Electronic states in cylindrical core-multi-shell nanowire
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electronic states in cylindrical core-multi-shell nanowire
A. O. Rudakov and I. A. Kokurin
1, 2, 3, ∗ Institute of Physics and Chemistry, Mordovia State University, 430005 Saransk, Russia Ioffe Institute, 194021 St. Petersburg, Russia St. Petersburg Electrotechnical University “LETI”, 197376 St. Petersburg, Russia (Dated: January 19, 2021)The recent advances in nanowire (NW) growth technology have made possible the growth of morecomplex structures such as core-multi-shell (CMS) NWs. We propose the approach for calculationof electron subbands in cylindrical CMS NWs within the simple effective mass approximation.Numerical results are presented for GaAs / Al . Ga . As radial heterostructure with AlGaAs-coreand 4 alternate GaAs and AlGaAs shells. The influence of an effective mass difference in heterolayersis discussed.
I. INTRODUCTION
The recent progress in nanowire (NW) growth technol-ogy gives an opportunity to made the NW-based complexstructures with so-called axial [1] and radial (core-shell)heterostructuring. Now there is a possibility to grow alarge number of shells (see, for instance, reviews [2, 3]and references therein). Such structures known as core-multi-shell (CMS) NWs attract attention of researchersdue to interesting properties and possible applications.Due to a special geometry and the possibility to gov-ern carrier states by means of external fields (electric,magnetic or deformation) the devices based on NW-structures [4] are very attractive for modern electronicsand photonics. For example, there are lasers [5, 6] andlight-emitting diodes [7] with CMS NW as a work item.The carrier mobility can increase in CMS structures[8] comparative to continuous NW, that means the pos-sibility of conductance quantization in such structures.Thus, CMS NWs are good candidates for utilizing as theworking part of field effect transistors including spin ones.The realization of multiple quantum wells in CMSNWs can lead to the spatial separation between the elec-tron and hole, that allows one to control the lifetime ofindirect excitons [9], which can travel over large distancesbefore recombination, and cool down close to the latticetemperature and below the temperature of quantum de-generacy.Thus, there is a need to know electron and hole sub-band spectrum in such structures. Here we develop asimple approach to find electronic states in III − V CMSNW with zinc-blende crystal lattice within single bandeffective mass approximation (EMA). ∗ E-mail:[email protected]
II. HAMILTONIAN AND NUMERICALDIAGONALIZATION
Now we propose the approach for calculation of elec-tron subband spectrum of CMS NWs. We use a single-band envelope function approximation (EFA) to findelectronic states in cylindrical CMS NW with zinc-blendecrystal structure (starting from bulk Γ -band states withscalar effective mass).In cylindrical NW there are translational and rota-tional invariance, that means the following form of en-velope wavefunctionΨ mnk ( r, ϕ, z ) = R mn ( r ) 1 √ π e imϕ √ L e ikz , (1)where k is the longitudinal momentum, m = 0 , ± , ± , ...n = 1 , , , ... . The energy spectrum of one-dimensional(1D) subbands is given by E mn ( k ) = ǫ mn + ¯ h k m ∗ , (2)where m ∗ is the scalar conduction band effective mass,and ǫ mn is the energy of 1D-subband bottom.For the case of uniform cylindrical NW of radius R applying zero boundary conditions (hard-wall potential)we have for R mn in Eq. (1) and ǫ mn in Eq. (2) well knownresult R mn ( r ) = √ R | J | m | +1 ( j mn ) | J | m | (cid:16) j mn rR (cid:17) , (3)and ǫ mn = ¯ h j mn m ∗ R . (4)Here J m ( x ) is the first kind Bessel function of order m ,and j mn is the n -th root of function J m , i.e. J m ( j mn ) = 0,with n = 1 , , , ... .Now consider the spectral problem for CMS NW.The one-electron Hamiltonian within single-band EMAis H + V , with H being the Hamiltonian of uniform (a) z j AlGaAsGaAs r r infinity0 B B BA A V(r)V rrrrr (b) FIG. 1: (a) The sketch of CMS NW. The case of two-well-three-barrier structure is depicted, that can be realized bymeans of usual hetero-pair GaAs-Al x Ga − x As. (b) An effec-tive confining potential which replicates the conduction bandprofile for the structure depicted in (a). Character sizes (radii)discussed in text are depicted. The conduction band offset isequal to V . NW, that corresponds to wavefunction (3), and V ( r ) isthe potential giving the difference between NW and CMSNW. The conduction band offset serves as effective radialpotential V ( r ) = (cid:26) V in barriers (5)is depicted in Fig. 1b. In reality such a structure musthave GaAs outer capping layer in order to avoid Al oxida-tion and the following degradation of the structure. Wesuppose that such a capping layer usually thin and we donot make a mistake in wave function behavior neglectingthem.Translation and rotational invariance are conserved inthis case and wave function will have form (1) but with R mn different from R mn of Eq. (3). However, we canlook for wave function as a series on basis functions ofEq. (3). R mn ( r ) = X l C mnl R ml ( r ) . (6)We can find the Hamiltonian H + V ( r ) matrix in basis(3). It will contain ǫ ml in diagonal and matrix elementsof (5) V ml ′ l = V X i =0 r i +1 Z r i drrR ml ′ ( r ) R ml ( r ) (7)in all positions. Where r i ( i = 1 −
5) is the radius of i -thradial heterointerface, and r = 0, r ≡ R . In chosenbasis the matrix elements V ml ′ l can be found analytically[10], but we do not write them here due to their cumber-some form.The numerical diagonalization was performed for CMSNW with 45 nm radius and the widths of well and bar-rier regions depicted in Fig. 2 (transparent and shaded E n e r gy , m e V E Radial distance , nm rm =0 m =1 m =2 m =3 m =4 m =5 m =6 FIG. 2: The CMS NW subband energies ǫ mn and correspond-ing radial probability densities r | Ψ | . The first 10 subband en-ergies presented for the structure with r i = 0 , , , , , V = 230 meV, m ∗ = 0 . m . The areas correspondingto wells and barriers are transparent and shaded, respectively. areas, respectively). The finite barriers are of 230 meVheight, that approximately corresponds to conductionband offset at heterointerface GaAs / Al x Ga − x As with x = 0 . m ∗ = 0 . m . We used a finite dimension Hamiltonianof 40 ×
40 dimension, that gives the perfect precision (bet-ter than 0.1%) for first 10 subbands in each block withfixed m . Besides 1D-subband bottoms the coefficients C mnl found giving coordinate dependence of wavefunc-tion in accordance with Eqs. (1),(6),(3). III. DISCUSSION
The results of numerical diagonalization for first ten1D-subband bottoms are depicted in Fig. 2 as wellas corresponding probability densities r | Ψ | found fromEqs. (6),(3). One can see expected behavior of wave func-tion: for low-lying states belonging to the family with thesame m the ground and first excited state wave functionsare predominantly localized in different wells. For high-lying subbands wave functions will have nodes inside eachwell region. As in uniform NW the subband ground statecorresponding to higher | m | -value has a higher energy.However, relative position of subband with ǫ | m | +1 , andsubband with ǫ | m | , crucially depends on V and relationbetween r i .Till now we supposed the equal effective masses both inwells and in barriers. In real structures it is necessary alsoto take into account a difference of effective masses in dif-ferent cylindrical layers. In Al x Ga − x As with x = 0 . . m [11]. This difference doesnot distort the translational and rotational symmetry ofthe structure. However, in this case the motion along andacross CMS NW is not formally separated. This meansthat Hamiltonian matrix remaining diagonal in m and k will parametrically depend on the longitudinal momen-tum k . In order to include difference in masses into ourscheme we have to use instead of Eq. (5) the followingoperator e V ( r ) = ¯ h (cid:18) m A − m B (cid:19) X j =1 ( − j δ ( r − r j ) ∂∂r + ( − (cid:16) − m A m B (cid:17) H + V in barriers , (8)where m A and m B are effective masses in wells andbarriers, respectively. The second term takes into ac-count the difference in the kinetic energy that is dueto the difference of effective masses in layers. The firstterm ensures the Hermiticity of the operator, and itarises from the standard form of the kinetic energy op- erator − ¯ h ∇ m ∗ ( r ) ∇ in the systems with spatially in-homogeneous mass (the step-like dependence of m ∗ ( r )leads to δ -function contribution). It is obvious, that at m A = m B we have the same result as before. Due to thetranslational invariance along the NW axis one can sim-ply replace in H the longitudinal momentum operator p z = − i ¯ h∂/∂z by its eigenvalue ¯ hk .The spectral problem in this case can be solved in thesame manner but in order to find E ml ( k ) one have todiagonalize total Hamiltonian at each k value, giving ǫ ml to be the function of k . This leads to change in subbandbottom energy and renormalization of effective massesin different subbands and nonparabolicity as well due todifferent penetration of wave function into the barriers.These results will be published elsewhere. IV. CONCLUSION
We proposed the approach for calculation of conduc-tion band states in CMS NW. The numerical results forfive radial-layer GaAs / Al . Ga . As CMS NW of 45 nmradius are presented. This approach can be easily gen-eralized to describe hole states in complex valence bandor multiband Hamiltonian in CMS NW of narrow-gapsemiconductor. [1] G. Nylund, K. Storm, S. Lehmann, F. Ca-passo, L. Samuelson, Designed Quasi-1D PotentialStructures Realized in Compositionally GradedInAs − x P x Nanowires. Nano Lett. , 1017 (2016). doi:10.1021/acs.nanolett.5b04067 .[2] C. M. Lieber, Z. L. Wang, Functional Nanowires. MRSBulletin , 99 (2007). doi:10.1557/mrs2007.41 .[3] M. Royo, M. De Luca, R. Rurali, I. Zardo, A review onIII–V core–multishell nanowires: growth, properties, andapplications. J. Phys. D: Appl. Phys. , 143001 (2017). doi:10.1088/1361-6463/aa5d8e .[4] O. Hayden, R. Agarwal, W. Lu, Semiconduc-tor nanowire devices. Nano Today , 12 (2008). doi:10.1016/S1748-0132(08)70061-6 .[5] D. Saxena, N. Jiang, X. Yuan, S. Mokkapati, Y.Guo, H. H. Tan, C. Jagadish, Design and Room-Temperature Operation of GaAs/AlGaAs MultipleQuantum Well Nanowire Lasers. Nano Lett. , 5080(2016). doi:10.1021/acs.nanolett.6b01973 .[6] T. Stettner, P. Zimmermann, B. Loitsch, M. D¨oblinger,A. Regler, B. Mayer, J. Winnerl, S. Matich, H. Riedl,M. Kaniber, G. Abstreiter, G. Koblm¨uller, J. J. Finley,Coaxial GaAs-AlGaAs core-multishell nanowire lasers with epitaxial gain control. Appl. Phys. Lett. , 011108(2016). doi:10.1063/1.4939549 .[7] K. Tomioka, J. Motohisa, S. Hara, K. Hiruma, T. Fukui,GaAs/AlGaAs Core Multishell Nanowire-Based Light-Emitting Diodes on Si. Nano Lett. , 1639 (2010). doi:10.1021/nl9041774 .[8] S. Funk, M. Royo, I. Zardo, D. Rudolph, S. Mork¨otter, B.Mayer, J. Becker, A. Bechtold, S. Matich, M. D¨oblinger,M. Bichler, G. Koblm¨uller, J.J. Finley, A. Bertoni,G. Goldoni, G. Abstreiter, High Mobility One- andTwo-Dimensional Electron Systems in Nanowire-BasedQuantum Heterostructures. Nano Lett. , 6189 (2013). doi:10.1021/nl403561w .[9] L.V. Butov, Excitonic devices. Superlatt. Microstruct. , 2 (2017). doi:10.1016/j.spmi.2016.12.035 .[10] A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integralsand Series: Special Functions , vol. 2, (Gordon & Breach,New York, 1986).[11] I. Vurgaftman, J. R. Meyer, L. R. Ram-Mohan,Band parameters for III-V compound semiconductorsand their alloys, J. Appl. Phys. , 5815 (2001). doi:10.1063/1.1368156doi:10.1063/1.1368156