A Solution of the Dirac equation

Aharonov-Bohm effect in relativistic and nonrelativistic 2D electron gas: a comparative study

Abstract

We carry out a comparative study of electronic properties of 2D electron gas (2DEG) in a magnetic field of an infinitesimally thin solenoid with relativistic dispersion as in graphene and quadratic dispersion as in semiconducting heterostructures. The problem of ambiguity of the zero mode solutions of the Dirac equation is treated by considering of a finite radius flux tube which allows to select unique solutions associated with each point of graphene’s Brillouin zone. Then this radius is allowed to go to zero. On the base of the obtained in this case analytical solutions in the Aharonov-Bohm potential the local and total density of states (DOS) are calculated. It is shown that in the case of graphene there is an excess of LDOS near the vortex, while in 2DEG the LDOS is depleted. This results in excess of the induced by the vortex DOS in graphene and in its depletion in 2DEG. We discuss the application of the results for the local density of states for the scanning tunneling spectroscopy done on graphene.

pacs:
03.65.-w, 73.20.At, 72.10.Fk

I Introduction

Physics of graphene perception begins when one compares Landau levels in two dimensional Schrödinger and Dirac theories. Such spectacular phenomenon as unconventional quantum Hall effect (1); (2) is caused by the anomaly of the lowest Landau level (LLL) (3) which for Dirac fermions in graphene is field independent and can accommodate only half the usual number of the states from the conduction band and takes the other half from the valence band. The easiest way to accomplish this peculiar feature of the LLL is to solve a pair of the Dirac equations that describe excitations near two inequivalent points of graphene’s Brillouin zone. Normally this is done in a constant homogeneous magnetic field, although this property of the LLL for Dirac fermions is topologically protected for inhomogeneous field configurations and in the presence of ripples (4).

The simplest inhomogeneous field configuration which contains nontrivial Aharonov-Bohm physics can be created by an infinitesimally thin solenoid. In practice such magnetic field configuration may be obtained when a type-II superconductor is placed on top of graphene or semicondunducting heterojunction hosting a 2D electron gas (2DEG) with quadratic dispersion. While graphene devices still have to be fabricated, devices like this with a superconducting film grown on top of a semiconducting heterojunction (such as GaAs/AlGaAs) hosting a 2DEG have in fact been fabricated twenty years ago (5); (6) and theoretically well studied (see e.g. Refs. (7); (8); (9); (10); (11)).

Theoretically a problem of the Dirac fermions in the field of Aharonov-Bohm flux was encountered in the context of cosmic strings by Gerbert and Jackiw (12). While for the solutions of the Dirac equation with nonzero angular momentum 1 the square integrability requirement specifies which of the two independent solutions should be taken, they noticed this is not the case for the zero angular momentum. For the zero momentum there is an ambiguity as both solutions are square integrable, but divergent as at the origin, where is the space coordinate. The ambiguity of the solution selection is caused by the singular nature of the infinitesimally thin solenoid vector potential at the origin. This problem has initiated a vast theoretical literature which addresses interesting aspects related to the rigorous treatment of the solutions of the Dirac equation with the Aharonov-Bohm potential (see e.g. Refs. (13); (14); (15) for a review). In the condensed matter context the Dirac fermions in the field of solenoid emerged during the study of the Dirac-Bogolyubov-de Gennes quasiparticles in the vortex state of -wave superconductors (16) (see also (17) for a review). Due to the divergence of the zero mode solutions theory predicts a formation of nonzero local density of states (LDOS) near the vortex center. However, this theoretical prediction based on the Dirac nature of quasiparticles in -wave superconductors does not agree with the results of scanning tunneling spectroscopy (STS) measurements (18) in high-temperature superconductors. Finally we mention a related problem of the description of topological defects in graphene based on the Dirac equation with a pointlike pseudomagnetic vortex which has also been studied intensively, see e.g. Refs. (19); (20).

The purpose of the present paper is to study the electronic excitations in graphene in the field of the Aharonov-Bohm flux and compare them with the corresponding results for 2DEG with a quadratic dispersion. We rely on the existing studies of the Dirac fermions in the Aharonov-Bohm potential, but focus on the specific features of graphene such as the presence of two inequivalent points which implies that one should consider the solutions for both inequivalent irreducible representations of the Dirac matrices. Also to avoid unnecessary formal complications we consider the physical regularization of the problem modeling a finite radius flux tube created by the Abrikosov vortex. We utilize the simplest case of magnetic field concentrated in a thin cylindrical shell of small but finite radius when (21); (22). In contrast to high-temperature superconductors Dirac description of the quasiparticles in graphene is proven valid under the different conditions. In particular, STS measurements of graphene flakes on graphite (23) exhibit the structural and electronic properties expected of pristine graphene such as the development of a single sequence of pronounced Landau level peaks corresponding to massless Dirac fermions in a homogeneous magnetic field. We propose to perform STS measurements for graphene penetrated by vortices from a type-II superconductor, because the Dirac theory predicts rather peculiar behavior of LDOS not expected for the 2DEG with a quadratic dispersion of carriers.

The paper is organized as follows. In Sec. II we introduce the model Hamiltonians and discuss the regularization of the Aharonov-Bohm potential used in this work. Sec. III is devoted to the nonrelativistic case, and the relativistic case is discussed in detail in Sec. IV. The structure of both sections is the same: we consider the solution of the corresponding Schrödinger or Dirac equation, construct the Green’s function (GF) with coinciding arguments, obtain the DOS and study the behavior of the LDOS. In Sec. V our final results are summarized.

Ii Models and main notations

We consider nonrelativistic and relativistic Hamiltonians. The 2D nonrelativistic (Schrödinger) Hamiltonian has the standard form

(1)

where , with the vector potential , Planck’s constant and the velocity of light describes a spinless particle with a mass and charge . The Dirac quasiparticle in graphene is described by the Hamiltonian

(2)

where the matrices and are defined in terms of the Pauli matrices as

(3)

Here labels two unitary inequivalent representations of gamma matrices, so that one considers a pair of Dirac equations corresponding to two inequivalent points of graphene’s Brillouin zone. In Eq. (2) is the Fermi velocity and is the Dirac mass (gap), which is introduced in the Hamiltonian for generality. Note that we consider the simplest case when the gap has the same sign for [see Ref. (24) for a discussion of more general cases]. While tight binding calculations show that the quasiparticle excitations in graphene have a linear dispersion at low energies (25) and are described by the massless Dirac equation with (26), recent STS measurements revealed a mass gap near the Dirac point in a single layer graphene sample suspended above a graphite substrate (23). Since this gap and its origin are intensively studied both theoretically and experimentally in the last few years, here we consider a generic case with a finite value of .

The vector potential of a vortex at the origin directed in the direction is

(4)

where is the flux of the vortex expressed via magnetic flux quantum of the electron with , 2 where the value corresponds to the flux created by the Abrikosov vortex. The magnetic field is then

(5)

The essential difference between the Schrödinger

(6)

and Dirac

(7)

equations in this case can be seen if one squares the latter:

(8)

where the commutator

(9)

which introduces pseudo-Zeeman term which is related to the sublattice rather than the spin degree of freedom. It should be mentioned that in the case of graphene the components of the spinor are associated with a sublattice rather than a spin degree of freedom. Since the Hamiltonian (2) originates from a nonrelativistic many-body theory, the Zeeman interaction term has to be explicitly added to this Hamiltonian. This resembles the case of the nonrelativistic Hamiltonian (1) which becomes Pauli one when the interaction between the magnetic moment of the spin and an external magnetic field is added. In the present paper we do not include the spin degree of freedom neither in (1) nor in (2).

Eqs. (8) and (9) identify the origin of complications (22); (27) in the problem with a singular vortex (4) when a singularity in occurs at a singular point of the differential equation (8). To avoid these complications one can consider a vortex with a finite radius flux tube (9); (21); (22), i.e. with the magnetic field and vector potential written in cylindric coordinates :

(10)

where is a profile function with a compact support satisfying the normalization and connected to the profile function by the relation .

The simplest choice of the field distribution which regularizes the problem with the solutions solely expressed in terms of Bessel functions is a magnetic field concentrated on the surface of the cylinder of radius , . Then, the corresponding profile function

(11)

In the limit we recover the Aharonov-Bohm potential (4) but avoiding formal complications. As shown in Ref. (22), there is no dependence on the detailed form of in the limit provided that . In the present paper we restrict ourselves by considering the profile function (11).

Iii Nonrelativistic case

In this section we recapitulate the results of Refs. (9); (10) for nonrelativistic case. They are important not only for comparison with relativistic case, but also because the relativistic result is constructed using the nonrelativistic one.

iii.1 Solutions of the Schrödinger equation in Aharonov-Bohm potential and general representation for LDOS

In the limit the admissible solution of the Schrödinger equation is always a regular solution which in polar coordinates takes the form

(12)

where is the Bessel function with the wave vector which is related to the quasiparticle energy via .

The eigenfunction expansion for the retarded Schrödinger GF reads

(13)

or after substituting the wave function (12) it becomes

(14)

where . Since an analytic continuation of the GF (14) on the imaginary axis in the complex momentum plane, is free of singularities, it is convenient to work with the corresponding GF

(15)

where

(16)

In Eq. (15) we already set two arguments coinciding, , because in the present work we consider the DOS only. As we will see below, the function is also used in the representation of the DOS for the Dirac fermions. After the calculation of the GF (16) is done, the LDOS per spin projection can be found by returning back to the real momentum axis

(17)

The GF was calculated in Ref. (28) using the contour integration technique. A weak point of this calculation was discussed in (9), where the same method was applied to obtain the -dependent contribution to the GF, with coinciding arguments when the approach of (28) is valid. Referring to the derivation of (28), here we simply start from the corresponding expression for obtained in Ref. (28)

(18)

Notice that Eq. (18) coincides with the corresponding expression from (9) up to a coefficient. Changing the variables to , one can obtain from (18) the final expression

(19)

which we will use in what follows. It turns out that the investigation of the full DOS is simpler than the analysis of the LDOS. Thus in the next Sec. III.2 we firstly consider the DOS and return to the LDOS (17) below in Sec. III.3.

iii.2 The density of states

The full DOS per spin projection is obtained from the LDOS (17) by integrating over the space coordinates

(20)

Since we have the integral representation (19) for , it is straightforward to calculate directly the perturbation of DOS, induced by the Aharonov-Bohm potential. Here is a free DOS of 2DEG per unit area and is the 2D volume (area) of the system. Firstly integrating over space, one obtains (9)

(21)

Then integrating over and we obtain that

(22)

Returning to the real axis we reproduce the usual Aharonov-Bohm depletion of the DOS with respect to the free DOS (10); (9):

(23)

Writing Eq. (23) we have also included a case of the opposite field direction.

iii.3 The local density of states

Now we come back to the LDOS (17). As in the case of the full DOS, it is convenient to consider the excess LDOS, . Then the value can be obtained by calculating the function given by Eq. (19) and substituting the result to Eq. (15). The analytic continuation described by Eq. (17) has to be done at the very last step of the calculation.

Thus our purpose is to derive a simple representation for the function . First, one can rewrite it in the form

(24)

where we used that . Differentiating (19) we get the integrand of the last expression

(25)

Using the integral representation for the modified Bessel function (29)

(26)

and the formula (2.16.13.2) from (30)

(27)

we come to the equation

(28)

Integrating the last expression with Mathematica, we get

(29)

where is the generalized hypergeometric function and is the logarithmic derivative of the gamma function . After the analytic continuation is made only two terms in the square brackets (with the hypergeometric function itself remaining real) and logarithmic term contribute in , so that

(30)

with

(31)

Note that the last term in Eq. (30) arises from the logarithmic term of (29). In the important limiting cases the last expression is greatly simplified. For example, in the limit one obtains

(32)

Also in the physically important case the LDOS is expressed in terms of the sine integral, as follows

(33)

Using the asymptotic of the integral sine (29) for from (33) we recover the previous expression (32) valid for and . In the opposite limit for , we see that .

In Fig. 1 we show the dependence (33) of the induced LDOS on the distance from the center of the vortex .

Figure 1: (Color online) The normalized LDOS function as a function of the dimensionless variable for .

We observe that in the case of nonrelativistic 2DEG the presence of the vortex induced the depletion of the LDOS for small . The function crosses zero near and for the function it reaches the maximal value . In Fig. 2 we model a situation when the STM tip is positioned at some distance from the center of vortex and a bias voltage is swept to explore the energy dependence of the LDOS. To take into account the presence of a finite carrier density in 2DEG, we introduce a finite Fermi energy , so that the LDOS at zero energy, , corresponds its value at the Fermi level, i.e. . To choose the appropriate units we set the distance scale to be the order of the lattice constant. Then the energy scale, is the order of the bandwidth. The dimensionless variable can now be rewritten as follows .

Figure 2: (Color online) The normalized LDOS function as a function of energy for three values of and .

The dependence of is shown in Fig. 2 for three values of : for – solid (blue) curve, for – long-dashed (red) curve and for – short-dashed (black) curve. The chemical potential is taken . We observe that only for the smallest ratio the values of the LDOS are significantly depleted below the free LDOS . As we saw, the depletion of the LDOS occurs for . Since the presence of the Fermi surface makes the value of large, so that the region of small is accessible only for . Indeed, we observe that only for the smallest ratio the values of the LDOS are depleted to a half of the value of free LDOS . Since the realistic values of the vortex core size are at least of the order of magnitude larger than , this implies that the region of a significant depletion of the LDOS is not accessible experimentally. Still, due to the slow decay of even for the amplitude of oscillations is of order of , so that this behavior can be probably observed experimentally.

Iv Relativistic case

In this section we examine the density of states for Dirac particles in the potential of the infinitesimally thin solenoid. To avoid formal complications we consider the physical regularization of the problem with the magnetic field concentrated in a thin cylindrical shell of small but finite radius and take the limit at the end of the calculation. All answers are presented in the form convenient for comparison with the case of the Schrödinger equation.

iv.1 Solutions of the Dirac equation in Aharonov-Bohm potential and general representation for LDOS

The Dirac equation (7) in the field of the regularized vortex (10), (11) (see also (72)) is solved in Appendix A. In our consideration we follow Refs. (22); (21). The profile (72) implies that for the particle obeys the Dirac equation (74) for a free particle, while for the particle moves in the field of Aharonov-Bohm vortex (75). Accordingly, for the squared Dirac equation (8) is equivalent to free Schrödinger equations for the components of the spinor

(34)

Notice that in the Appendix A the definition (69) for explicitly includes the factor . For the components satisfy the Schrödinger equations with Aharonov-Bohm potential and the commutator (9) is singular at :

(35)

The solution of the problem can found by matching the solutions obtained in the domains and [see e.g. (78)]. The radial components of the spinor have to be continuous:

(36)

and the singularity of the commutator (35) is taken into account by a condition on the derivatives

(37)

We stress that in contrast to the Dirac equation case, for the nonrelativistic case when the solution (12) is obtained using the same regularization procedure, both the wave function and its derivative should be continuous. This as we saw from Eqs. (8), (9) and (70) is related to the pseudo-Zeeman term.

After the limit is taken we obtain for the case the following solutions:

(38)

for positive value of the energy and ;

(39)

for negative value of energy and . Here the upper and lower sign of the second spinor’s component corresponds to and solutions, respectively. The solution turns out to be special, while the upper component is regular at , the lower component diverges as :

(40)

for and

(41)

for . One can check this property by using the matching conditions for component and then finding from Eq. (75b). On the other hand, one can discover this singularity in the limit by directly analyzing the matching conditions for and then finding nonsingular from Eq. (75a). Comparing Eqs. (40) and (41) we observe that a singular at zero-mode solution is hole-like, because a singular electron-like solution vanishes due to the factor. Concluding the discussion of the solutions for the case we stress that for the opposite field direction it is the upper component of the spinor which is singular in the solution equivalent to Eqs. (40) and (41). Moreover, the nonvanishing zero mode is now electron-like. As it was firstly noticed in Ref. (21), this behavior under the change in the field direction breaks the symmetry under (see also Ref. (31) for a discussion).

The set of solutions for the case is the following:

(42)

for and

(43)

for . The prescription for the upper and lower sign of the second spinor’s component is the same as for . The solutions in the case are the following:

(44)

for and

(45)

for . We observe that in this case it is the upper component of the spinor which is singular at and nonvanishing is the electron-like zero mode.

Now we are at the position to construct the GF for the Dirac fermions using the presented above solutions. However, before going to this, we should stress that as shown in(21); (22) the field configuration with confined to the surface of a cylinder of radius is not essential for the main result. In simple words, an more general and not singular at the origin potential can be considered as a set of concentric shells. Or putting more formally, a profile function should satisfy the condition given below Eq. (11) which excludes delta-funtion at the origin.

The eigenfunction expansion for the retarded Dirac GF’s now includes both positive and negative energy solutions

(46)

where depending on the sign of the sum is taken over either or solutions. Accordingly, for the diagonal matric elements of , for instance, in the case we obtain

(47)

and

(48)

where . Then the GF with coinciding arguments, acquires the following form

(49)

for , and

(50)

for . Here the function is related to the function defined in Eq. (16) by analytic continuation . Note that while the function is identical for both nonrelativistic and relativistic cases, the real momentum function has a different analytical properties in these cases reflecting the fact that in contrast to the nonrelativistic case the relativistic spectrum contains positive and negative energy branches. Similarly, the function

(51)

is the analytic continuation to the imaginary axis of the function . The contribution to originates from the zero mode solutions of the Dirac equation. Using the integral [see Eq. (2.12.32.12) from(30)]

(52)

where is the modified Bessel function and is the Macdonald function, we obtain the following simple result for

(53)

iv.2 The density of states

Due to the matrix structure of the GF and the presence of the valley degree of freedom, , in contrast to the nonrelativistic expression (20), the full DOS in the relativistic case involves not only the integration over area, but also summing over the diagonal components of the GF and valleys. For a better understanding of the final result for the full DOS, it is instructive to consider separate expressions for the DOS corresponding separate points

(54)

Recall that for the free DOS of the Dirac quasiparticles per spin and one valley is equal to .

Using Eqs. (49) and (50) one can express the DOS (54) via the functions and defined by Eqs. (16) and (51), respectively, as follows

(55)

The integral of used in the nonrelativistic case function , or to be precise of the function , over the space coordinates is calculated in Eq. (22). The corresponding space integration of the function given by Eq. (53) produces the result

(56)

Notice that since for , there is no need to introduce a function . Having these results for the spatial integration we can calculate the imaginary part. Since both Eqs. (22) and (56) depend on , the imaginary part is obtained by using a simple prescription . This gives the final expression for the perturbed DOS, by the Aharonov-Bohm vortex

(57)

The first term of (57) which is originates from part of Eq. (55) and resembles the nonrelativistic result (23). For it turns out to be twice larger than (23) simply because in Eq. (57) we summed over the diagonal components of the GF which are related to the two sublattices of graphene. The last term of (57) originates from part of Eq. (55) and thus is related to the zero mode solution of the Dirac equation. The mentioned above fact that the singular component of the zero mode solution is hole-like for [see Eq. (41)] and it is electron-like for [see Eq. (44)] finds its reflection in the asymmetric form of the last term of Eq. (57) which also corresponds to the holes (electrons) for (). In accordance with the behavior of the zero mode solutions described in Sec. IV.1, when the direction of the field is reversed the expressions and are interchanged. The full excess DOS

(58)

is obviously symmetric in energy. In contrast to the nonrelativistic case, in the Dirac case the Aharonov-Bohm vortex induces the excess of the states which is related to the presence of the last term of Eq. (57) and caused by the zero modes. We note that in Ref. (13) the corresponding term of Eq. (57) has a wrong sign. The positiveness of can also be understood by the following simple argument. For the free DOS . Therefore, since the DOS has to be positive, the value should also be positive.

iv.3 The local density of states

Now we investigate the LDOS for the Dirac case. While it was useful to consider each valley separately, especially because field theoretical studies of the problem often involve only one unitary inequivalent representation of gamma matrices [see e.g.(13)], the LDOS measurement picks up both valleys together. On the other hand, LDOS distinguishes sublattices. Thus we consider separately the LDOS for and sublattices which are defined as follows

(59)

Again we consider the perturbation of the LDOS induced by the Aharonov-Bohm potential, . Using Eqs. (49) and (50) we rewrite the LDOS’s in terms of the functions and as follows

(60)

Here the upper (lower) sign corresponds to () sublattice. To obtain the final expression for LDOS we should make the analytic continuation . For the function given by Eq. (29) one obtains

(61)

where the function is defined in Eq. (31). One can easily see that because in the nonrelativistic case the analytic continuation to the real momentum