Band Gap Bowing of Binary Alloys: Experimental Results Compared To

Home , Lattice constant

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Academy, manian B n n ∗ - - 1 − n edCzycholl Gerd and x n iepedapiain stereetoi bulk electronic their as applications widespread find C t a ftealy.Ti ae tdfcl ogo homo- grow to difficult miscibil- it makes the This and Cd alloyed effects alloys. geneously the separation of phase gap with ity realizati deal experimental cubi to the the the has in Also, is crystallize CdSe will pure modification. ZnSe of zincblende pure phase while stable the phase, that wurtzite fact or the considerably from discrepancy inates this view, of point perimental bet erdc oigefcsadaeeati h disorde the in exact are and effects a bowing reproduce costful, to computationally able albeit models, These nowadays. between h usrt nwihtesml a grown. was sample on the depend which often on will substrate and the difficult very is CdSe inv zincblende calculations ing for parameters input hence mat and of properties determination experimental the as to side, directly theoretical over carry problems These range. concentration d ieeprclpseudopotential empirical me like highly-sophisticated ods from systems, alloye bulk of III-V gaps band and the II-VI calculate to applied been has methods vreprcltgtbnigmdl (ETBM) models tight-binding empirical over oaie ai e nafiiesupercell finite a on set quantitativel basis give localized to fail results, also accurate approaches CPA standard the eta prxmto,CA.BcueteVAi nw to cases, most known in bowing is gap band VCA the underestimate the vastly po- Because (coherent CPA). methods approximation, function tential Green VC sophisticated approximation, more crystal interpolat or (virtual linear parameters simple relevant a of using either by translatio effective system and invariant an alloyed onto the systems map implicitly disordered and therefore set basis plane-wave a use admodels band dsprelsz,a ela ntencsaynumber necessary the on as well as size, supercell nd slsotie rmelpoercmaueet of measurements ellipsometric from obtained esults to fteconcentration the of ction kCd lk hsmcnutraly yamultiband a by alloys semiconductor ch ocluaeteeetoi rpriso uhsemiconduc- such of properties electronic the calculate To nteps,abodsetu fdfeetcomputational different of spectrum broad a past, the In h adgpo h lo.Ti ramn allows treatment This alloy. the of gap band the e x Zn b 1 x − = Zn x 9 ea eldfie oe ytmand system model well-defined a as Se 39Bee,Germany Bremen, 8359 and 0 ihnteEB,oecmo prahi to is approach common one ETBM, the Within . uiecne(L esrmnson measurements (PL) luminescence 1 9Bee,Germany Bremen, 59 − 11,12 rdt hoeia tight-binding theoretical to ared x Se b ih-idn T)mtosta s a use that methods (TB) tight-binding = x fHommel ef Zn 1 x . 1 n seati h alloy- the in exact is and 6e sreported. is eV 26 − x ebl ape vrtewhole the over samples bulk Se 6 rDT+LDA + DFT or sp 13 ito of riation 3 r oepromising more are tight- 2–5 4,8 osml two simple to rmteex- the From Romania 7 models, 10 nally erial olv- and ion the th- ig- A) on re r. d y c 2 tor alloys, we start from multiband sp3 tight-binding mod- from Ref.17. This Hamiltonian has already been used suc- els for the pure bulk semiconductor materials AC and BC cesfully for bulk as well as nanostructure calculations for sev- and then perform calculations on a finite ensemble of su- eral material systems. For detailed comparisons with other percells. We will make recommendations for the cell size well-established models, we refer to Refs. 18,19. Our spe- as well as for the necessary number of realizations, based cific choice of values for the material parameters for CdSe on the convergence of the bowing parameter. These recom- and ZnSe has been proven to be reliable in calculations for 20 mendations will then be verified by comparison to band gaps pure quantum dots as well as for mixed CdxZn1−xSe col- as determined from photoluminescence data for catalytically loidal nanocrystals.21 3 grown CdxZn1−xSe nanowires by Venugopal et al. We also present new experimental results obtained for CdxZn1−xSe films grown by MBE, for which the band gap is determined B. Treatment of alloying in the effective bond-orbital model by an analysis of absorption data. A good agreement between these experimental results and the theoretical results of the TB The usual approach for translationally invariant systems supercell approach is obtained. like pure bulk crystals is the construction of a plane-wave basis set from Eq. (2), which then fulfills the Bloch condition. II. THEORY For the CdxZn1−xSe alloy system, we will now assume uncor- related substitutional disorder and hence lose the translational symmetry. In analogy to calculations for zero-dimensional A. Empirical tight-binding model for pure bulk nanostructures, 18,19,22 we will use a finite supercell with peri- semiconductors: The effective bond-orbital model odic boundary conditions, with the localized basis, Eq.(2), on each lattice site. Each primitive cell will either be occupied Linear combinations of atomic orbitals can be used as by the diatomic CdSe or ZnSe basis, where the probability for ansatz for the Wannier functions that are localized within the the respective species is directly given by the concentrations unit cell of the Bravais lattice. From these Wannier func- x and 1 − x. Accordingly, we will use the matrix elements, tions, the itinerant Bloch functions can be determined by a Eq. (3) of the pure CdSe or ZnSe material for the correspond- unitary transformation. Because neither the atomic states nor ing lattice sites and use a linear interpolation for hopping ma- the Wannier functions are explicitly used in an ETBM, one trix elements between unit cells of different material. Due to can directly assume a basis of Wannier-like functions, which the resolution on the Bravais lattice, all influences which stem is the basic idea of the so-called effective bond-orbital model from effects on a smaller length scale, like the difference in (EBOM). This resolution on the scale of unit cells is especially the Cd–Se and Zn–Se bond lengths, are absorbed into the val- convenient for the case of binary semiconductor compounds ues of the correspondingTB matrix elements between the “ef- of the type AxB1−xC, as each unit cell can either be occupied fective“ orbitals. Furthermore, we incorporate an appropriate by the AC- or BC-material, respectively. valence band offset (VBO) between the two materials by shift- In order to adequately reproduce one s-like conduction ing the respective site-diagonal matrix elements by the value band (CB) and three p-like valence bands, the heavy hole ∆Evb = 0.22 eV, as established in Ref. 20. Although this treat- (HH), light hole (LH) and split-off (SO) band, we use a lo- 3 ment of the intersite hopping and the VBO has merely been calized sp basis per spin direction: well-proven for interfaces in quantum dot systems,18,19,22 our results for the bowing have turned out to be insensitive not |R,αi, α ∈ s ↑, p ↑, p ↑, p ↑,s ↓, p ↓, p ↓, p ↓ . (2) x y z x y z only to the choice of the interface treatment but also to the ∆ Here R labels the N sites of the fcc lattice, which is the under- specific value of Evb (not shown). lying Bravais lattice of the zincblende crystal structure. The numerical diagonalization of the corresponding Hamil- The TB matrix elements of the bulk Hamiltonian Hbulk are tonian for each concentration and microscopic configuration then given by gives the density of states (DOS) of the finite ensemble, from which the band gap for each concentration can be determined RR′ bulk ′ ′ Eαα′ = hR,α|H R ,α . (3) as the difference between the overall lowest conduction band and highest valence band energy. Fig.1 shows the corre- When restricting the non-vanishing matrix elements, Eq. (3), sponding band gap over the concentration in dependence on up to second nearest neighbors, a one-to-one correspondence the number of lattice sites (for fixed number of realizations), between the TB matrix elements and a set of material pa- where each lattice site corresponds to one unit cell and thus rameters can be determined. In the present work, we use the two atoms. Obviously, too small supercells produce an arti- parametrization scheme of Loehr,14 which fits the bulk band ficially high band gap bowing even with periodic boundary structure to a set of material parameters, namely the bulk band conditions, and this finite size effect cannot be compensated gap Eg, the conduction band effective mass me, the spin-orbit for by a higher number of realizations (not shown here). This splitting ∆so, the conventional lattice constant a and addition- could be attributed to clustering effects, as a local accumu- ally to the X-point energies of the conduction band X1c, the lation of the small band gap material will lead to an overall HH and LH bands X5v (which are degenerate at the Brillouin smaller band gap from the DOS. Our studies reveal that statis- zone boundary),and the SO band X3v. We use aset ofmaterial tically relevant convergence for the bowing b is safely reached parameters from Refs.15,16, augmented by X-point energies with 4000 lattice sites (8000 atoms) and N = 50 realizations 3

3.0 Cd Zn Se x 1−x 108 sites, N=30 256 sites, N=30 2.8 2048 sites, N=30 Cd Zn Se, 4000 sites, N=50 x 1−x 2.6

2.4 (eV)

g 2.2 E

2.0 0 1.8 rel. freq. (arb. units) −0.1 0.5 −0.05 1.6 0 0.05 1 0.8 0.6 0.4 0.2 0 1 Ei (x) − Eav(x) (eV) 0.1 x x g g

FIG. 1: (Color online) Dependance of the bulk band gap Eg on the FIG. 2: (Color online) Distribution of the deviation of the band gap i th concentration x of CdxZn1−xSe for different supercell sizes. Each Eg(x), as calculated from the i microstate, from the average value av lattice site corresponds to one unit cell and hence two atoms, while Eg (x) for each concentration x. The supercell contains 4000 lattice N = 30 gives the number of distinct microscopic configurations. sites and N = 50 realizations per concentration were chosen. when seeking for a resolution of 10 meV, which corresponds scopic configurations and the supercell was chosen to contain 4000 sites in order to reach convergence. The distribution of to the last digit of the input energies. With a cubiform super- i cell, which does not artificially confine the degrees of freedom the Eg(x) is quite narrow for most concentrations 0 < x < 1 of the microscopic disorder, this correspondsto an edge length and, of course, δ-like for the pure system (x = 0 and x = 1), of 10a. Of course, these large supercells and high number of where all microstates are identical. The biggest deviations, i av configurations require large computational costs. approximately 20 meV, between Eg(x) and Eg (x) occur for x 0 1. Further studies (not shown) reveal that only the con- From our results, we cautiously conclude that ETBM cal- = . duction band edge is broadened due to the disorder, while culations which use a lesser number of atoms on rectangular the valence band edge is δ-like over the whole concentration supercells (like Ref. 10, where an approximate band struc- range within our finite resolution, which is in agreement with ture is reconstructed with a 40a×2a×2a supercell, i. e.1280 CPA calculations for other II-VI alloys.11 The overall band atoms) could therefore overestimate the bowing of the alloy gap of the finite ensemble is hence given by band gap. We performed first test calculations for other material systems (e.g. GaxAl1−xN) as well and the convergence i Eg(x)= minE (x) . (4) criterion of 4000 sites and 50 realizations remained still valid. g av Finally, we want to dicuss the distributions of the band gap In the present work, we used Eg for the comparison to PL values from the different configurations. In theory, the dif- data and the overall band gap of the ensemble DOS Eg for ference between the lowest overall conduction band energy the absorption experiments. Nevertheless, the distributions and the highest overall valence band energy corresponds to are so narrow that the bowing parameters calculated in this the band gap Eg(x) of the mixed AxB1−xC crystal, as approx- work only change within the error boundaries when switching imated by the finite ensemble. This is also the usual way to between both approaches, provided that the above mentioned determine band gaps in the CPA,11,12 which can reproduce convergence criteria are fulfilled. broadening effects due to disorder. When referring to the energetic positions obtained from PL peak positions (other effects like the Stokes shift left aside for the moment), one III. COMPARISON TO EXPERIMENTAL RESULTS av should in principle compare to the ensemble-average Eg (x)= i 1/N ∑i Eg(x) of the band gap, because the effects of configu- A. Comparison to PL on bulk-like nanowires rational and concentrational disorder, amongst others, are al- i ready absorbed into the finite emission linewidth. Eg is here Venugopal et al. performed photoluminescence (PL) the difference between the lowest conduction band state and and Raman-scattering measurements on catalytically grown th the highest valence band state for the i microscopic configu- CdxZn1−xSe nanowires (60 - 150 nm in diameter and several ration. tens of micrometers in length) in Ref.3. From these data, they In Fig.2, we have depicted the relative frequency of determined the band gap as a function of the concentration x i av the deviation of Eg(x) from the average value Eg (x) for and the corresponding bowing parameter. Our TB parameter CdxZn1−xSe. As discussed above, we used 50 distinct micro- set was chosen so that their respective experimental band gaps 4

3.0 Cd Zn Se x 1−x 4000 sites, N=50 (224) quadratic fit, b=(0.49 ±0.05) eV 2.8 experiment Venugopal et al.

2.6 GaAs

2.4 (eV)

g 2.2 E

2.0

1.8 CdSe 1.6

1 0.8 0.6 0.4 0.2 0 x

FIG. 3: (Color online) Dependance of the band gap Eg on the concentration x of CdxZn1−xSe bulk-like nanowires. The black squares give the ETBM ensemble results for 4000 lattice sites and 50 real- FIG. 4: (Color online) Reciprocal space map of the (224) reﬂex for izations, while the red triangles give the results of PL measurements the binary CdSe sample, measured by high-resolution X-ray diffrac- 3 on bulk-like CdxZn1−xSe nanowires by Venugopal et al. Although tion. The CdSe layer is fully relaxed, since it lies on a straight line only ﬁtted to x = 1 and x = 0, the ETBM results show a very good that passes through the origin of the graph. agreement over the whole concentration range.

flection high-energy electron-diffraction (RHEED) system.A at room temperature for the pure systems, x = 0 and x = 1, 2×1 reconstruction indicates group VI-rich growth, while a are reproduced in our calculations. As opposed to the films c(2×2) reconstruction can be assigned to metal-rich (i.e.Zn discussed in Sec. III B, the Cd-richer nanowires still tend to and Cd) conditions. Stoichiometry is achieved when a super- crystallize in the wurtzite phase. Regardless, the EBOM re- position of both reconstructions can be identified within the sults from the localized basis set, Eq. (2), for the band gap RHEED pattern. The growth rate of the Cd Zn Se alloy 23 x 1−x will remain valid when the only slightly larger energy gap depends sensitively on the II/VI flux ratio. Since the deposi- of hexagonal CdSe is used in the calculations, especially as tion time for CdxZn1−xSe was kept constant (2 hours) for the the experimental data does not show any discontinuity. relevant samples, the thickness of the different CdxZn1−xSe The resulting curve and the PL results are given in Fig. 3 layers show a significant variation. However, this does not and show a very good agreement over the whole concen- influence the accurate determination of the band gap as dis- tration range. A quadratic fit yields a bowing parameter of cussed below. The Cd content and the respective strain state b = (0.49 ± 0.05) eV, which is in excellent agreement with of the CdxZn1−xSe layers has been determined by the evalu- the value of b = (0.45 ± 0.02) eV from Ref. 3. ation of reciprocal space maps (RSM) in the vicinity of the (224) reflection recorded using high-resolution X-ray diffraction (HRXRD). The RSM measured for the binary CdSe sam- B. Comparison to epitaxial films ple (x = 1) is examplarily shown in Fig.4. The peak of the CdSe layer is located on a straight line that passes through The samples have been prepared by molecular beam epi- the origin of the graph, indicating that the layer is fully re- taxy (MBE) on a GaAs (100) substrate at a temperature of laxed and therefore unstrained. The same result is found for ◦ 280 C. The binary ZnSe and CdSe layers have been grown the other CdxZn1−xSe films. Merely the layer of the binary directly on the substrate, while for the ternary alloys first a ZnSe sample (x = 0) is not fully relaxed on the GaAs sub- ZnSe buffer layer of 20 nm thickness has been deposited, fol- strate and therefore remains compressively strained. The ab- lowed by the CdxZn1−xSe layer. In order to realize differ- sence of strain is especially important for the TB modelling ent Cd concentrations of the ternary compound, the operating for the system under consideration, as it does not have to in- temperature of the Cd Knudsen cell was varied from 226 - clude the effect of strain and piezoelectricity on the electronic 240◦C during deposition. Special care was taken to ensure states, as it is often the case in calculations for quantum dot stoichiometric or slightly Se-rich growth conditions for each systems.24,25 case by adjusting the respective flux using both a Se Knud- In order to evaluate the optical constants, the band gap and sen cell and a valved cracker source at the same time. A the thickness of the CdxZn1−xSe films, optical measurements valved cracker source offers the advantage of rapid variations were carried out using a J.A.Woollam variable-angle spec- in elemental flux due to the availability of a motorized valve. troscopic ellipsometry (VASE) setup.26 The spectra cover the The stoichiometry has been controlled in-situ employing a re- wavelength range from 193 nm to 1700 nm (6.42 eV - 0.7 5

Wavelength (nm) TABLE I: Selected parameters of the CdxZn1−xSe ﬁlms determined 900 800 700 600 500 400 by spectroscopic ellipsometry and high-resolution X-ray diffraction. 1.5

0.0 % Cd Cd content (%) thickness (nm) roughness (nm) Eg (eV)

20.0 % Cd

26.5 % Cd 0.0 281.2 7.1 2.64 ± 0.02 2

32.5 % Cd 20.0 1396.7 7.8 2.37 ± 0.03

56.0 % Cd

1.0

100.0 % Cd 26.5 1490.3 6.9 2.30 ± 0.02

(eV/cm) 32.5 1821.0 7.3 2.25 ± 0.03 10

56.0 1170.4 4.4 2.07 ± 0.02 100.0 304.0 2.6 1.69 ± 0.02 x 10

0.5 2 E) ( 3.0 Cd Zn Se x 1−x 4000 sites, N=50 quadratic fit, b=(0.60 ±0.06) eV 2.8 0.0 experiment epitaxial films

1.5 2.0 2.5 3.0

Energy (eV) 2.6

2.4 FIG. 5: (Color online) Absorption curves of CdxZn1−xSe layers in dependance of the Cd content. The black dashed lines indicate the (eV) tangent used for determination of the bandgap energy Eg. g 2.2 E

2.0 eV) with wavelength steps of 2 nm. To get more accurate data, spectra for three different angles of incidence (65◦, 70◦ 1.8 and 75◦) were registered and analyzed for each sample. All measurements were perfomed at room temperature. Beside 1.6 the GaAs (100) substrate, three layers were taken into ac- 1 0.8 0.6 0.4 0.2 0 count, namely the ZnSe buffer, the CdxZn1−xSe-layer and a x thin additional layer to include the surface roughness into the model. The latter one was simulated by a Bruggemaneffective FIG. 6: (Color online) Dependence of the bulk band gap Eg on the medium approximation(EMA) using 50% voids in a matrix of concentration x of CdxZn1−xSe epitaxial films. The black squares the II-VI semiconductor material. The optical constants were again give the ETBM ensemble results, while the red triangles give derived from the curve fits using a model which combines a the results as obtained from Fig. 5. Although only material parame- Herzinger-Johs “P-semi oscillator” (four connected functions ters for x = 1 and x = 0 from the literature were taken as input, we to model each peak)27 with four Gauss oscillators. still have a good agreement between this specific experimental and the theoretical data, with exception of the value for x = 0.56. The band gap Eg of the CdxZn1−xSe layers was obtained using the formula28

2 (αE) ∝ (E − Eg), (5) retical results. We get a good agreement within the error boundaries between the experiment and the theoretical data, and the equation α(E)= 4πκ(E)/λ for the attenuation co- except for one data point with the concentration x = 0.56, efficient α, where κ(E) denotes the extinction coefficent, E which clearly does not fit the general trend in the data. When the energy and λ the wavelength of the incident light. The this point is omitted, the experimental bowing coefficient is spectral extinction coefficent κ(E) is determined by the spec- b = (0.41 ± 0.09) eV, which lies below the theoretical result troscopic ellipsometry measurement of the samples and Eg is of b = (0.60 ± 0.06) eV from the TB calculations for the sys- obtained graphically by evaluating the intersection of the tan- tem under consideration. This discrepancy could for exam- gent of (αE)2 with the energy axis (see Fig.5). A compre- ple be explained by the different experimental conditions, as hensive overview of the relevant experimental parameters for especially the production of high-quality CdxZn1−xSe films, the samples is given in Table I. but also the determination of the band gap from the absorp- For the theoretical calculations, we determined the TB pa- tion edges may be afflicted with a slightly higher degree of rameters of the pure systems (i.e. for x = 1 and x = 0) from uncertainty than the shift-adjusted PL measurement of the standard band structure data in the literature,17,29 valid at room nanowires of Ref.3. For example, a look at Tab.I reveals that temperature, as the input parameters for the pure cubic CdSe the literature value of Eg = 2.68 eV for pure ZnSe at room were determined under the same experimental conditions and temperature17 cannot be exactly reproduced. For the sake of on the same substrate as in the present measurements. completeness, we would like to note that one obtains an ex- The values for the band gap in dependence on the Cd frac- perimental bowing value of b = (0.6 ± 0.2) eV, which fits the tion x are depicted in Fig.6 in comparison with the theo- theoretical result within the error range, if the literature value 6

tions in order to determine the band gap bowing coefficient b TABLE II: Summary of the results for the bowing parameter b for with the same degree of accuracy as the input parameters, as CdxZn1−xSe at room temperature (all numbers given in eV). Note that the crystal structure of the nanowires changes from zincblende Fig.1 showed that the results depend strongly on these model to wurtzite for larger x. parameters. Furthermore, we compared our results for the band gap system experiment supercell ETBM bowing of CdxZn1−xSe with two slightly distinct experimen- nanowires 0.45 ± 0.023 0.49 ± 0.05 tal realizations. At first, we compared our results to pho- epitaxial films 0.41 ± 0.09 0.60 ± 0.06 toluminescence data from catalytically grown CdxZn1−xSe nanowires by Venugopal et al. Then we used our TB model for an analysis of absorption data from unstrained MBE- grown Cd Zn Se films, and also gave a detailed descrip- is used for the binary ZnSe sample instead. x 1−x tion of the experimental realization of this data set. In both cases, our model fits the experimental data well. For the nanowires PL data, even the experimental band gap bow- C. Comparison of the bowing results ing parameter can be reproduced within the error margin, while the measurements on the epitaxially grown films show For the sake of comparison, the results for the bowing of a slightly higher deviation from the theoretical predictions. CdxZn1−xSe for both experiments are summarized in TableII. Nevertheless, within the experimental accuracy the bulk band In spite of the overall good agreement, the results from the gaps from all but one MBE samples fit the theoretical curve TB supercell calculations for the disordered systems (1 > x > well. Dependent on the specific experimental conditions, we 0) are all slightly shifted to lower energies in comparison to carefully approve to narrow down the range of the bowing pa- the experimental values, which result in the higher bowing rameter for bulk CdxZn1−xSe at room temperature to between parameter in the theory. b ≈ 0.4 eV and b ≈ 0.6 eV. When we have a closer look at the values for the bowing In contrast to common tight-binding models, which treat coefficient b, we can conclude that, in spite of the different the disorder within a mean-field framework, like the virtual experimental conditions and therefore boundary values at x = crystal approximation, our simple TB supercell approach is 1 and x = 0, all bowing values lie within the comparatively able to satisfactorily reproduce the experimental findings and narrow range between b ≈ 0.4 eV and b ≈ 0.6 eV. This stands can thus be used to coherently calculate bowing coefficients in contrast to several previous studies, who reported a broad 2–5 for other material systems and growth conditions, which in variety of values between b = 0 and b = 1.26 eV, as already addition may not be experimentally accessible over the whole mentioned in the introduction. Therefore, we recommend a concentration range. For this purpose, our model can eas- critical review of literature values outside the range given in ily be transferred to other material systems like Cd Zn S, Tab.II before using them for further purposes. x 1−x AlxGa1−xAs, InxGa1−xAs, GaxAl1−xN, and many others in order to calculate the bowing of the band gap. Also, materials with a wurtzite structure can be simulated, as only the corre- IV. CONCLUSION AND OUTLOOK sponding TB Hamiltonian in the effective bond-orbital model has to be replaced by the one given in Ref.22. First results in- In this paper, we presented a combined theoretical and ex- dicate that our minimal criteria for the numerical convergence perimental approach for the determination of the band gap of the band gap bowing will still remain valid, although this bowing of binary compound semiconductor alloys of the type will have to be tested carefully for each system under consid- AxB1−xC. On the theoretical side, we used a multiband em- eration, in combination with the respective experimental data pirical tight-binding approach with a localized basis set on the obtained by means of epitaxy. lattice sites of a finite supercell with periodic boundary conditions. By calculating the band gap for each concentration x from the density of states of a finite ensemble with different microscopic configurations, our approach is exact in the Acknowledgments alloy-induced disorder. The input parameters are a set of material properties of the pure AC or BC system, i.e. for the The authors would like to thank Jan-Peter Richters and To- cases x = 1 and x = 0. Using the example of CdxZn1−xSe, we bias Voß for fruitful discussions. The financial support of gave recommandations on the minimal size of the supercell the Deutsche Forschungsgemeinschaft(Project No. 436 RUM and the minimal necessary number of microscopic configura- 113/27/0-1) is gratefully acknowledged.

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