The Induced Charge Generated By The Potential Well In Graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r The Induced Charge Generated By The Potential Well InGraphene
Alexander I. Milstein and Ivan S. Terekhov
Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
Abstract
The induced charge density, ρ ind ( r ), generated in graphene by the potential well of the finiteradius R is considered. The result for ρ ind ( r ) is derived for large distances r ≫ R . We alsoobtained the induced charges outside of the radius r ≫ R and inside of this radius for subcriticaland supercritical regimes. The consideration is based on the convenient representation of theinduced charge density via the Green’s function of electron in the field. PACS numbers: 81.05.Uw, 73.43.Cd . INTRODUCTION As known, the induced charge density, ρ ind ( r ), in the external electric field appears dueto vacuum polarization. In the field of heavy nucleus, this important effect of QuantumElectrodynamics (QED) was investigated in detail in many papers, see, e.g., Refs. . Newpossibilities to study vacuum polarization in QED at large coupling constant have appearedafter recent successful fabrication of a monolayer graphite (graphene), see Ref. and recentReview . The single electron dynamics in graphene is described by a massless two-componentDirac equation so that graphene represents a two-dimensional (2D) version of masslessQED. On the one hand, this version is essentially simpler than conventional QED becauseeffects of retardation are absent due to instant Coulomb interaction between electrons. Onthe other hand, the “fine structure constant” α = e / ~ v F is of order of unity since theFermi velocity v F ≈ m/s ≈ c/
300 ( c is the velocity of light), and therefore we have astrong-coupling version of QED. Below we set ~ = c = 1.Screening of charged impurity in graphene can also be treated in terms of vacuumpolarization . Investigation of impurity screening is important for understanding of thedependence of transport properties on the impurity concentration. There are two regimes forthe Coulomb impurity in the gapless graphene, subcritical and supercritical. In the subcrit-ical regime, it is shown in the leading order in α and exactly in the Coulomb potential thatthe induced charge is localized at the impurity position, see . In the supercriticalregime, vacuum polarization in the Coulomb field has been recently considered in Refs. .In this case, the induced charge density is not localized at the impurity position due to theeffect similar to that of e + e − pair production in 3D QED in the electric field of supercriticalheavy nuclei. In the present paper, we answer to the question whether the phenomenon ofthe induced charge localization also exist in the potential well of finite size R and depth U .Namely, we calculate the asymptotics of ρ ind ( r ) in the field of an azimuthally symmetricpotential well at large distances r ≫ R . We apply the method suggested in Ref. for calcu-lation of the finite nuclear size effect on the induced charge density at large distances in astrong Coulomb field in 3D QED. We show that there are also subcritical and supercriticalregimes in this problem. However, the induced charge is not localized at r . R in thesubcritical regime and has power ”tail” in contrast to the case of the Coulomb field. In thevicinity of transition from the subcritical regime to the supercritical one, small variation of2he potential parameters drastically changes the induced charge density. We demonstratethat this fact is not related to the smoothness of the potential well. The attempt to calculatethe induced charge distribution in the potential well in graphene was previously performedin Ref. . The authors of this paper used the method which akin to that used at calculationof conventional Friedel oscillations. However, our results for the induced charge densitydiffer from that obtained in Ref. mainly due to the mistake performed in Ref. at thecalculation of the phase shift.The paper is organized as follows. In Section II we derive the general expression forthe induced charge density convenient for calculation of the asymptotics at large distances.In Section III we consider the Green’s function of electron in an azimuthally symmetricpotential and use this function in calculations of ρ ind ( r ) in Section IV. Critical values of g are discussed in Section V calculating the scattering phase shifts of electron wave functionin the field of the potential well. The induced charges outside of the radius r ≫ R andinside of this radius for subcritical and supercritical regimes are considered in Section VI.Finally, in Section VII the main conclusions of the paper are presented. II. GENERAL DISCUSSION
In graphene, the induced charge density in the potential U ( r ) have the form ρ ind ( r ) = − ieN Z C dǫ π Tr { G ( r , r | ǫ ) } , (1)where N = 4 reflects the spin and valley degeneracies, and the Green’s function G ( r , r ′ | ǫ )satisfies the equation [ ǫ − U ( r ) − v F σ · p ] G ( r , r ′ | ǫ ) = δ ( r − r ′ ) I. (2)Here σ = ( σ , σ ), and σ i are the Pauli matrices; p = ( p x , p y ) is the momentum operator, r = ( x, y ), and I = diag { , } . The matrixes σ act on the pseudo-spin variables and the spindegrees of freedom are taken into account in a factor N . According to the Feynman rules,the contour of integration over ǫ goes below the real axis in the left half-plane and abovethe real axis in the right half-plane of the complex ǫ plane. Using the analytical propertiesof the Green’s function, we deform the contour of integration with respect to ǫ so that itcoincides finally with the imaginary axis. Then we follow Ref. and write the equation for3he Green’s function in the form G ( r , r ′ | iǫ ) = G (0) ( r , r ′ | iǫ ) + Z d r d r G (0) ( r , r | iǫ ) [ U ( r ) δ ( r − r )+ U ( r ) G ( r , r | iǫ ) U ( r )] G (0) ( r , r ′ | iǫ ) , (3)where G (0) ( r , r ′ | iǫ ) is the solution of Eq.(2) at zero external field.It is convenient to represent ρ ind ( r ) as a sum ρ ind ( r ) = ρ (1) ind ( r ) + ρ (2) ind ( r ) , (4)where ρ (1) ind ( r ) is the linear in U ( r ) contribution and ρ (2) ind ( r ) is the contribution of high orderin U ( r ) terms. It follows from Eqs.(1) and (3) that ρ (1) ind ( r ) = eN Z ∞−∞ dǫ π Z d r Tr n G (0) ( r , r | iǫ ) U ( r ) G (0) ( r , r | iǫ ) o , (5) ρ (2) ind ( r ) = eN Z ∞−∞ dǫ π Z d r d r Tr n G (0) ( r , r | iǫ ) U ( r ) G ( r , r | iǫ ) U ( r ) G (0) ( r , r | iǫ ) o . (6)Formulas (5) and (6) are very convenient for calculation of the induced charge density atlarge distances. III. GREEN’S FUNCTION IN AN AZIMUTHALLY SYMMETRIC POTENTIAL
The free Green’s function G (0) ( r , r ′ | iǫ ) is given by G (0) ( r , r ′ | iǫ ) = − iǫ π (cid:20) K ( | ǫ | ξ ) − sign( ǫ ) ( σ · ξ ) ξ K ( | ǫ | ξ ) (cid:21) , (7)where ξ = r − r ′ , and K , ( x ) are the modified Bessel functions of the third kind. Letus represent the electron Green’s function G ( r , r ′ | ǫ ) in an azimuthally symmetric potential U ( r ) in the form G ( r , r ′ | ǫ ) = 12 π ∞ X m = −∞ e im ( φ − φ ′ ) A m ( r, r ′ | ǫ ) − ie − iφ ′ B m ( r, r ′ | ǫ ) ie iφ C m ( r, r ′ | ǫ ) e i ( φ − φ ′ ) D m ( r, r ′ | ǫ ) , (8)and use the relation δ ( r − r ′ ) = δ ( r − r ′ )2 π √ rr ′ ∞ X m = −∞ e im ( φ − φ ′ ) . (9)4hen, from Eq.(2) we obtain the equations( ǫ − U ( r )) A m − ∂C m ∂r − m + 1 r C m = δ ( r − r ′ ) √ rr ′ , ( ǫ − U ( r )) C m + ∂A m ∂r − mr A m = 0 , (10)and the relations D m = A − m − and B m = − C − m − . Therefore, to find the Green’s functionin a azimuthally symmetric potential, it is sufficiently to solve equations (10). IV. AN INDUCED CHARGE DENSITY AT LARGE DISTANCES
To calculate the asymptotics of the function ρ (1) ind at distances r ≫ R , where R is a typicalsize of the potential, we can put r = 0 in the arguments of the free Green’s functions inEq.(5). After that we take the integral over ǫ and obtain: ρ (1) ind ( r ) = eN r Z dr ′ r ′ U ( r ′ ) . (11)One can see that the induced charge density in the leading order in the external field goesto zero at large distances as 1 /r .Let us consider the function ρ (2) ind ( r ) at r ≫ R . We substitute Eqs.(8) and (7) to Eq.(6),put r = 0 and r = 0 in the arguments of the free Green’s function, and take the integralover angels of the vectors r and r . Then we obtain ρ (2) ind ( r ) = − eN π ∞ Z −∞ dǫ ǫ (cid:2) K ( | ǫ | r ) − K ( | ǫ | r ) (cid:3) ∞ Z ∞ Z dr dr r r U ( r ) U ( r ) A ( r , r | iǫ ) . (12)Here A ( r , r | iǫ ) is the solution of Eq.(10) at m = 0. Note that Eq.(12) includes thecontributions of the terms with m = 0 and m = − D − = A . It is convenient to introduce the functions a ( r, ǫ ) = Z ∞ dr ′ r ′ U ( r ′ ) A ( r, r ′ | iǫ ) , c ( r, ǫ ) = Z ∞ dr ′ r ′ U ( r ′ ) C ( r, r ′ | iǫ ) , (13)Let us multiply both sides of the equations (10) by r ′ U ( r ′ ), and then take the integral over r ′ from zero to infinity. As a result we obtain the following equations for the functions a ( r, iǫ )and c ( r, iǫ ): ( iǫ − U ( r )) a ( r, ǫ ) − ∂c ( r, ǫ ) ∂r − c ( r, ǫ ) r = U ( r ) , ( iǫ − U ( r )) c ( r, ǫ ) + ∂a ( r, ǫ ) ∂r = 0 . (14)5he boundary conditions for these equations are a (0 , ǫ ) , c (0 , ǫ ) < ∞ , and lim r →∞ a ( r, ǫ ) =lim r →∞ c ( r, ǫ ) = 0. In terms of the function a ( r, iǫ ), Eq.(12) has the form ρ (2) ind ( r ) = − eN π Z ∞−∞ dǫ ǫ (cid:2) K ( | ǫ | r ) − K ( | ǫ | r ) (cid:3) Z ∞ dr ′ r ′ U ( r ′ ) a ( r ′ , ǫ ) . (15)Then we pass in this equation from the variable ǫ to the variable E = rǫ and replace a ( r ′ , E/r ) on a ( r ′ ,
0) at r ≫ R . We can do that because the integral over E converges at E ∼ K - functions. After this replacement we take the integralover E and arrive at the following expression for the asymptotics of ρ (2) ind ( r ): ρ (2) ind ( r ) = eN r Z ∞ dr ′ r ′ U ( r ′ ) a ( r ′ , . (16)Thus, the function ρ (2) ind ( r ) has the same behavior at large distances as ρ (1) ind ( r ).Let us consider a simple example of the potential, U ( r ) = − U θ ( R − r ), where θ ( x ) isthe step function, R is the radius of the potential well. The solution a ( r,
0) of Eq. (14) is a ( r,
0) = J ( U r ) J ( U R ) − , r < R , , r > R , (17)where J n ( x ) is the Bessel function. Using this solution, we find the sum of the contributionsEq.(11) and Eq.(16), ρ ind ( r ) = − eN J ( g ) R J ( g ) r , (18)where g = U R is the effective dimentionless coupling constant. The induced charge density(18) is the odd function of the parameter g , which corresponds to the Furry theorem inQED. The formula (18) contains singularities at the critical values of g = g c satisfying theequation J ( g c ) = 0. In our case, the first three values are g c ≈ . , . , .
65. Existence ofsuch singularities is not related to strong variation of our potential around the point r = R .We found numerically the first three critical values of g = U R for the smooth potentials U ( r ) = − U e − r/R and U ( r ) = − U e − r /R . In the first case, g c ≈ . , . , .
0. In thesecond case, g c ≈ . , . , .
0. We see that the corresponding numerical values of g c areclose to each other.Actually, singularities in Eq.(18) have appeared as a result of substitution a ( r ′ , E/r ) → a ( r ′ ,
0) in Eq.(16), which is not valid in the vicinity of g = g c since Eq.(17) has no sense at g = g c . In the vicinity of g = g c , it is necessary to perform calculation of the integrals in6q.(15) more accurately. For the step-like potential U ( r ) = − U θ ( R − r ), the solution ofEq. (14) at ǫ = 0 has the form a ( r, ǫ ) = γ J (( U + iǫ ) r ) − U / ( U + iǫ ) , r < R ,βK ( | ǫ | r ) , r > R , (19) c ( r, ǫ ) = γ J (( U + iǫ ) r ) , r < R , − iβ sign( ǫ ) K ( | ǫ | r ) , r > R . (20)Taking into account continuity of the functions a ( r, ǫ ) and c ( r, ǫ ) at r = R , we obtain γ = (cid:18) iǫU (cid:19) (cid:20) J (( U + iǫ ) R ) − i sign( ǫ ) J (( U + iǫ ) R ) K ( | ǫ | R ) K ( | ǫ | R ) (cid:21) . (21)Then we substitute Eqs.(19) and (21) to Eq.(15) and take the integral over r ′ . As above,the main contribution to the integral over ǫ at r ≫ R is given by the region ǫ . /r , so thatwe can use the relations ǫR ≪ ǫ/U ≪
1. Finally we find the expression for the sumof ρ (1) ind ( r ) and ρ (2) ind ( r ) at large distances, ρ ind ( r ) = eN J ( g ) J ( g ) Rπ Z ∞ dǫ ǫ K ( ǫr ) − K ( ǫr ) J ( g ) + J ( g )( ǫR ) ln ( ǫR ) . (22)This expression is valid at arbitrary value of the coupling constant g = U R . If | J ( g ) | ≫ ( R/r ) ln( r/R ), then it is possible to neglect the second term in the denominator of theintegrand, and we return to the expression (18). If g is close to some g c so that | J ( g ) | ≪ ( R/r ) ln( r/R ) ≪
1, we obtain ρ ( r ) = − eN sign( g − g c )2 π r ln | g − g c | . (23)In this case the induced charge density diminishes as 1 /r and has opposite sign for g < g c and g > g c . In terms of distances, the asymptotics (23) is valid at1 ≪ r/R ≪ − ln | g − g c | / | g − g c | . At r/R ≫ − ln | g − g c | / | g − g c | and | g − g c | ≪
1, we have (see Eq.(18)) ρ ( r ) = eN R r ( g − g c ) , (24)In order to illustrate the transition from the asymptotics (23) to the asymptotics (24), weconsider the ratio ρ ind ( r ) /ρ ( r ) at r ≫ R and | g − g c | ≪
1. In this case this ration dependsonly on the variable η = − r | g − g c | / ( R ln | g − g C | ). The dependence of ρ ind ( r ) /ρ ( r ) on η isshown in Fig.1. We see that ρ ind ( r ) ≈ ρ ( r ) already at η ≃ .5 1 1.5 2 2.5 3 3.5 40.20.40.60.81 η ρ i n d ( r ) / ρ ( r ) FIG. 1: Ratio ρ ind ( r ) /ρ ( r ) at r ≫ R and | g − g c | ≪ η = − r | g − g c | / ( R ln | g − g C | ).The asymptotics ρ ( r ) is given by Eq.(24) and ρ ind by Eq.(22). V. CRITICAL VALUES OF g AND SCATTERING PROBLEM
It is possible to explain critical values of g using the approach based on the scatteringproblem, as it is usually performed at the consideration of Friedel oscillations, see Ref. .Writing the wave function of electron as ψ ( r ) = u m ( r ) e imφ id m ( r ) e i ( m +1) φ , (25)we obtain equations for the functions u m ( r ) and d m ( r ), cf. Eq.(10),( ǫ − U ( r )) u m − ∂d m ∂r − m + 1 r d m = 0 , ( ǫ − U ( r )) d m + ∂u m ∂r − mr u m = 0 . (26)The solution of this equations in the the step-like potential has the form (common nor-malization factor is omitted): u m ( r ) = J m ( | U + ǫ | r ) , r < R ,µ m J m ( | ǫ | r ) + ν m N m ( | ǫ | r ) , r > R , (27) d m ( r ) = sign( U + ǫ ) J m +1 ( | U + ǫ | r ) , r < R , sign( ǫ ) [ µ m J m +1 ( | ǫ | r ) + ν m N m +1 ( | ǫ | r )] , r > R . (28)Here N m ( x ) are the Bessel functions of the second kind. From continuity of the functions8 m ( r ) and d m ( r ) at r = R , we have µ m = − π | ǫ | R h J m ( | U + ǫ | R ) N m +1 ( | ǫ | R ) − J m +1 ( | U + ǫ | R ) N m ( | ǫ | R )sign( ǫ )sign( ǫ + U ) i ,ν m = − π | ǫ | R h − J m ( | U + ǫ | R ) J m +1 ( | ǫ | R )+ J m +1 ( | U + ǫ | R ) J m ( | ǫ | R )sign( ǫ )sign( ǫ + U ) i . (29)Using the asymtotics of the Bessel functions at large value of argument, we find the phaseshift δ m ( ǫ ) = − arctan( ν m /µ m ). Critical values of g are given by the solution of the equation J m ( g c ) = 0 at m ≥ J | m |− ( g c ) = 0 at m <
0. Taking into account the asymptotics N ( x ) ≈ xπ , N | m | ( x ) ≈ − | m | ( | m | − x | m | π at x ≪
1, we find for | ǫ | R ≪ | ǫ | ≪ U , and g = U R close to g c δ ( ǫ ) = arctan (cid:20) π ǫRǫR ln( | ǫ | R ) − ( g − g c ) (cid:21) ,δ m ( ǫ ) = − arctan (cid:20) π ( ǫR ) m +1 m m !( m − m + 1) ǫR + 2 m ( g − g c )] (cid:21) at m > , (30)and δ −| m | ( ǫ ) = δ | m |− ( ǫ ). If ǫ <
0, which corresponds to electrons inside Fermi surface, and g < g c , then δ m is always small. For g > g c , the phase shift δ m ( ǫ ) can be equal to ± π/ ǫ <
0. That means the appearance at g = g c of the additional quasi-bound state onthe Fermi surface.Calculation of the phase shift in the step-like potential was previously performed inRef. . However, the coefficients corresponding to µ m and ν m , Eq.(29), were found in Ref. by matching the function u m ( r ) and its first derivative at r = R , instead of matching thefunctions u m ( r ) and d m ( r ). It is easy to check that the first derivative of u m ( r ) is not acontinuous function in the point r = R . As a consequence, the asymptotics of the inducedcharge density at large distances obtained in Ref. is not correct. VI. AN INDUCED CHARGE
Let us consider the induced charge Q > ( r ) outside of the radius r ≫ R , Q > ( r ) = 2 π Z ∞ r dr ′ r ′ ρ ing ( r ′ )= − eN J ( g ) J ( g ) Rr π Z ∞ dǫ ǫ K ( ǫr ) + K ( ǫr ) K ( ǫr ) − K ( ǫr ) J ( g ) + J ( g )( ǫR ) ln ( ǫR ) . (31)9or | J ( g ) | ≫ ( R/r ) ln( r/R ), we have Q > ( r ) = − eN πRJ ( g )8 J ( g ) r . (32)In the case | g − g c | ≪ ( R/r ) ln( r/R ), we find with logarithmic accuracy Q > ( r ) = eN sign( g − g c ) (cid:18) r/R )ln | g − g c | (cid:19) . (33)Since N = 4, then Q > ( r ) /e tends to the integer number N sign( g − g c ) at g → g c , havingopposite sign for g < g c and g > g c .Let us discuss the induced charge Q < ( r ) inside of the radius r ≫ R . Since the totalinduced charge Q tot = Q > ( r ) + Q < ( r ) is zero for the potential well at g less than theminimal g c , we have Q < ( r ) = − Q > ( r ) for such value of g . Note that Q tot is not zerofor massless electron in graphene in the Coulomb potential U C ( r ) = − Zα/r even in thesubcritical regime
Zα < /
2, see Ref. , due to zero mass of a particle and slow decreasingof a Coulomb potential at large distances. For g larger than the minimal g c , the total inducedcharge is already not equal to zero due to the effect similar to e + e − pair production in theelectric field of superheavy nucleus . In this case Q tot = eN M , where M is a number of g c less than g , so that Q < ( r ) = − Q > ( r ) + eN M . The quantity M is nothing but the number ofthe quasi-bound states at a given value of g , see discussion in SectionV. The explicit valuesof g c are given by zeros of the Bessel functions, as it is pointed out in SectionV.In Section IV and this section we have considered the contributions of the angular mo-menta m = 0 and m = − Q > ( r )at large distances. Of course, the contributions of m > m < − R/r even in the vicinity of the correspondingcritical points. However, M in Q tot = eN M includes numbers of g c coming from m > m < − g c , it is convenient to represent Q > ( r ) and Q < ( r ) at r ≫ R as follows: Q > ( r ) = eN [sign( g − g c ) + F ( g, r )] ,Q < ( r ) = eN [sign( g c − g ) − F ( g, r ) + M ] , (34)where F ( g, r ) is some continuous function of g . The dependence of this function on g at R/r = 0 .
1, obtained from Eq.(31) in the vicinity of minimal value of g c , is shown in Fig.2(solid line), as well as its asymtotics, obtained with the use of Eq.(32) (dashed line).10 g F ( g , r ) FIG. 2: Dependence of the function F ( g, r ), defined in Eq.(34), on g at R/r = 0 . g c . Exact result obtained from Eq.(31) is shown as a solid line, the asymtotics,obtained with the use of Eq.(32), as a dashed line. It is seen that the region, where Eq.(32) is not applicable, is very narrow.
VII. CONCLUSION