather large exciton binding energy. A more systematic experimentation should be carried out to verify this property.
Adopting the band-structure calculation of Cu2ZnGeS4by Chen et al.[18], a schematic representation of the plausible assignments for the observed polarization-dependent PzR spectra of Cu2ZnSiS4
is presented inFig. 3. The observed near band-edge PzR spectra are attributed to the direct point transitions from the valence band maximum (VBM) to conduction band minimum (CBM). Con-struction of this schematic diagram at point is based upon two assumptions. First, the conduction and valence bands are assumed to be dominated by s-like and p-like, respectively. It is referred from the crystal structure’s tetrahedral bonding arrangement which arises from s-p3 hybridization. The degree of p-d hybridizations is assumed to be small and can be neglected. The s-like and p-like energy bands involved in the optical transitions shall have the same symmetry properties as the atomic functions s, p||zand p⊥z. The second assumption is that spin-orbit splitting is much less than the crystal-filed splitting and thus may be neglected. We infer this from the PzR spectrum which exhibits one transition for E⊥c and a distinctly higher transition for E||c. The energy of the transi-tion depends strongly on the orientatransi-tion of the polarizatransi-tion with respect to the crystallographic directions. This suggests that the orbital angular momentum of the p states is sufficiently quenched to render the spin–orbit interaction negligible. Thus the splitting between the E⊥c and E||c levels of the valence band shown inFig. 3 is attributed solely to the orthorhombic crystalline symmetry of Cu2ZnSiS4which is described by the space group Pmn21.
The temperature-dependent PzR spectra of Cu2ZnSiS4with E⊥c and E||c polarizations at 10, 77, 150, 225 and 300 K are shown in Fig. 4. The open-circles and triangles curves are, respectively, the experimental PzR spectra of E⊥c and E||c polarizations, while the solid curves are the least-squares fits to Eq.(1). Arrows below the curves inFig. 4show the positions of the two interband excitonic features. As the general property of most semiconductors, the exci-tonic transitions in the PzR spectra exhibit a redshift and lineshape broadening when the temperature is increased.
Fig. 4. The E⊥c polarization (open-circles) and E||c polarization (open-triangles) PzR spectra of Cu2ZnSiS4at 10, 77, 150, 225 and 300 K. The solid lines are the least-squares fits of experimental data to Eq.(1). The obtained transition energies are denoted by arrows.
Fig. 5shows the fitted data of temperature dependence of theE⊥ex andE||exexcitonic transition energies. The uncertainties in experi-mental data are expressed in the form of the representative error bars. The solid curves in Fig. 4are the least-squares fits to the Varshni empirical relationship[19]:
Eex(T) = Ei(0)− ˛iT2
ˇi+ T, (2)
where i =⊥ or ||, Ei(0) is the excitonic transition energy at 0 K, and
˛iandˇiare the Varshni coefficients. The constant˛iis related to the electron (exciton)–phonon interaction andˇiis closely related to the Debye temperature. The obtained values of Ei(0),˛iandˇi
corresponding to the excitonic transitions for Cu2ZnSiS4are listed inTable 2. For comparison purposes the numbers from previous reports on freestanding wurtzite WZ-GaN[20], GaAs[21], ZB-ZnSe [22]and ZB-ZnS[23]are listed inTable 2.
The temperature dependence of the interband excitonic tran-sition energiesEex⊥ and E||ex can also be fitted (dashed curve) by an expression containing the Bose–Einstein occupation factor for phonon[21,24]:
Eex(T) = Ei(0)− 2aBi
[exp(Bi/T) − 1], (3)
Fig. 5. Dependence on temperature of the excitonic transition energies of Cu2ZnSiS4. The open-circles and triangles are energies of theEex⊥ andEex|| excitonic transitions, respectively, with representative error bars. The dashed and solid curves represent the fit to the Varshni (Eq.(2)) and Bose–Einstein (Eq.(3)) expressions, respectively.
S. Levcenco et al. / Journal of Alloys and Compounds 506 (2010) 46–50 49
Table 2
Values of the Varshni and Bose Einstein parameters of Cu2ZnSiS4obtained by fitting the temperature dependence data of theEex⊥ andEex|| excitonic transition energies to Eqs.
(2)and(3), respectively. The corresponding values for the freestanding WZ-GaN, GaAs, ZB-ZnSe and ZB-ZnS are also listed for comparison.
Materials Feature E(0) (eV) ˛ (10−4eV/K) ˇ (K) aB(meV) B(K) represents the strength of the electron (exciton)–phonon interac-tion, andBicorresponds to the average phonon temperature. The fitted values for Ei(0), aBi, andBiare given inTable 2, and the corre-sponding values for freestanding WZ-GaN[20], GaAs[21], ZB-ZnSe [22]and ZB-ZnS[23]are also listed inTable 2for comparison.
The parameter ˛i in Eq. (2) can be related to aBi and Bi
in Eq.(3) by taking the high-temperature limit of both expres-sions, which yields to ˛i = 2aBi/Bi. Comparison of the values presented inTable 2shows that this relation is indeed satisfied.
From Eq.(3)it is straightforward to show that the high-temperature limit of the slope ofEex(T) versus T curve approaches the value of−2aBi/Bi, The calculated values of−2aBi/Bi forEex⊥ andEex||
equal to −0.43 and −0.41 meV/K for E⊥c and E||c, respectively, which agree well with the values of dEex⊥/dT = −0.41 meV/K and dEex|| /dT = −0.37 meV/K obtained from linear extrapolation in the higher temperature (200–300 K) PzR experimental data.
The fitted values of the broadening parameters of theE⊥exand Eex|| features for Cu2ZnSiS4with representative error bars are dis-played inFig. 6. The temperature dependence of the broadening parameters of semiconductors can be expressed as[21]:
i(T) = i0+ iLO
[exp(iLO/T) − 1], (4)
where i =⊥ or ||, the first term of Eq.(4) represents the broad-ening invoked from temperature-independent mechanisms, such as electron–electron interactions, impurity or dislocation, whereas the second term is caused by the electron (exciton)–longitudinal optical (LO) phonon (Fröhlich) interaction. The quantityiLO
rep-Fig. 6. Temperature-dependent broadening parameters of the excitonic features of Cu2ZnSiS4. The open-circles and triangles are broadening parameters of theE⊥exand E||exfeatures, respectively, with representative error bars. The solid curves represent the least-squares fits of the data to Eq.(4).
Table 3
Values of the0,LOandLOparameters of Cu2ZnSiS4obtained by fitting the temperature dependence of the broadening parameters of theE⊥exandEex|| features to Eq.(4). For comparison the numbers for GaAs and ZB-ZnSe are listed.
Materials Feature 0(meV) LO(meV) LO(K)
resents the strength of the electron (exciton)–LO phonon coupling whileiLOis the LO phonon temperature[21]. The solid curves in Fig. 6are least-squares fits to Eq.(4), which made it possible to eval-uatei0,iLOandiLOfor the excitonic transitions of Cu2ZnSiS4. The obtained values of these quantities are listed inTable 3together with the numbers for GaAs[25], and ZB-ZnSe[22].
As listed inTable 3, the values of⊥0and||0for the excitonic transitions are determined to be approximately 41 and 37 meV, respectively. The large values ofi0imply that the sample probably have many scattering centers such as compositional fluctuations, alloy scattering, electron–electron interaction, impurity and dislo-cation etc. FromTables 2 and 3, it can be seen thatiLOis higher than
iB. The smaller valuesiBto the relevant toiLOcan be caused by the contribution of the acoustic and optical phonons to the tem-perature variation of the excitonic gap energy wherebyiLOis due to the electron (exciton)–LO phonon coupling[21,24]. The values for the exciton–LO phonon-coupling parameters,⊥LOand||LO, obtained by the fit to Eq.(4)are 55 and 65 meV, respectively, which are considerably larger than those reported for GaAs (20 meV)[25]
and ZnSe (24 meV)[22]. The larger value ofLOmight be due to the much lager average LO phonon in comparison to that of ZnSe and GaAs. In addition, it is possible that a larger deformation poten-tial interaction, which may account for a significant fraction ofLO
in addition to the Fröhlich interaction, is responsible for the larger
LO. 4. Summary
The temperature dependence of the band-edge excitonic tran-sitions of Cu2ZnSiS4 single crystals were characterized by using polarization-dependent PzR in the temperature range between of 10 and 300 K. The single crystals of Cu2ZnSiS4 were grown by chemical vapor transport technique using iodine as transport agent. The PzR measurements were carried out on the as-grown basal plane with the normal along [2 1 0] and the c axis parallel to the long edge of the crystal platelet. The PzR spectra revealed polarization-dependentEex⊥ andE||exfeatures for E⊥c and E||c
polar-50 S. Levcenco et al. / Journal of Alloys and Compounds 506 (2010) 46–50
ization, respectively. The splitting of 90 meV between the E⊥c and E||c levels at the point of the valence band may be attributed solely to the orthorhombic crystalline symmetry of Cu2ZnSiS4
which is described by the space group Pmn21. From a detailed line-shape fit to the PzR spectra, the temperature dependence of the transition energies and broadening parameters of the band-edge excitons were determined accurately. The temperature depen-dence of the near band-edge excitonic transition energies has been analyzed by both Varshni- and Bose–Einstein-type expressions.
The parameters extracted from both expressions by extending into the high-temperature regime are found to agree reasonably well.
The parameters that describe the temperature dependence of the broadening function of the band-edge excitonic features have also been studied. The electron (exciton)–LO phonon-coupling constant,
LO, are determined to be considerably larger than that reported for GaAs and ZnSe. The larger value ofLOmight be due to the much lager average LO phonon in comparison to that of ZnSe and GaAs. In addition, it is possible that a larger deformation potential interac-tion, which may account for a significant fraction ofLOin addition to the Fröhlich interaction, is responsible for the largerLO. Acknowledgments
The authors acknowledge the support of National Science Coun-cil of Taiwan under Project Nos. NSC 098-2811-E-011-019, NSC 97-2112-M-011-001-MY3 and 98-2221-E-011-015-MY2.
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