ON THE INFLUENCE OF THE UNIT CELL TYPE ON THE WAVE FUNCTION IN ONE-DIMENSIONAL KRONIG-PENNEY MODELS

Transcription

1 ON THE INFLUENCE OF THE UNIT CELL TYPE ON THE WAVE FUNCTION IN ONE-DIMENSIONAL KRONIG-PENNEY MODELS M. G. Tashkova, A. M. Miteva ands.g.donev 3 ) Institute of Organic Chemistr with Center of Phtochemistr, Bulgarian Academ of Sciences, Acad. G.Bonchev Str. 9, 3 Sofia, Bulgaria Space Research Institute, Bulgarian Academ of Sciences, P.O.Box 799, 000 Sofia, Bulgaria 3 Institute for Nuclear Research and Nuclear Energ, Bulgarian Academ of Sciences, blvd.tzarigradsko chaussee 7, 784 Sofia, Bulgaria PACS w - Quantum mechanics PACS m - Methods of electronic strucure calculations PACS r - Electron states at surfaces and iterfaces Abstract.- We stud the influence of the different choice of unit cells on the Bloch solutions of Schrödinger equation for one-dimensional periodic Kronig-Penne models with rectangular potential barriers or potential wells and partiall constant effective mass. We obtain generalized Kronig-Penne relations for bulk states and exact expressions for the corresponding periodic parts of the Block wave functions for the two possible choices of the matching conditions in an unit cell. We show also that our analtic expressions reduce to the well known expressions for the Kronig-Penne relations and Bloch waves under constant effective mass and appropriate matching conditions sdonev@inrne.bas.bg

2 The superlattice SL) electronic structure is of primar importance and interest in both fundamental and applied aspects, and it has been extensivel studied recentl -0. Theoretical studies of the electronic properties of SLs ma be carried out in the framework of one-dimensional Kronig-Penne KP) models 4,, in particular, calculating the electronic surface states,4-6, the electronic interface states 3,7-9 and the electronic bulk states 0 in the semiconductor superlattices. The basic point in these considerations is to determine the general solution to the corresponding one-dimensional Schrödinger equation together with the conditions for existence of surface and interface states, which conditions depend strongl on the corresponding wave functions -9. It hhas been noted b several authors -4 that the kind of the wave function depends on the choice of the unit cell, therefore, it seems ver important to have a more detailed knowledge of this dependence between the choice of the unit cell and the wave function. The aim of this letter is to consider some aspects of this problem, in particular, how the periodic parts of the Bloch solutions to the corresponding Schrödinger equation with periodic potential and partiall constant effective mass depend on the choice of the unit cell, more precizel, on the matching points for the wave functions and their derivatives in this unit cell. We are going to consider two cases of infinite one-dimensional KP-models: with rectangular potential barriers given on Fig., and with rectangular potential wells given on Fig.. In the first case potential barriers) the potential V x), determined b the conditions: V const, Ω x) :nb +n +)a x n +)a + b), V x) 0, Ω ) x) :na + b) x nb +n +)a, n 0, ±, ±,...) is periodic with respect to x with a period equal to the lattice constant a + b), i.e. V x) V x +a + b)). For an electron with a partiall constant effective mass mx) m const, Ω x) :nb +n +)a x n +)a + b) mx) m const, Ω ) x) :na + b) x nb +n +)a, n 0, ±, ±,...) moving in the periodic potential field ), the one-dimensional stationar Schrödinger equation d Ψx) dx + mx) E V x) Ψx) 0 3) has, according to the Bloch-Floquet theorem 4,, the following periodic in x solutions with period a + b): ψ I x) Aexp i + γ)x+bexp i γ)x 4) in regions I : Ω x) :nb +n +)a x n +)a + b), and of the form ψ x) Cexpi + β)x+dexp i β)x 5) in regions : Ω x) :na + b) x nb +n +)a. Further in our consideration we assume the quantities β and γ, introduced b 4) and 5), to be real and positive and their explicit dependences on the electron energ E,0 <E<V), are given b β m E, γ m V E), 6)

3 and is determined b the matching conditions. To obtain the coefficients A, B, C and D of the wave functions 4) and 5) we make use of the matching conditions at three successive points x,x,x 3 ) in one unit cell such, that x x 3 a + b), and x x a, x x 3 b, or x x b, x x 3 a, and we assume x x x 3 Fig.). Since there exist onl two was to choose these three points on the x axis we introduce them b the following relations: x + a x, x + b x 3, 7) x + b x, x + a x 3, 8) Relations 7) and 8) specif the two tpes of unit cells: the unit cell of tpe well-barrier which we call KP- tpe cell Fig.a), and the unit cell of tpe barrier-well, called KP- tpe cell Fig.b), respectivel. Thus, if we use the matching conditions used in,3,5-7 and denote m m, the solutions 4) and 5) for the unit cells 7) and 8) must obe the following conditions at x x : ψ I x )ψ x ) dψ I dx x ) dψ dx x ). It follows from the periodicit of ψx) that the solutions 4) and 5) and their first derivatives must satisf also at the points x x and x x 3 the conditions 9) ψ I x 3 )ψ x ) dψ I dx x 3) dψ dx x ) for the KP- tpe unit cell defined b 7), and the boundar conditions 0) ψ I x )ψ x 3 ) dψ I dx x ) dψ dx x 3) for the KP- tpe unit cell defined b 8). From 9) and 0) and from 9) and ) we obtain two sstems of four linear homogeneous equations each, for the four constants A, B, C and D, reflecting the differences of the two tpes unit cells. These two sstems have nontrivial solutions onl if the corresponding determinants are equal to zero, which leads to the same equation for β and γ given b cos a + b) chγb)cosβa)+ ) + γ β shγb)sinβa). ) βγ This equation ) we call generalized Kronig-Penne relation. Since onl three of the four coefficients A, B, C and D are linearl independent we can express an three of them through the fourth. So, when A µ j I B and C λj D,wherethe index j, denotes the two tpes unit cells, the corresponding periodic wave functions 4) and 5) become } ψ j I x) B µ j I exp i + γ)x+exp i γ)x, j, ), 3) ) 3

4 ψ j x) D λ j exp i + β)x+exp i β)x }, j, ). 4) For the KP- unit cell the corresponding nonhomogeneous sstem of equations 9) and 0) gives the following expression for µ I : cosβa) i ) µ I β γ β sinβa) exp ia + b) γb exp γx ). 5) cosβa)+ i ) β + γ β sinβa)+exp ia + b)+γb Similarl, the nonhomogeneous sstem of equations 9) and 0) gives for λ : ) ) chγb) i λ γ + β γ shγb) exp i a + b)+βa} ) ) exp iβx ). 6) chγb)+i γ β γ shγb)+exp i a + b) βa} In the same wa for the KP- unit cell, from the corresponding nonhomogeneous sstem of equations 9) and ) for the coefficients µ I and λ we obtain i ) β γ β sinβa) exp ia + b)+γb cosβa)+ µ I cosβa) i ) β + γ β sinβa)+exp ia + b) γb exp γx ), 7) λ chγb)+i chγb) i ) γ + ) γ ) β γ ) β γ shγb) exp i a + b)+βa} shγb)+exp i a + b) βa} exp iβx ). 8) Hence, for the both tpes of unit cells 7) and 8) it is seen from 5) and 7) that the wave functions 3) in the region I are different: ψi ψ I ; and from 6) and 8) we see that the wave functions 4) in the region are also different: ψ ψ. Consequentl, the choice of matching points inside the unit cell through relations 7)-) determines the form of the periodic parts 3) and 4) of Bloch solutions for the model under consideration. Let s consider the case, i.e. when the electron effective mass ) is constant: m m m o.inthiscaseβand γ in 6) become mo β o E, γ mo o V E) 9) and equation ) is reduced to the well-known Kronig-Penne equation. If for the KP- model Fig.a) we set x a in relation 7), and then, making use of 6) and 9), we find for the coefficient λ I λ I ) B the following value: ) λ I B ch b) i βo γ o sh b) exp i a + b)+β o a} ch b) i βo sh b)+exp i a + b) β o a} exp iβ oa), 0) which coincides with the corresponding expression 7) in Bloss paper. For the KP- model Fig.b) if x 0 in 8) we have the classical KP-model. In this case from 7) and 4

5 9) and from 8) and 9) we find the following expressions for the corresponding coefficients µ I µ I) KP and λ λ ) KP: µ I) KP cosβ oa) γo β o sinβ o a) expia + b)+ b cosβ o a) γo β o sinβ o a)+expia + b) b ) λ )KP chb)+i βo sh b) exp ia + b)+β o a} ch b)+i βo sh b)+exp ia + b) β o a} ) These values ) of µ I )KP and ) of λ )KP have been implicitl used in. However, choosing x b in 8), from 7) and 9) and from 8) and 9) we find that the corresponding coefficients µ I µ I )G and λ λ )G,givenb µ I) G cosβ oa) γo β o sinβ o a) expia + b)+ b cosβ o a) γo β o sinβ o a)+expia + b) b exp b) 3) λ )G chb)+i βo sh b) exp ia + b)+β o a} ch b)+i βo sh b)+exp ia + b) β o a} exp iβ ob) 4) are identical to Gubanov s results. Comparing ) with 3), and ) with 4), we immediatel obtain the following relations between them: µ I )G µ I )KP exp b), λ )G λ )KP exp iβ o b). Note that the exponential factors in 3) and 4) is due onl to the different choice of the coordinate origin in the unit cell. The case of the Kronig-Penne models with rectangular potential wells and partiall constant effective mass ) we treat in the same wa, and the corresponding results are obtained taking into account the periodic potential V x) Fig.): 0, Ω x) :na + b) x nb +n +)a V x) V, Ω 5) x) :nb +n +)a x n +)a + b), n 0, ±, ±,...) where V>0. Relations 7) and 8) again specif two tpes of unit cells: the unit cell of the tpe well-barrier called KP-a tpe cell Fig.a), and the unit cell of the tpe barrier-well called KP-a tpe cell Fig.b). In the regions with constant potentials the quantities θ and ϕ, which now pla the roles of γ and β in 4) and 5), are related to the electron energ E b means of the equations θ m E, ϕ m E + V ). In the case θ > 0, ϕ > 0 the Schrödinger equation 3) for an electron with effective mass ), moving in the potential field 5), for the both tpes of unit cells has nontrivial solutions onl if the quantities θ and ϕ satisf the corresponding generalized Kronig-Penne relation cos a + b) cosθa)cosϕb)+ ) θ + ϕ ) sinθa)sinϕb), 6) θϕ 5

6 where m m. Now the periodic part x) of the Bloch wave function is of the form } j I x) B ν j I expi + θ)x+expi θ)x, j, ), 7) in regions I barriers): Ω x) :na + b) x nb +n +)a, and of the form } j x) D χ j expi + ϕ)x+expi ϕ)x, j, ) 8) in regions wells): Ω x) :nb +n +)a x n +)a + b). For the KP-a unit cell, determined b condition 7), the corresponding nonhomogeneous sstem of equations 9) and 0) gives the following expressions for νi and χ : cosϕb) i ) νi ϕ θ ϕ sinϕb) exp i a + b)+θa} exp iθx ), 9) cosϕb)+i ) ϕ + θ ϕ sinϕb)+exp i a + b) θa} χ cosθa) i cosθa)+i ) ) θ + ϕ θ sinθa) exp i a + b)+ϕb} ) ) exp iϕx ), 30) θ ϕ θ sinθa)+exp i a + b) ϕb} For the KP-a unit cell, determined b 8), the coefficients νi and χ are obtained from the nonhomogeneous sstem of equations 9) and ) are given b: cosϕb)+i ) νi ϕ θ ϕ sinϕb) exp i a + b)+θa} exp iθx ), 3) cosϕb) i ) ϕ + θ ϕ sinϕb)+exp i a + b) θa} χ cosθa)+i cosθa) i ) ) θ + ϕ θ sinθa) exp i a + b)+ϕb} ) ) θ ϕ θ sinθa)+exp i a + b) ϕb} exp iϕx ). 3) It is clearl seen that the corresponding wave functions 7) and 8) satisf the inequalities I I and. If the electron energ is negative: E ε, ε > 0and V < E < 0, we have θ m ε<0 and ϕ m E + V ) > 0, and denoting m ε k, the generalized Kronig-Penne relation 6) becomes cos a + b) chka)cosϕb)+ ) k + ϕ ) shka)sinϕb). 33) ϕk In this case the quantit γi 9) for the KP-a model becomes ) ν ± cosϕb) i ) ϕ ± k ϕ sinϕb) exp ia + b) ± ka I exp±kx ), 34) cosϕb)+ i ) ϕ k ϕ sinϕb)+exp ia + b) ka 6

7 and from 30) we obtain also χ ) + ) χ ) χ ± ) ) ) χ ± chka) i k + ϕ k chka)+i ) k shka) exp i a + b)+ϕb} ) ϕ k shka)+exp i a + b) ϕb} exp iϕx ). 35) For the KP-a model the quantit νi 3) becomes ) ν ± cosϕb)+ i ) ϕ ± k ϕ sinϕb) exp ia + b) ka I exp±kx ), 36) cosϕb) i ) ϕ k ϕ sinϕb)+exp ia + b) ± ka and from 3) we again have χ ) + χ ) χ ) ±: χ ) ± chka)+i chka) i ) k + ) k ) ϕ k shka) exp i a + b)+ϕb} ) ϕ k shka)+exp i a + b) ϕb} Hence, we have the relations I ) + I ) I ) + I ), ) + ) ) + ). exp iϕx ). 37) When our generalized Kronig-Penne relations 6) and 33) reduce to the corresponding Kronig-Penne relations given in 8. In conclusion we can sa that the differences between the wave functions 3) and 4) and between the wave functions 7) and 8), which finall depend on the choice of unit cell and the matching conditions, should be taken into account when we compute surface and interface electronic states. In other words, varing the position of the barrier-well interface within an unit cell, we will influence the conditions for the existence of surface and interface states. For the two potentials ) and 5) our generalized Kronig-Penne relations ) and 6), and the particular case 33), are different, as it is in the original Kronig-Penne relations, and the corresponding Bloch wave functions for these two potentials are also different. 7

8 REFERENCES Bastard G., Wave Mechanics Applied to Semiconductor Heterostructures, LesEditions de Phsique, Les Ulis) 988. Bloss W. L., Phs. Rev. B, 4499) Shen M. R., Shen W. Z. and Li Z. Y., phs. stat. sol. b) 77993) K7. 4 Steślicka M., in Progr. Surf. Sci., vol. 5, edited b S. G. Davison Pergamon Press Ltd) 975, Stȩślicka M., Kucharczk R. and Glasser M. L., Phs. Rev. B 4990) Stȩślicka M., Progr. Surf. Sci., ) 07 7 Kandilarov B. D., Tashkova M., Petrova P. and Detcheva V., phs. stat. sol b) ) Kandilarov B. D., Tashkova M., and Petrova P., phs. stat. sol b) ) Kandilarov B. D., Detcheva V. and Primatarova M. T., J. Phs. C, 979) Carpena P., Gasparian V. and Ortuno, European Phsical Journal B, 8 999) 635. Kronig R. de L. and Penne W. G., Proc. Ro. Soc. A, 30 93) 499. Gubanov A. I. and Sharapov B. N., Izvestia LETI, ) 5 3 Metodieva M. and Detcheva V., Comp. rend. Acad. bulg. Sci., ) Bejenari I., Malkova N and Kanster V. G., phs. stat. sol. b) 3 00) Harrison W. A., Phs. Rev., 3 96) Ben Daniel D. J and Duke C. B., Phs. Rev., 5 966) Bastard G., Phs. Rev. B, 4 98) Haug A., Theoretical Solid State Phsics, vol., Chp., Oxford) 97. 8

9 Fig.: One-dimensional Kronig-Penne model with rectangular barriers and lattice constant a + b): b-the width of the barriers, V -the height of the barriers: a) unit cell of KP- tpe: region - region I well-barrier); b) unit cell of KP- tpe: region I- region barrier-well). Fig.: One-dimensional Kronig-Penne model with rectangular wells and lattice constant a + b): b-the width of the wells, V -the depth of the wells: a) unit cell of KP-a tpe: region - region I well-barrier); b) unit cell of KP-a tpe: region I- region barrier-well). 9