Scattering problems and boundary conditions for 2D electron gas and graphene

Structure and coordinate dependence of the reflected wave, as well as boundary conditions for quasi-particles of graphene and the two dimensional electron gas in sheets with abrupt lattice edges are obtained and analyzed by the Green's function technique. In particular, the reflection wave fu...

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Published in:	Физика низких температур
Date:	2018
Main Author:	Kadigrobov, A.M.
Format:	Article
Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2018
Subjects:	Електронні властивості низьковимірних систем
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Cite this:	Scattering problems and boundary conditions for 2D electron gas and graphene / A.M. Kadigrobov // Физика низких температур. — 2018. — Т. 44, № 12. — С. 1598-1605. — Бібліогр.: 16 назв. — англ.

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spelling	Kadigrobov, A.M. 2021-02-04T16:52:32Z 2021-02-04T16:52:32Z 2018 Scattering problems and boundary conditions for 2D electron gas and graphene / A.M. Kadigrobov // Физика низких температур. — 2018. — Т. 44, № 12. — С. 1598-1605. — Бібліогр.: 16 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176442 Structure and coordinate dependence of the reflected wave, as well as boundary conditions for quasi-particles of graphene and the two dimensional electron gas in sheets with abrupt lattice edges are obtained and analyzed by the Green's function technique. In particular, the reflection wave function contains terms inversely proportional to the distance to the graphene lattice edge. The Dirac equation and the momentum dependence of the wave functions of the quasi-particles near the conical points are also found by the perturbation theory with degeneracy in terms of the Bloch functions taken at the degeneracy points. The developed approach allows to formulated the validity criteria for the Dirac equation in a rather simple way. Використано техніку функцій Гріна для отримання граничних умов та для аналізу координатної залежності хвильової функції відбитої квазічастинки у графені та двовимірному електронному газі у разі, коли кристалічна гратка має різку межу. Показано, що відбита хвильова функція містить доданки, величина яких оберненопропорційна відстані до межі гратки графена. На основі теорії збурень у виродженому випадку виведено рівняння Діраку та отримано імпульсну залежність хвильової функції квазічастинки поблизу кінцевої точки спектру в термінах незбурених функцій Блоха в точках виродження. Розвинене наближення дозволяє сформулювати критерій справедливості рівняння Діраку відносно простим способом. Использована техника функций Грина для получения граничных условий и для анализа координатной зависимости волновой функции отраженной квазичастицы в графене и двумерном электронном газе в случае, когда кристаллическая решетка имеет резкую границу. Показано, что отраженная волновая функция содержит слагаемые, величина которых обратно пропорциональна расстоянию до границы решетки графена. На основе теории возмущений в вырожденном случае выведено уравнение Дирака и получена импульсная зависимость волновой функции квазичастицы вблизи конической точки спектра в терминах невозмущенных функций Блоха в точках вырождения. Развитое приближение позволяет сформулировать критерий справедливости уравнения Дирака относительно простым способом. The author thanks A.F. Volkov for useful discussions. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Електронні властивості низьковимірних систем Scattering problems and boundary conditions for 2D electron gas and graphene Задача розсіяння та граничні умови для двовимірного електронного газу і графена Задача рассеяния и граничные условия для двумерного электронного газа и графена Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Scattering problems and boundary conditions for 2D electron gas and graphene
spellingShingle	Scattering problems and boundary conditions for 2D electron gas and graphene Kadigrobov, A.M. Електронні властивості низьковимірних систем
title_short	Scattering problems and boundary conditions for 2D electron gas and graphene
title_full	Scattering problems and boundary conditions for 2D electron gas and graphene
title_fullStr	Scattering problems and boundary conditions for 2D electron gas and graphene
title_full_unstemmed	Scattering problems and boundary conditions for 2D electron gas and graphene
title_sort	scattering problems and boundary conditions for 2d electron gas and graphene
author	Kadigrobov, A.M.
author_facet	Kadigrobov, A.M.
topic	Електронні властивості низьковимірних систем
topic_facet	Електронні властивості низьковимірних систем
publishDate	2018
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
title_alt	Задача розсіяння та граничні умови для двовимірного електронного газу і графена Задача рассеяния и граничные условия для двумерного электронного газа и графена
description	Structure and coordinate dependence of the reflected wave, as well as boundary conditions for quasi-particles of graphene and the two dimensional electron gas in sheets with abrupt lattice edges are obtained and analyzed by the Green's function technique. In particular, the reflection wave function contains terms inversely proportional to the distance to the graphene lattice edge. The Dirac equation and the momentum dependence of the wave functions of the quasi-particles near the conical points are also found by the perturbation theory with degeneracy in terms of the Bloch functions taken at the degeneracy points. The developed approach allows to formulated the validity criteria for the Dirac equation in a rather simple way. Використано техніку функцій Гріна для отримання граничних умов та для аналізу координатної залежності хвильової функції відбитої квазічастинки у графені та двовимірному електронному газі у разі, коли кристалічна гратка має різку межу. Показано, що відбита хвильова функція містить доданки, величина яких оберненопропорційна відстані до межі гратки графена. На основі теорії збурень у виродженому випадку виведено рівняння Діраку та отримано імпульсну залежність хвильової функції квазічастинки поблизу кінцевої точки спектру в термінах незбурених функцій Блоха в точках виродження. Розвинене наближення дозволяє сформулювати критерій справедливості рівняння Діраку відносно простим способом. Использована техника функций Грина для получения граничных условий и для анализа координатной зависимости волновой функции отраженной квазичастицы в графене и двумерном электронном газе в случае, когда кристаллическая решетка имеет резкую границу. Показано, что отраженная волновая функция содержит слагаемые, величина которых обратно пропорциональна расстоянию до границы решетки графена. На основе теории возмущений в вырожденном случае выведено уравнение Дирака и получена импульсная зависимость волновой функции квазичастицы вблизи конической точки спектра в терминах невозмущенных функций Блоха в точках вырождения. Развитое приближение позволяет сформулировать критерий справедливости уравнения Дирака относительно простым способом.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/176442
citation_txt	Scattering problems and boundary conditions for 2D electron gas and graphene / A.M. Kadigrobov // Физика низких температур. — 2018. — Т. 44, № 12. — С. 1598-1605. — Бібліогр.: 16 назв. — англ.
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first_indexed	2025-11-25T22:45:16Z
last_indexed	2025-11-25T22:45:16Z
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fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12, pp. 1598–1605 Scattering problems and boundary conditions for 2D electron gas and graphene A.M. Kadigrobov Theoretische Physik III, Ruhr-Universitaet Bochum, D-44801 Bochum, Germany E-mail: kadig@tp3.rub.de Received July 3, 2018, published online October 26, 2018 Structure and coordinate dependence of the reflected wave, as well as boundary conditions for quasi-particles of graphene and the two dimensional electron gas in sheets with abrupt lattice edges are obtained and analyzed by the Green's function technique. In particular, the reflection wave function contains terms inversely propor- tional to the distance to the graphene lattice edge. The Dirac equation and the momentum dependence of the wave functions of the quasi-particles near the conical points are also found by the perturbation theory with de- generacy in terms of the Bloch functions taken at the degeneracy points. The developed approach allows to for- mulated the validity criteria for the Dirac equation in a rather simple way. Keywords: graphene, 2D electron gas, Dirac equation, boundary conditions, validity criteria, Green functions. 1. Introduction Dynamical and transport properties of various mesoscopic systems have been attracting much attention during the last decades [1,2]. Among them are quantum dots, quantum nano- wires, tunneling junctions and 2D electron gas based nanostructures. Fascinating dynamic and kinetic phenomena arise in graphene which is a two-dimensional (2D) semi-metal having no energy gaps between two bands of electrons and holes at six points of the hexagonal Brillouin zone. Electronic properties of graphene can be described by the two dimensional differential Dirac equation [3,4] sup- plemented by boundary conditions. Details of the boundary conditions depend on microscopic characteristics of the concrete structure of the sample boundary [5]. Theoretical derivations of the boundary conditions for Dirac equations are usually based on various models such as tight bound model (see, e.g., review papers [6,7] and references there), the effective mass model [8], tight-binding model with a staggered potential at a zigzag boundary and with the boundary orientation intermediate between the zigzag and armchair forms [9]. In this paper dynamics of quasi-particles in 2D electron gas and graphene are considered in the frame of the con- ventional approach to the scattering problems for finite lattices in terms of the electron Bloch functions and band energies without usage of the above-mentioned models. Using the Green's function technique the boundary condi- tions and the coordinate dependence of the wave functions of quasi-particles in 2D electron gas and graphene lattices with an abrupt edges are obtained. Criteria of the validity of the Dirac equation are formulated in a rather simple way. It is also shown that the wave function of the reflect- ed quasi-particle contains slow varying terms inverse pro- portional to the distance to the edge of the graphene sheet. The outline of this paper is as follows. In Sec. 2 the per- turbation theory with degeneracy is used to obtain the Di- rac equation and the wave functions in terms of the Bloch functions taken at the degeneracy point in the reciprocal lattice. In Sec. 3 scattering of quasi-particles by an external potential in graphene is considered in the Bloch function representation. The Dirac equation is derived and its validi- ty criteria are formulated. In Sec. 4 the Green's function approach to the problem of scattering of quasi-particles in a lattice sheet with an abrupt edge is developed. In Sec. 4.1 the wave function structure and boundary conditions for the 2D electron gas in a lattice sheet with an abrupt edge are found. In Sec. 4.2, the structure of the wave function and its dependence on the distance to the lattice edge are found. In Sec. 5 concluding remarks are presented. 2. Derivation of Dirac equation and Bloch functions for graphene by perturbation theory The Schrödinger equation for electrons is 0 , , ˆ ( ) = ( ) ( )s s sH ϕ ε ϕp pr p r (1) where 0Ĥ is the Hamiltonian for electrons moving in the periodic lattice potential ( )U +r a with the period a . This Hamiltonian reads as follows: © A.M. Kadigrobov, 2018 Scattering problems and boundary conditions for 2D electron gas and graphene 2 2 0 2 ˆ = ( ) 2 H U m ∂ − + ∂ r r  . (2) The wave function , ,( ) = e ( ) i s suϕ pr p pr r (3) is the Bloch function while the Bloch periodic factor , ( )su p r has the translation periodicity of the lattice, p is the electron quasi-momentum and s is the band number. In order to find the dependence of Bloch functions and the dispersion law of the quasi-particle in graphene on their momentum p one may use the perturbation theory in \| \| / 1b <<p with the degeneracy [10] at = 0p (here b is the characteristic period of the reciprocal lattice). Presenting Bloch functions as a superposition of the unper- turbed wave functions of the zero approximation ( )1 1,0 2 2,0( ) = exp ( ) ( )i g u g uϕ + prr r r  (4) (here 1,2;0u (r) are the periodic factors of the Bloch func- tions of the degenerated bands taken at the point of degen- eracy p = 0) and inserting it in the Schrödinger equation Eq. (1), after taking the matrix elements one gets a set of algebraic equations for the expansion constants 1,2g . In the first approximation in the momentum p these equations are , , , =1,2 = 0 ' g g′ ′α α α α α −ε + ∑p ppv , (5) where = 1,2α is the band number of the degenerated band while ( )αε ≡ ε p ; the matrix elements of the velocity opera- tor ˆ = ( / ) /i m− ∂ ∂v r are , ,0 ,0ˆ= ( ) ( )u u d′ ′α α α α∫ r r rv v . (6) Equating the determinant of Eq. (5) one gets the con- ventional dispersion law of quasi-particles near the degen- eration point: 2 2 12( ) 4 \| \| ( ) = 2 + − ± ± + ε pv pv pv p , (7) where 11 22=± ±v vv . From Eq. (7) it follows that the dispersion law of quasi- particles in the vicinity of the band intersection is of the graphene-type 2 2( ) = =x yp p p±ε ± + ±p v v (8) if the lattice symmetry imposes the following conditions on the velocity matrix elements at the point = 0p of the degeneration 1 2(0) = (0)ε ε : 11 22 12(0) = (0) = 0, \| (0) \|= ,v v v v ( ) 12 12(0) = (0),y xi±v v (9) where 6= 1 10F ≈ ⋅v v m/s for graphene. Inserting these values in Eq. (5) one gets equation for dependence of the expansion coefficients on the momen- tum p as follows: 1, 2,( ) = 0;x yg p ip g−ε + −p pv 1, 2,( ) = 0x yp ip g g+ − εp pv . (10) Using Eq. (10) one finds the dispersion law Eq. (8) and the Bloch functions of quasi-particles in graphene: ( ) 1,0( ; ) , 1 ( ) 2,0 ( ) 0 1 ( ) = exp e0 ( ) gr i u g i u ± ± α θ±    ϕ        p rprr r   , (11) where 1g is the normalizing constant and the phase = arctan( / )x yp pθ . In the next section, scattering of quasi-particles by an external potential in 2D gas and graphene is considered. 3. Scattering of quasi-particles by an external potential and derivation of the Dirac equation In this section, the scattering problem of electrons by a potential ( )V r (the characteristic properties of which are later described) in 2D gas and graphene is investigated. The Schrödinger equation of the system under consid- eration is ( )0 ˆ ( ) ( ) = ( )H V+ Ψ εΨr r r . (12) It is assumed that two energy bands are closely spaced or intersect in a certain point = 0p of the reciprocal space as it takes place in graphene while the energy ε is in the vi- cinity of the degenerate energy. In order to investigate dy- namics of electrons in such a situation it is convenient to expand Ψ in the series of the following functions: ,0 , , ( ), = 1, 2 = exp ( ), 1, 2,s s u s i u s α ≡ α χ   ≠  p p rpr r  (13) where band numbers = 1,2α designate the bands close to each other, the periodic Bloch factors of which are taken at = 0p . As ,sχ p constitute a complete set of functions the following expansion is satisfied for all values of p. , ,( ) = ( )s s s g dΨ χ∑∫ p pr r p. (14) Inserting this expansion in Eq. (12) one gets , =1,2 ˆexp ( (0)d g iα α α   ε + +    ∑ ∫ p prp pv  2 ,0( ) ) ( ) 2 V u m α+ + − ε + p r r , , 1,2 ˆ( ( ) ) ( ) = 0s s s s d g ≠ + ε + − ε ϕ∑ ∫ p pp p pv r , (15) where ˆ = ( / ) /i m− ∂ ∂v r is the velocity operator. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1599 A.M. Kadigrobov Multiplying this equation on the left by ,αχ p and ,sχ p in turns and integrating one gets a set of coupled algebraic equations for the expansion factors , :gα p 2 , , , =1,2 ( (0) ) 2 ' g g m ′ ′α α α α α α ε + − ε + +∑p p p pv , , , 1,2 ( ( ) ) ( )p p s s s s V g d a g ∞ ′ ′− α α ≠−∞ ′+ + ε − ε +∑∫ p pp p p , , 1,2 ( ) = 0p p s s s s V a g d′ ′ ′− ≠ ′ ′+ ∑ ∫ pp,p p , (16) 2 , , , =1,2 ( ( ) ) ( (0) ) 2s s sg a g mα α α α ε − ε + ε + − ε +  ∑p p pp * , , , ,( )s p p sg V a g d ∞ ′ ′ ′α α − α α −∞ ′ ′+ + +  ∫p ppv p p , , 1,2 ( ) = 0,p p s s s s V a g d′ ′ ′ ′− ≠ ′ ′+ ∑ ∫ pp,p p (17) where the matrix elements of the velocity operator ˆ = ( / ) /i m− ∂ ∂v r are * , ,0 ,0( ) ˆ= ( ) ( )u u d′ ′α α α α∫ a v r v r r  , , , ,0( ) ˆ= ( ) ( )s su u d′α α∫ pa v r v r r  (18) and , ,0 ,( ) ( ) = ( ) ( )s sa u u drα α∫ pa p r r  , * , , ,( ) ( ) = ( ) ( )s s s sa u u dr′ ′ ′′ ∫ p pa p,p r r  . (19) Integration in Eqs. (18), (19) is over a unit cell. The above equations are valid for all values of the electron momen- tum p and for an arbitrary form of the potential ( )V r . The equations which describe dynamics of electrons in graphene and analogous conductors (Dirac equations) in the vicinity of the degeneration energy are readily obtained from Eqs. (16), (17) in the following limits: \| \| = 2 /b aπp  (a is the characteristic atomic spacing) while the potential is assumed to be small and slowly vary- ing that is gap\| \| = \| ( ) ( ) \|sV α∆ ε − εp p , s ≠ α, and the characteristic interval lδ of the variation of ( )V r is l aδ  . Indeed, under the above assumptions one may neglect the dependence of the matrix elements in Eq. (19) on p and obtain , ,= = 0,s sa sα αδ ≠ α. Inserting this equality in Eq. (17) one gets 2, , gap \| \| \| \|, ( /2 )= 1 \| \| sg V p m gα γ ∆ p p   (20) and hence equations Eq. (16) and Eq. (17) are decoupled in the zero approximation in 1γ . Therefore, in this appro- ximation the Schrödinger equation in the χ-representation (see Eq. (14)) for electrons in the vicinity of the intersec- tion of two bands, 1 2(0) = (0) = 0ε ε , reads as follows: , , =1,2 ( (0) ) ' ' g g ′α α α α ε − ε + +∑p a,a ppv , = 0p pV g d ∞ ′ ′− α −∞ ′+ ∫ p p . (21) While writing this equation we assumed 2 /2 \| \|p m vp ). Using equalities Eq. (9) one gets the set of equations that describes dynamics of quasi-particles in the presence of potential ( )V x : 1, 1, 2,( ) = 0;p p x yg V g d p ip g ∞ ′ ′− −∞ ′−ε + + −∫p p pp v 1, 2, 2,( ) = 0x y p pp ip g g V g d ∞ ′ ′− −∞ ′+ − ε + ∫p p p pv , (22) where for the sake of certainty ( ) 12 12=y xi−v v is chosen. Ex- panding the wave functions in Eq. (22) into the Fourier series 1,2; 1,2= ( )exp pg i d Φ −   ∫p rr r  (23) one find the equation for the Fourier factors: 1 1 2 2 ( ) = ( ) x y x y V i i i V ∂ − ∂  Φ Φ    − ε      Φ Φ∂ + ∂      r r v . (24) The above equation describes dynamics and, in particular, quantum tunnelling of quasi-particles between intersecting energy bands in the vicinity of the point of degeneration. This set of differential equations (with proper changes of parameters) arises in all cases in which the unperturbed energy spectrum has points of degeneration (or points of close approach of energy bands), e.g., in graphene (see review papers [6,7,13]), in the cases of Landau–Zener tun- nelling (see Ref. 12 and references there) and the magnetic breakdown — quantum tunnelling in metals under a strong magnetic field (see Refs. 11, 14, 15). Note, that the tunnel- ling transmission of electrons between intersecting energy bands without back-scattering (“Klein tunneling” takes place in many cases, in particular, in the cases of grapene [6,7,13] (normal incident of the electron to barrier) and the magnetic breakdown [11]. As it follows from the above derivation of Eq. (24) the Dirac equation [6,13] is valid only in the limit of small and smooth potentials (see Eq. (20) and the text around it) and hence it can not be used for investigation of the problem of electron scattering by sharp boundaries of the sample. In the next section the Green function approach is developed to solve this problem for the cases of 2D gas and graphene. 1600 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 Scattering problems and boundary conditions for 2D electron gas and graphene 4. Scattering of electros in lattice with abrupt edge (Green function approach) Boundary conditions for Dirac Fermions in graphene were derived in Refs. 8, 9, 13 (see also, e.g., Review pa- pers [6,7]) in the tight-binding model. In this section the boundary conditions for two dimensional electron gas and grapene are derived by use of the Green’s function tech- nique in terms of the general properties of electron spectra and proper wave functions. Below a half infinite two dimensional sheet of 2D gas or graphene in the half plane 0x ≥ with the edge line at = 0x is considered. In this case the Schrödinger equation is 2 2 2 ( ) ( ) = 0 2 U m  ∂ − + − ε Ψ  ∂  r r r  , (25) where ( )U r is the periodic lattice potential. For the sake of certainty the boundary conditions ( ) = 0, = 0xΨ r , ( ) ,0 0 ( ) = ( ),in s xΨ ϕ → +∞pr r (26) are assumed. Here ( ) ,0 0 in sϕ p is the Bloch function (see Eq. (3)) incident to the boundary and (0) 0 = ( , )xp qp where ( )= in yq p is the conserving momentum projection. Below, to investigate the problem of scattering by the abrupt edge at = 0x Green’s function for the infinite lat- tice is used, that is the needed Green function satisfies the equation 2 2 2 ( ) ( , ) = ( ) 2 U G m  ∂ ′ ′− + − ε δ −  ∂  r r r r r r  (27) in which the lattice potential ( )U r covers the whole plane ( , )x y . Expanding ( , )G ′r r in the series of wave functions of electrons in the infinite lattice and using Eq. (27) one finds * , ,( ) ( ) ( , ) = ( ) s s ss G d i ′ϕ ϕ ′ ε − ε + δ∑∫ p pr r r r p p , (28) where 0δ → + Below we also assume that along the edge line = 0x the lattice is periodic with the period a and hence the mo- mentum projection yp conserves. Taking into account this requirement and using Eqs. (25), (27) together with Eq. (28) and the boundary conditions for ( )Ψ r one finds the wave function on the right half-plane 0x ≥ as follows: ( ) 2,0 0 , 0 ( ) = ( 0, ) 2 in s x x s dy y mv ∞ −∞ ϕ ′Ψ + Ψ − ×∫ p r  , , , ,( 0, ) ( ) ( , ) s p q s p qx x y x s x ys y dp p q i ϕ − ϕ × ε − ε + δ∑∫ r (29) where , 0 < 0x sv is the x-projection of the velocity of the in- cident electron that normalizes its wave function to the unity flux; 0 y yq p≡ is the conserving y-projection of the momen- tum 0p of the incident electron; ( 0, ) = ( )/x y x′Ψ − ∂Ψ ∂r at = 0x − ; as the value of Ψ-function in Eq. (29) exactly on the boundary contour is a matter of convention (see Ref. 16) the boundary contour is assumed to be shifted to = 0 0 ,x − ≡ − δ 0δ → while ( )x′Ψ r is defined on the half-plane 0x ≥ . It is now necessary to introduce the integral equation for ( 0, )yΨ − solution of which completes the definition of the wave function ( )Ψ r : ( ) 2,0 0 =0 , 0 1(0, ) = \| (0, ) 2 in s x x x x s y dy y x m ∞ −∞ ∂ϕ ′ ′Ψ + Ψ × ∂ ∫ p  v , , , ,(0, ) (0, ) . ( , ) s p q s p qx x y x s xs y y dp p q i ′ϕ ϕ × ε − ε + δ∑∫ (30) Here ( ) = ( )/f f x′ ∂ ∂r r . In the general case and without usage of an approximate model this integral equation can not be solved. However, important general features (in terms of (0, )x y′Ψ ) of the quasi-particle scattering by the abrupt lattice edge follow from Eq. (29). Indeed, let us consider one-dimensional integrals in Eq. (29) presenting them in the form //2 , , , , /2 (0, ) ( )e = ( , ) ixpb xx s p q s p qx y x y x s x ybx u y u I dp p q i − ε − ε + δ∫ r  . (31) Here xb is the period of the reciprocal lattice in the x-di- rection. In the complex plane the dispersion law ( , )s xp qε considered as a function of the complex variable = xz p i+ ξ is a multi-valued function which has branching points in the complex plane and hence this integral is a sum of the resi- dues minus sum of integrals along the brunch cuts in the upper complex half-plane 0ξ ≥ inside the contour schemati- cally shown in Fig. 1. The left and right vertical lines of the contour are separated by the reciprocal period period xb and hence the integrals along them cancel each other because the integrands are periodic functions of the same period. The integral along its upper horizonal part exponentially goes to zero as this contour part goes to i∞ . The poles and branching cuts of the integrand which con- tribute to the integral Eq. (31) are separated in two types: 1. Poles lying on the upper side of the real axis ( ) ( )= , = 1,2,..., 0,x x x p p iα α δ + α δ→ v where their real parts ( ) xp α are determined by the equation = ( , )x yp qαε ε (32) while α defines the number of the band which are present in the infinite lattice at the energy ε and the momentum projection =yp q (in Fig. 1 they are shown as pockets Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1601 A.M. Kadigrobov with the band numbers = 1,2α ). One easily sees that these poles are inside the integration contour if the x-projections of the velocity , ( ) ( , ) = \| > 0x y x pxx p q p α α α ∂ε ∂ v (33) and, therefore, they correspond to the states of electrons reflected back by the boundary. 2. Poles lying high in the upper complex plane (0) (0)= s s x sp p ib+ which are determined by the equation = ( , ),s x yp qε ε s ≠ α where the energy bands ( , )s x yp qε do not overlap bands α (in which the energy ε lies). As the dispersion law ( , )s xp qε is a multi-valued func- tions of xp (a circuit around the branching point changes the band number s) there are branching cuts in the upper half plane = xz p i+ ξ , schematically shown in Fig. 1, which pass from the branching points ( ) ( ) ( ) ,=br br br s x s sp p i+ ξ ( ) , br x sp i+ ∞). Taking into account the above-mentioned poles and branch cuts one easily carried out integration in Eq. (31) and, inserting the result in Eq. (29), one writes the required wave function as follows: ( ) , ,0 0 , ,0 ( , ) ( ) = in s px x s x x y C αα α αα ϕϕ Ψ + +∑p r v v (0) (0), / , ( , ) e s p b xs ss x ss x y C − ≠α  ϕ  + +   ∑  v / ( ), ( ) ( ) ( ) ( , )e x s brs pss s br B x y d ∞ − ζ ζ ξ   + ζ ϕ ζ   ∑ ∫  , (34) where summation = ,s s α (that is summation with respect to s is over all electron bands including all α-bands), the integral is taken along the s -branching cut in which the variable change = iζ ξ has been done; the functions in square brackets are ( ) (0) (0), , = exp s x s p s ps s xp qy u i  + ϕ      , ( ) , ( ) ( ), ( ) , ( ) ) = exp br x s br brs p s ps s xp qy u i ζ ζ  + ϕ       , (35) where ( ) ( ) ,( ) =br br s x sp pζ + ζ; constants ,sA Cα and ( )sB ζ are presented in Appendix, Eqs. (38), (39). 4.1. Scattering of electrons by abrupt edge in 2D electron gas As one sees from Eq. (34) the functions in square brackets exponentially decrease with an increase of the x- coordinate. In the general case the energy gaps between non-overlapping electron bands are of the order of the band widths E∆  1–10 eV and hence the imaginary parts of the pole and branch momenta are of the order of the / FE∆ v that is ( )(0) /s s brb aξ   where a is the atomic spacing. From the above considerations and Eq. (34) it follows that inside the layer x a adjacent to the sample boundary the electron wave function is a superposition of Bloch wave functions ϕs of all energy bands including those vir- tual which are above and below the band of the incident electron (that is their band numbers s ≠ α). At the distances much larger than the atomic spacing, x a , all the virtual wave functions exponentially drop out from the superposition and the electron wave function Ψ reduces to ( ),( ) / ,0 0 , ( , ) ( ) = (e ) pin x ax s x x y C O αα − α αα ϕ Ψ ϕ + +∑pr v . (36) According to the calculations the Bloch functions under the summation sign belong to the states with the positive x- projections of the electron velocity (see Eq. (33), Fig. 1(а)). Therefore, they are the Bloch functions of the electron scattered back by the sample boundary into all the availa- ble energy bands at the energy of the incident electron ε and the conserving y -projection of the momentum while constants Cα are the amplitude probability of this many- channel scattering (an example of such the two-channel scattering is presented in Fig. 1(а)). The above general scattering scenario requires a special treatment in the case of the generation when the top and the bottom of two energy bands are very close or coincide in some point of the reciprocal space as it takes place in graphene. In the next section the scattering of quasi- particles by a sharp graphene boundary is considered. Fig. 1. (a) An example of the Brillouin zone with 5 contours of equal energy of energy bands ( ) =αε εp with the band number = 1, 2, 3, 4, 5α . The arrows show the directions of the velocities at the points of intersections of the equal energy contours with the line of the constant y-projection q . The incident electron has the con- serving momentum projection ( )= in yq p and the negative direction of the velocity x -projection, xv . (b) Contour of integration of Eq. (29), (31). Crosses on the real axis xp and those in the upper complex half-plane show positions of the poles corresponding to two points with positive xv and to virtual states ( ) ;s sε ≠ ε ≠ αp . Thick vertical lines are brunch lines corresponding to the brunch- ing points (thick dots) in the electron spectrum. 1602 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 Scattering problems and boundary conditions for 2D electron gas and graphene 4.2. Scattering of quasi-particles by abrupt edge of graphene sheet In this section scattering of quasi-particles in graphene by an abrupt edge is considered. Graphene fills the half- plane 0x ≥ while the boundary condition for the quasi- particle wave function is ( ) (0, ) = 0gr yΨ . In the general approach to the scattering problem devel- oped above, the only peculiarity of the scattering of quasi- particles in graphene lies in their dispersion law whereas all the equations of the previous section remain valid. The incident electron in graphene with a negative x- projection of the velocity in the Bloch state ( , ) ( ) gr in inϕ p and the energy > 0ε belonging (for the sake of certainty) to the electronic band ( ) =p p+ε +v is considered (see Fig. 2(a)). Inserting the graphene quasi-particle dispersion law Eq. (8) in Eq. (34) one finds the electron wave function at the distances much greater that the deBroglie’s wave length = /Bx λ εv as follows (details of calculations are given in Appendix B): ,x ε v  ( ; )( , ) ( )( ) , ,( ) ( ) ,=1,2 ( , ) ( ) = grgr in inin p px ygr gr x x x y C + αα α αα ϕϕ Ψ + +∑pr v v ( ) / ( ; )( ; ) ( ) ( ); ,= =1,2 e( , ) inxpy grgr gr inp px y B x y x − νν α αν ± α + ϕ∑ ∑  . (37) Here Bloch functions ( ) , gr αϕ p (see Eq. (11)) are ( ) ( ) ( ) 1,( ; ) , 1 ( ) 2, ( ) 0 1 ( ) = exp e0 ( ) gr gr gr i u g i u α α ± ± α θ±      ϕ         p p p r prr r   , where gα is the normalizing constant and ( ) 1 = 0grp for 1,α = while for 2,α = it is equal to the coordinate of the second cone deg 2p ; note that ( )in yp is the conserving y- projection of the incident quasi-particle momentum. Therefore, as it follows from Eqs. (34), (37), near the graphene lattice boundary, inside the layer x a (a is the atomic spacing), the quasi-particle wave function is a superposition of Bloch wave functions belonging to all energy bands (including those virtual). At the distance much larger than de Brouglie’s wavelength = /Bλ εv the superposition reduces to the sum of the Bloch functions of the reflected electron, Eq. (11), (note, it was assumed that an electron was the incident quasi-particle) of the infinite graphene sample plus additional terms propor- tional to the graphene Bloch functions with the momen- tum the both projections of which are equal to the con- serving y -projection of the incident quasi-particle ( )in yp . The latter terms slowly decrease at the distances ( )/ \| \| /in yx pδ ε v . If the normal incidence of the quasi- article on the graphene boundary takes place, ( ) = 0in yp , these terms decease as 1/x . 5. Conclusion In this paper the problem of scattering of quasi-particles by an abrupt edge in the 2D electron gas and in graphene lattice is considered by the Qreen's function technique. This approach allows to find the coordinate dependence of the wave function of the quasi-particle reflected at such an edge and the boundary conditions in a rather simple way. In particular, it is shown that the wave function of the re- flected quasi-particle in graphene contains terms slowly decreasing with an increase of the distance to the edge. In the case of the transverse incidence they are inverse pro- portional to this distance. For graphene the Dirac equation, the momentum de- pendence of the wave functions near the conic points and the dispersion law are derived by the perturbation method with degeneracy in terms of the Bloch functions the periodic factors of which are taken at the degeneracy point (the conic point). This approach allows to formu- late the lattice symmetry and external field properties needed for validity of the Dirac equation in grahene and other two-dimensional conductors with degenerated en- ergy bands. The author thanks A.F. Volkov for useful discussions. Fig. 2. (a) Shematic representation of two equal energy contours 2 2 =x yv p p+ ε and 2 2( ) =deg x x yv p p p− + ε . The arrows show directions of the velocities at the points of intersections of the equal energy contours with the line of the constant y-projection q . The incident electron has the conserving momentum projec- tion ( )= in yq p and the negative direction of the velocity x -projec- tion, = /x xp pv v . (b) Contour of integration of Eqs. (29), (31). Crosses on the real axis xp and those in the upper complex half- plane show positions of the poles corresponding to two points with positive xv and to virtual states ( ) ;sε ≠ εp s ≠ α . Thick vertical lines are brunch lines corresponding to the brunching points (thick dots) in the quasi-particle spectrum. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1603 A.M. Kadigrobov Appendix A After taking the integral in Eq. (31) and the use of Eq. (29) one finds constants , sC Cα and function ( )sB ζ as follows: 2 ( ), , (0, ) = 2 (0, ) 2 s px x x y C i dy y m ∞ α α α−∞ ϕ ′π Ψ∫  v , 2 (0), , (0, ) = 2 (0, ) 2 s ps s x x s y C i dy y m ∞ −∞ ϕ ′π Ψ∫  v (A1) and ( )2 , ( ) ( ) , (0, ) (0, ) ( ) = . 2 ( , )) x brs ps s br s x s y y B dy m p q i ∞ ζ −∞ ′Ψ ϕ ζ ε − ε + ζ + δ∫  (A2) Appendix B. Coordinate dependence of the integral along the cut for graphene Using the grapene dispersion law Eq. (8) and Eq. (A2) one re-writes the terms in the last sum in Eq. (34) related to the energy bands with the grapene dispersion laws, = 1, 2α , as follows: 2 ( ) ( )/ (0, ) e 2 inip y yy xdy y m ∞ − −∞ ′Ψ ×∫   ( )* ( ) ( ) ( ) ( ); ; (0, ) ( ) ( , )gr gr gr gr gru y u J yα α α × p p r r , (B1) where / ( ) 2 2 e= x gr q J d v q i ∞ − ζ α ζ ε −ζ + + δ ∫   . (B2) Changing the variables qζ − → ζ one gets / ( ) / 0 e= e . ( 2 x gr qxJ d iv q ∞ − ζ − α ζ ε ζ ζ +∫    (B3) As one sees from Eq. (B3) the main contribution of the integrand to the integral is at /xζ  . This inequality means that the square root in the integral denominator is much less than /ε v (note that ( )\| /in yp ε v). Therefore, neglecting the term with the square root one easily takes the integral and obtains the result presented in Eq. (37) of the main text in which constants Bα are 2 ( ; )( ; ) ( ) ( ); , = (0, ) (0, ) 2 grgr x gr inp py y B dy y y dy m ∞ ±± α α −∞ ′Ψ ϕ∫  . (B4) ________ 1. Th. Heinzel Mesoscopic, Electronics in Solid State Nanos- tructurs, Wiley, New York (2003). 2. Th. Dittrich, P. Hänggi, G.-L. Ingold, B. Kramer, G. Schön, and W. Zwerger, Quantum Tranasport and Dissipation, Wiley, New York (1998). 3. P.R. Wallace, Phys. Rev. 71, 622 (1947). 4. D.P. DiVincenzo and E.J. Mele, Phys. Rev. B 29, 1685 (1984). 5. Y.-W. Son, M.L. Cohen, and S.G. Louie, Phys. Rev. Lett. 97, 216803 (2006). 6. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, Rev. Mod. Phys. 81, 109 (2009). 7. S. Das Sarma, Shaffique Adam, E.H. Hwang, and Enrico Rossi, Rev. Mod. Phys. 83, 109 (2011). 8. E. McCann and V.I. Fal’ko, J. Phys.: Condence. Matter 16, 2371 (2004). 9. A.R. Akhmerov and C.W.J. Beenakker, Phys. Rev. B 77, 085423 (2008). 10. L.D. Landau and E.M. Lifshitz, Qauntum Mechanics, § 79, The intersection of electron terms, Oxford: Butterworth- Heinemann (1998); Such an approach was used by A.A. Slutskin for investigation of the magnetic breakdoun effect [11]. 11. A.A. Slutskin, Zh. Eksp. Teor. Fiz. 53, 767 (1967) [Sov. Phys. JETP 26, 474 (1968)]. 12. L.D. Landau and E.M. Lifshitz, Qauntum Mechanics, § 90, Pre-dissociaion, Oxford: Butterworth-Heinemann (1998). 13. C.W.J. Beenakker, Rev. Mod. Phys. 80, 1337 (2008). 14. E.I. Blount, Phys. Rev. 126, 1636 (1962). 15. A.A. Slutskin and A.M. Kadigrobov, Fiz. Tverd. Tela 9, 184 (1967) [Sov. Phys. Solid State 9, 138 (1967)]. 16. P.M. Morse and H. Feshbach, Methods of Rheoretical Physics, Part I, §7.2, 805, McGraw-Hill Book Company, New York, Toronto, London (1953). ___________________________ Задача розсіяння та граничні умови для двовимірного електронного газу і графена А.М. Кадигробов Використано техніку функцій Гріна для отримання гранич- них умов та для аналізу координатної залежності хвильової функції відбитої квазічастинки у графені та двовимірному елек- тронному газі у разі, коли кристалічна гратка має різку межу. Показано, що відбита хвильова функція містить доданки, вели- чина яких оберненопропорційна відстані до межі гратки графе- на. На основі теорії збурень у виродженому випадку виведено рівняння Діраку та отримано імпульсну залежність хвильової функції квазічастинки поблизу кінцевої точки спектру в термінах незбурених функцій Блоха в точках виродження. Роз- винене наближення дозволяє сформулювати критерій справедливості рівняння Діраку відносно простим способом. Ключові слова: 2D електронний газ, рівняння Дірака, граничні умови, критерії обгрунтованості, функції Грина. 1604 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 https://doi.org/10.1103/PhysRev.71.622 https://doi.org/10.1103/PhysRevB.29.1685 https://doi.org/10.1103/PhysRevLett.97.216803 https://doi.org/10.1103/RevModPhys.81.109 https://doi.org/10.1088/0953-8984/16/13/016 https://doi.org/10.1103/PhysRevB.77.085423 https://doi.org/10.1103/RevModPhys.80.1337 https://doi.org/10.1103/PhysRev.126.1636 Scattering problems and boundary conditions for 2D electron gas and graphene Задача рассеяния и граничные условия для двумерного электронного газа и графена А.М. Кадигробов Использована техника функций Грина для получения граничных условий и для анализа координатной зависимости волновой функции отраженной квазичастицы в графене и двумерном электронном газе в случае, когда кристалличе- ская решетка имеет резкую границу. Показано, что отражен- ная волновая функция содержит слагаемые, величина кото- рых обратно пропорциональна расстоянию до границы решетки графена. На основе теории возмущений в вырож- денном случае выведено уравнение Дирака и получена им- пульсная зависимость волновой функции квазичастицы вблизи конической точки спектра в терминах невозмущен- ных функций Блоха в точках вырождения. Развитое прибли- жение позволяет сформулировать критерий справедливости уравнения Дирака относительно простым способом. Ключевые слова: 2D электронный газ, уравнение Дирака, граничные условия, критерии обоснованности, функции Грина. Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 12 1605 1. Introduction 2. Derivation of Dirac equation and Bloch functions for graphene by perturbation theory 3. Scattering of quasi-particles by an external potential and derivation of the Dirac equation 4. Scattering of electros in lattice with abrupt edge (Green function approach) 4.1. Scattering of electrons by abrupt edge in 2D electron gas 4.2. Scattering of quasi-particles by abrupt edge of graphene sheet 5. Conclusion Appendix A Appendix B. Coordinate dependence of the integral along the cut for graphene

Scattering problems and boundary conditions for 2D electron gas and graphene

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