We show that the large band offsets between GaN and InN and the heavy carrier
effective masses preclude the use of the Virtual Crystal Approximation to
describe the electronic structure of Ga_(1-x)In_(x)N/GaN heterostructures while
this approximation works very well for the Ga_(1-x)In_(x)As/GaAs
heterostructures.Comment: submitted to Applied Physics Lette

Bastard, Gerald

Ferreira, Robson

Nguyen, Duc Phuong

Regnault, Nicolas

English

arXiv

submitted to Applied Physics LetterWe show that the large band offsets between GaN and InN and the heavy carrier effective masses preclude the use of the Virtual Crystal Approximation to describe the electronic structure of Ga_(1-x)In_(x)N/GaN heterostructures while this approximation works very well for the Ga_(1-x)In_(x)As/GaAs heterostructures

Bastard, Gérald

Hal-Diderot

Alloy effects in GaInN/GaN heterostructuresDuc Phuong Nguyen, Nicolas Regnault, Robson Ferreira, Ge´rald BastardTo cite this version:Duc Phuong Nguyen, Nicolas Regnault, Robson Ferreira, Ge´rald Bastard. Alloy effects inGaInN/GaN heterostructures. Solid State Communications, Elsevier, 2004, 130, pp.751. <hal-00000821>HAL Id: hal-00000821https://hal.archives-ouvertes.fr/hal-00000821Submitted on 6 Nov 2003HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00000821 (version 1) : 6 Nov 2003Alloy effects in Ga1−xInxN/GaN heterostructuresDuc-Phuong Nguyen,∗ N. Regnault, R. Ferreira, and G. BastardLaboratoire Pierre Aigrain ENS, 24 rue Lhomond, 75005 Paris, FranceWe show that the large band offsets between GaN and InN and the heavy carrier effec-tive masses preclude the use of the Virtual Crystal Approximation to describe the electronicstructure of Ga1−xInxN/GaN heterostructures while this approximation works very well for theGa1−xInxAs/GaAs heterostructures.PACS numbers: 73.22., 78.40.FyThe use of semiconductor alloys proves to be neces-sary to adjust the electronic parameters to the designof specific devices. Ga1−xInxN alloys inserted betweenGaN [1] would be ideally suited to cover the entire nearinfrared/ultra-violet spectrum [2]. Despite the lack fortranslation invariance, one may very often describe theelectronic behavior of an alloy by means of a band struc-ture, the so-called Virtual Crystal Approximation [3](VCA). Close to it, the Coherent Potential Approxima-tion allows to introduce part of the disorder effects in theVCA scheme, which leads to a damping of the VCA Blochstates. An example of alloys which is well described bythe VCA/CPA is Ga1−xInxAs. The VCA has also beenused to describe the electronic properties of Ga(In)N het-erostructures [4]In this letter, we shall show that the Ga1−xInxN sys-tem behaves in a radically different way. A numericalcomputation of the electronic states of Ga1−xInxN/GaNquantum wells and quantum dots will show very largedifferences from the VCA predictions. Instead, we shallshow that Ga1−xInxAs/GaAs electronic states are veryclose to the VCA predictions. The difference betweenthe two alloy families arises from the much larger bandoffsets between GaN and InN than those between GaAsand InAs and from the heavier effective masses in the ni-tride system than in the arsenide system. We note thatthe large potential offset between In and Ga has led Kentand Zunger [5] to predict that nitride based alloys cannotbe described by models which neglect fluctuations.In the following, we use one band effective mass Hamil-tonians to describe the electron and hole kinematics.The effective mass is slightly anisotropic in the conduc-tion band (mez = 0.184 m0, me// = 0.166 m0) andanisotropic in the valence band (mhz = 1.1 m0, mh// =0.5045 m0) [6, 7]. The conduction ∆Ec (valence ∆Ev)band offsets between GaN and InN is taken equal to1.8 eV (0.9 eV) while those between GaAs and InAs aretaken equal to 0.41 eV and 0.29 eV respectively [8]. Thus,there is a much larger energy fluctuation in the nitridesthan in the arsenides when in a given unit cell a Ga atomreplaces an In atom. The two heterostructures we shallbe dealing with are a 3.2 nm thick Ga0.83In0.17N/GaNquantum well and a Ga0.83In0.17N truncated pyramidwith hexagonal basis (6 nm side), 2.6 nm height anda basis angle of 30◦ embedded into a GaN matrix andfloating on a 1.1 nm thick wetting layer (see Fig.(1)). Apyramid with the same geometrical parameters made ofGa0.5In0.5As and embedded into GaAs will be consid-ered for comparison (50% of In instead of 17% was con-sidered to get bound states to the pyramid). It is wellknown that nitride heterostructures contain huge inter-nal fields [6, 7, 9]. This will be modeled by assumingthat there exist piecewise constant electric fields whichare oriented along the Z direction and have a magnitudeof 2.45 MV/cm in Ga0.83In0.17N and -0.1 MV/cm in GaN[6, 7].3.2 nm3.5 eV       ZFigure 1: Left: VCA band diagram and wave functions inquantum well in the growth direction. Right: geometry of aquantum dot with its wetting layerThe quantum well problem will be reduced to a twodimensional (2D) calculation by considering an effectivein - plane potential which is the average of the actual 3Dpotential weighted by |Ψ(Z)|2where Ψ (Z) is the enve-lope function solution of the 1D problem in the presenceof the electric field and a square well confining poten-tial with a depth of 0.31 eV and 0.15 eV (= 0.17∆Ecor 0.17∆Ev) (see Fig.(1)). This effective 2D potential isthen supplemented by a hard box potential which con-fines the carrier in the layer plane into a square withdimensions L × L = 36 nm × 36 nm. A given sample isthen randomly generated by filling each unit cell eitherby In or by Ga corresponding to a potential energy of ei-ther 0 (In) or ∆Ec (Ga) for electrons and either 0 (In) or∆Ev (Ga) for holes. For the quantum dots a complete 3Ddiagonalization was undertaken enclosing the pyramid ina box with a square basis with a side of 14.9 nm and aheight of 7.9 nm. The box is then subdivided into smallerboxes of atomic size where the potential is supposed to beconstant. The wave functions are then developed on the2sinus solutions leading to a Hilbert space of dimension75000.For the QDs, two situations have been considered: ei-ther uncorrelated sites where the probability that a givencell is occupied by In is x ( = 17%) irrespective of the oc-cupancy of the other sites or site correlations which favorthe In cluster formation [10, 11] by allowing a probabilityp > x for a cell to be occupied by an In if its first neigh-borings are already occupied by at least one In. The onlysamples retained in the analysis are those such that theIn fraction in the whole QD lies in the interval [0.17, 0.18].The first few eigenstates of those 2D (40 states) and3D (6 states) problems are evaluated by means of exactdiagonalizations using the La´nczos algorithm. Once theeigenstates are known one can calculate various averageslike the mean position, the mean square deviation to it,etc..While the properties of a single sample have virtuallyno chance to be compared to any experiment, the aver-age properties, obtained by taking the arithmetic meanof the outcomes of a given physical quantity in a series ofsamples, have more physical substance since they can (inprinciple) be compared to the results obtained on ensem-bles of quantum dots or quantum wells. This is for theseaverage properties that one can discuss the effectivenessor failure of the VCA. One important quantity is De, theaverage electron density of states which is :De(ǫ) =1πNN∑j=1∑iΓ/2(ǫ− ǫei (j))2+ (Γ/2)2(1)where N is the number of samples (which differ from oneanother by the locations of the In atoms) while ǫei(j) isthe ith eigenvalue of the jth sample and Γ has been takenequal to 1 meV. Here we replace the usual delta peaks byLorentzian distributions for clarity. Equally important(and measurable) is the average electron - hole opticaldensity of states defined as :Deh(ǫ) =1πNN∑j=1∑k,l(Γ/2) |〈Ψek|Ψhl〉|2(ǫ− ǫek(j)− ǫhl(j))2 + (Γ/2)2(2)In the finite QW structure the VCA results in peaks re-lated to the confinement box which fulfill:ǫnp = V0 + ǫ1z +h¯2π22m//L2(n2 + p2); n, p = 1, 2, ...(3)where V0 = x∆Ec is the VCA average potential, ǫ1z is thelowest confining energy in the Z direction and L =36 nm.Fig.(2) shows for the QW structure the mean squaredeviation δσ plotted versus the mean location 〈X〉 for theelectron (upper panel) and the hole (lower panel) alongthe X axis (the results are equivalent for the Y axis) for50 samples, together with the VCA results. The In dis-tribution is at random (x = 0.17). It is obvious that 〈X〉4060801001200 50 100 150 200 250 300 3500 50 100 150 200 250 300 350406080100120   <X> (Å)c. Hole406080100120  (Å)    (Å) b. Electron    a. VCAFigure 2: Calculated mean square deviations δσ plotted ver-sus mean carrier position 〈X〉 for the VCA (a), electrons (b)and holes (c) in a finite Ga0.83In0.17N/GaN quantum well.Random alloys and 50 samples.2.70 2.72 2.74 2.76 2.78 2.80 2.82 2.84 2.860.000.050.100.150.200.250.300.35  Optical density of states (a.u.)Energy (eV)  Random alloy  VCA peaksFigure 3: Optical density of states averaged over N = 50Ga0.83In0.17N/GaN quantum wells. Random alloys. Dashedlines: VCA peaks.and δσ for holes differ strongly from the VCA predic-tions while the electrons with their lighter mass are lessreadily confined by In rich fluctuations and thus displayδσ(〈X〉) which are closer from the VCA results than theholes. Fig.(3) shows the electron - hole optical density ofstates Deh(ǫ) versus the energy ǫ averaged over N = 50samples. Deh is proportional to the light absorption co-efficient if we neglect electron-hole interaction, which isjustified here due to the strong e-h separation by the elec-tric field (770 meV compared to the Coulomb interactionenergy of the order of 44 meV). The VCA results in arounded (by Γ) staircase - like Deh for an infinite QWwhile the large box produces VCA peaks whose positionfollows eq. (3). We see that the random alloy results in a3broadening of these peaks which is quite important. Notefor instance the significant (Deh > 0.05) band tail whichdevelops down to ≈ 15 meV below the nominal edge ofthe ground peak. This bandtail corresponds to increas-ingly localized states around In clusters, in particular forthe holes. The high energy decrease of the optical densityof states is unphysical. It reflects the high energy cut-offof the electron and hole eigenvalues. Notice that the pho-toluminescence line is found experimentally at ≃2.6 eVin Ga0.83In0.17N/GaN [6] with which our results are co-herent.-1.00 -0.95 -0.90 -0.85 -0.80 -0.75 -0.700100200300400500  <n> = 2.1<n> = 1.6Mean DOS (1/eV)Energy (eV)VCA peaksRandom alloy<n> = 1.2Figure 4: Electron density of states averaged over N = 100Ga0.83In0.17N/GaN pyramids. 〈n〉 is the mean number of firstIn neighbors for an In. From top to bottom: VCA results(divided by 6), random alloy 〈n〉 = 1.0, and increasingly seg-regated alloys 〈n〉 = 1.2; 〈n〉 = 1.6; 〈n〉 = 2.1. The energyzero is taken at the barrier monolayer next to the bottom ofthe wetting layer.0.00 0.05 0.10 0.15 0.200100200300400500600700  Mean DOS (1/eV)Energy (eV) Random alloy VCA peaksFigure 5: Electron density of states averaged over N = 50Ga0.5In0.5As/GaAs pyramids. Solid lines: random alloy.Dashed lines:VCA results.Fig.(4) shows the calculated electron density of statesfor Ga0.83In0.17N/GaN pyramid (100 samples); the opti-cal density of states displays similar results although thehole states can be unbound in the pyramid. The VCAresults shows a twofold degenerate excited state locatedsome 90 meV above the ground state. It lies some 80 meVbelow a narrow triplet. The random alloy case showsboth a redshift and a considerable broadening. Note inparticular the tendency towards the closing of the gapsbetween the remnants of the VCA peaks. This effect ismore pronounced when In segregation is introduced inthe calculations. In addition, increasing the segregationproduces densities of states which bear less and less re-semblance with the VCA result. In contrast (fig.(5)) theGa0.5In0.5As/GaAs pyramids (50 samples) display den-sities of states which are very close to the VCA results.Even though the 50% alloy should display the maximumof disorder, we see that the calculated DOS features arevery well separated from each others, and much narrowerthan the ones found in the 17% nitride alloy. Note inparticular that the bandtail extends only a few meV’sbelow the VCA peaks. Thus, if one knows the shape ofa Ga0.5In0.5As pyramid, fig.(5) shows that it does makesense to attempt to identify the peaks, to fit their energydifference etc. On the other hand, the huge deviationsfrom the VCA results in the 17% nitride dot demonstratethat any peak attribution in this system is rather elusive.We would like to thank Pr. Runge and Pr. Lefe`bvrefor fruitful discussions and insights. The LPA-ENS isLaboratoire associe´ au CNRS et aux Universite´s Paris 6et Paris 7.∗ Electronic address: Duc-Phuong.Nguyen@lpa.ens.fr[1] C. Adelmann, J. Simon, G. Feuillet, N. 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Alloy effects in GaInN/GaN heterostructures

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