Electronic states in nanowires with hexagonal cross-section
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electronic states in nanowires with hexagonal cross-section
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 electron spectrum in a uniform nanowire with a hexagonal cross-section is calculated bymeans of a numerical diagonalization of the effective-mass Hamiltonian. Two basis sets are utilized.The wave-functions of low-lying states are calculated and visualized. The approach has an advantageover mesh methods based on finite-differences (or finite-elements) schemes: non-physical solutionsdo not arise. Our scheme can be easily generalized to the case of multi-band (Luttinger or Kane) k · p Hamiltonians. The external fields (electrical, magnetic or strain) can be consistently introducedinto the problem as well.
I. INTRODUCTION
Semiconductor nanowires (NWs) are currently of greatinterest due to the possibility of their application in elec-tronics. NWs can be used as a work item of field-effecttransistors [1, 2], photodetectors [3]. Moreover, there isa wide variety of NW-based photonic devices includinglight-emitting diodes, chemical and gas sensors, waveg-uides, solar cells and nonlinear optical converters [4, 5].The NW-based structures are also of fundamental inter-est. The topological states of the matter and Majoranafermions are realized in NWs due to the proximity ef-fect [6–9].Usually, NWs of III-V materials with a zinc-blendelattice are grown in [111] crystal direction, that leadsto the hexagonal shape of NW’s cross-section (Fig. 1a).Early, simple models of NW with circular or square cross-sections were used for the calculation of the charge carrierspectrum and wave functions. However, for optical andtransport applications it is necessary to know the carriersubband spectrum with higher precision, i.e., take intoaccount a real NW’s shape.The NW’s translation invariance in longitudinal direc-tion simplifies the problem: one needs simply to solve thespectral problem for a two-dimensional electron boundedin a hexagon. Usually, the finite-difference (or finite el-ements) method is used for this purpose [10–12]. Thisproblem is nontrivial even for the case of the electronin a non-degenerate band described by the scalar effec-tive mass. We propose an alternative approach based onthe numerical diagonalization of the matrix Hamiltonianwritten in an appropriate basis.
II. HAMILTONIAN AND BASIS FUNCTIONS
The effective potential barrier bounding an electronin NW is equal to the electron affinity χ (several eV). ∗ E-mail:[email protected]
Within the effective mass approximation, such a height isequivalent to an infinite barrier. To find electronic statesin NW with a hexagonal cross-section we propose to usethe matrix mechanics. It is convenient to choose theeigenfunctions of the Hamiltonian H , which describeselectrons in NW with a circular or rectangular cross-section, as the basis functions. The corresponding cir-cle or rectangle is chosen to be circumscribed around thehexagon (see Fig. 1b,c). The spectral problem is reducedto the problem with the Hamiltonian H = H + V ( r ),where V ( r ) is nonzero in shaded areas of Fig. 1b,c. Theheight of this potential cannot be chosen infinite at cal-culation, however, we do not make a big mistake puttingit to be finite but high, e.g., V ∼ χ . The envelope func-tion approximation in a single band with a scalar effectivemass m ∗ is used. The spin-dependent terms are excludedfrom consideration.The eigenfunctions and eigenenergies for the electronin the infinite circular potential well of radius R are well-knownΨ mn ( r, ϕ ) = √ RJ | m | +1 ( j mn ) J | m | (cid:16) j mn rR (cid:17) √ π e imϕ , (1) E mn = ¯ h j mn m ∗ R , (2)where m = 0 , ± , ± , ... , n = 1 , , ... ; J m ( x ) is the firstkind Bessel function, and j mn is the n th zero of J m ( x ).In the case of a potential well of rectangular shapecircumscribing the same hexagon, the eigenfunctions andeigenenergies are given byΨ mn ( x, y ) = √ / R sin (cid:16) πmx R (cid:17) sin (cid:18) πny √ R (cid:19) , (3) E mn = π ¯ h m ∗ R (cid:18) m + 43 n (cid:19) (4)with m, n = 1 , , ... . [111](a) xyz r j R (b)( )с y x R Ö R FIG. 1: (a) The sketch of NW with a hexagonal cross-section.(b) The use of circular basis set (1). The shaded areas be-tween the circle and the hexagon serve as additional potentialbarriers. (c) The same as in (b) for the rectangular basis (3).
We will search for the electron wavefunctions in hexag-onal NW (h-NW) as a series in above basis setsΨ j ( r ) = X mn C jmn Ψ mn ( r ) . (5)The spectral problem is reduced to finding the eigen-values of the Hamiltonian H = H + V ( r ) matrix.For the matrix elements we have, h m ′ n ′ | H | mn i = E mn δ m ′ m δ n ′ n + h m ′ n ′ | V ( r ) | mn i . The latter term is pro-portional to the overlap integral I m ′ n ′ ; mn of the basisfunctions in the single barrier segment.We can use some symmetry arguments for the matrixelements calculation. They are given by h m ′ n ′ | V ( r ) | mn i = 6 V δ m ′ ,m +6 M I m ′ n ′ ; mn , (6)and h m ′ n ′ | V ( r ) | mn i = 4 V δ m ′ ,m +2 M δ n ′ ,n +2 N I m ′ n ′ ; mn , (7)for the case of a circular and rectangular basis, respec-tively. Here M, N = 0 , ± , ± , ... . In the latter casethe overlap integral can be found analytically, but in theformer, only numerically. III. NUMERICAL DIAGONALIZATION
For the numerical diagonalization of derived matrixHamiltonians one needs to truncate the matrix dimen-sion. At the same time, we have to choose a matrixsize so as to ensure acceptable accuracy. The maximalvalues of m max and n max determine the size of the trun-cated matrix. In the circular basis the matrix dimen-sion is (2 m max + 1) n max , while in rectangular one we have m max n max . The position of calculated subbandbottoms in h-NW is depicted in the central section ofFig. 2a. The results are depicted for the truncated ma-trix of dimension 775 ×
775 and 1000 × m max = 15, n max = 25 and m max = 40, n max = 25, respectively. The energies are scaled to thevalue E = ¯ h / m ∗ R , that for the case of GaAs NW( m ∗ = 0 . m [13]) with R = 20 nm is equal to 1.41meV. The barrier height V was set to 10 E .The energy levels are a single or twofold degenerate(excluding spin). This is especially easy to trace whenconsidering a circular basis. In this case the degeneratestates arise even at diagonalization of the Hamiltonianmatrix of small size, which does not provide a good pre-cision. In this sense the use of Cartesian basis is moreappropriate (there are no degenerate states) in order totrack the convergence of the method with a growing ma-trix dimension. Nonetheless, the use of a Cartesian basisrequires a larger matrix size to attain the same preci-sion as for a circular basis. Moreover, for the case of anon-degenerate Cartesian basis, the real twofold degen-eracy of states is reached only in the limit of V → ∞ , m max , n max → ∞ . This is due to the lack of 6-th ordersymmetry axis in the model described in Fig. 1c com-pared to that in Fig. 1b.The calculated coefficients C jmn give us the opportu-nity to find the spatial behavior of wave functions [seeEq. (5)]. The electron distributions | Ψ | correspondingto energy levels of Fig. 2a (central panel) are depictedin Fig. 2b. The wave functions of degenerate states cal-culated in the Cartesian basis, in general, do not possesshexagonal symmetry. However, the total electron densityat degenerate levels has this property. IV. CONCLUSIONS