On the inhomogeneous magnetised electron gas - Thesis

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On the inhomogeneous magnetised electron gas

Kettenis, M.M.

Publication date

2001

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Final published version

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Kettenis, M. M. (2001). On the inhomogeneous magnetised electron gas. Ridderprint

offsetdrukkerij b.v.

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II MARK KCTTCNI51

Onn the Inhomogeneous Magnetised

Electronn Gas

Onn the Inhomogeneous Magnetised

Electronn Gas

ACADEMISCHH PROEFSCHRIFT terr verkrijging van de graad van doctor

aann de Universiteit van Amsterdam, opp gezag van de Rector Magnificus

prof.. dr. J. J. M. Franse

tenn overstaan van een door het college voor promoties ingesteldee commissie, in het openbaar te verdedigen

inn de Aula der Universiteit

opp donderdag 6 december 2001 te 12.00 uur

door r

Markk Martinus Kettenis

Promotor:: prof. dr. H.W. Capel Co-promotor:: dr. L.G. Suttorp

Faculteitt der Natuurwetenschappen, Wiskunde en Informatica

Fromm Alban Bergs Siebenfrühe Lieden lm Zimmer by Johannes Schlaf ©© 1928 by Universal Edition A.G., Wien

renewedd 1956 by Helene Berg

Thiss work is part of the research programme of the "Stichting voor Fundamenteel Onderzoekk der Materie (FOM)", which is financially supported by the "Nederlandse Organisatiee voor Wetenschappelijk Onderzoek (NWO)".

lmm Zimmer

Herbstsonnenschein. Herbstsonnenschein. DerDer liebe Abend blickt so still herein. EinEin Feuerlein rot

knistertknistert im Ofenloch und loht. So,So, mein Kopfauf deinen Knien, soso ist mirgut.

WennWenn mein Auge so in deinem ruht, wiewie leise die Minuten ziehn.

Johanness Schlaf

Thiss thesis is based on the following papers:

M.. M. Kettenis and L. G. Suttorp, Charge and current density profiles of a degenerate

magnetizedmagnetized freeelectron gas near a hard wall, J. Phys. A: Math. Gen. 31 (1998), 6 5 4 7

-65600 (Chapter 2).

M.. M. Kettenis and L. G. Suttorp, Path-decomposition expansion and edge effects in a

confinedconfined magnetized free-electron gas, J. Phys. A: Math. Gen. 32 (1999), 8209-8223

(Chapterr 3).

M.. M. Kettenis and L. G. Suttorp, Correlations in a confined magnetized free-electron

Contents s

11 Introduction 1 1.11 Diamagnetism 1 1.22 Basic techniques 5 1.33 Outline 13 22 Density profiles 15

2.11 Green Junctions; charge and current density 16

2.22 Explicit form of Green function; parabolic cylinder functions 18 2.33 Asymptotic expansions 22

2.44 Discussion 28

2.. A Appendix: Asymptotics of en ( K ) 30

2.BB Appendix: Asymptotics of the normalisation factor 30

33 More density profiles 35

3.11 Path-decomposition expansion 36 3.22 Magnetic field 42

3.33 Asymptotics (non-degenerate case) 44 3.44 Asymptotics (degenerate case) 50

3.AA Appendix: Higher-order terms in the PDXseries 53

44 Weak magnetic fields 57

4.11 Magnetic field perpendicular to the wall 57 4.22 Magnetic field parallel to the wall 66 4.AA Appendix: Properties of J(w, u) 71 55 Correlations 75

5.11 Correlation Junctions 77 5.22 Correlations in the bulk 78 5.33 Path-decomposition expansion 82 5.44 Parabolic cylinder Junctions 86

Vlll l _{Contents Contents}

5.55 Correlations for large separations |r — r ' | 92 5.66 Correlations near the wall 98

5.77 Discussion and conclusion 101

5.AA Appendix: Zeroes of the parabolic cylinder junctions 103 Epiloguee 105

Bibliographyy 107 Samenvattingg 111 Dankwoordd 115

Chapterr 1

Introduction n

Inn this thesis, we will investigate an inhomogeneous gas of charged particles in the pres-encee of a hard wall. From the point of view of physics one would like to study a "real" plasma,, taking into account the inter-particle (Coulomb) interactions. Unfortunately, thiss would be a very difficult task. Since the interaction-less case could serve as a valu-ablee reference system, and various aspects of it have not been studied before, a study of aa gas of charged non-interacting particles in a magnetic field close to a hard wall seems appropriate.. As we will see in this thesis, this study turns out to yield some surprising resultss that describe a richer structure than one would expect for such a rather simple model. .

1.11 Diamagnetism

Sincee we do not consider the inter-particle interactions in the gas, our study essentially becomess a study of the inhomogeneous magnetised free-electron gas. This subject is closelyy related to the study of diamagnetism in electron theory, which has an interesting history. .

Inn the early days of (classical) electron theory, various attempts at explaining the mag-neticc properties of materials were made. One of the questions raised was, whether free electronss in such materials could cause diamagnetism. J J . Thomson [49] reasoned that pathss of electrons in a magnetic field are curved and that this curvature of the path will producee a magnetic field in the direction opposite to the external field. A piece of metal containingg free electrons will, therefore, act as a diamagnetic body. This view seemed too be generally accepted at the time. Unfortunately, this view is not correct. It is not entirelyy clear who was the first to point this out. In his Master's thesis [9], Bohr gives 1 1

2 2 ChapterChapter 1. Introduction

aa qualitative argument, which he clarifies later in his Ph.D. thesis [10], In her Ph.D. thesis,, van Leeuwen refers to lectures given by Lorentz in 1910—1911 in which Lorentz gavee two proofs for the absence of diamagnetism in a free-electron gas. Both Bohr and vann Leeuwen argue that the presence of the magnetic field does not change the statisti-call distribution of the particles in the gas, and that therefore there will be no magnetic effectt whatsoever.

Inn modern-day language we can see this by calculating the free energy F of the system andd then use the thermodynamic relation

MM = - g (1.1) too calculate the magnetisation. In the canonical ensemble, the free energy is given by

d3p d3r e -p H ( p>r )) (1.2)

PP Ni

FF = — — In

wheree H ( p , r ) is the Hamiltonian. In the presence of a (constant) magnetic field the Hamiltoniann is given by

H ( p , r )) = - ^ - ( p - ^ A )2 (1.3)

ln\,ln\, c

wheree A is the vector potential related to the magnetic field B = V x A. If we substitute 7tt = p — | A as the integration variable in (1.2), we immediately see that F becomes independentt of A and therefore independent of the strength B of the magnetic field. Thereforee the magnetisation must be zero. An essential problem with this reasoning is thatt it does not take into account the influence of the boundary on the system.

Therefore,, it was considered to be essential to look at the issue from the viewpoint off electron orbits, and include the influence of the wall on those orbits. Bohr [9, 10] andd van Leeuwen [37, 38] did just that. Under the assumption that electrons are re-flectedflected like elastic spheres from the walls of their container, they were able to prove thatt the magnetic effects of electrons moving in so-called skipping orbits close to the wall,, exactly cancel the contribution of the bulk electrons moving in closed orbits in thee interior. Figure 1.1 (taken from Bohr's Ph.D. thesis) shows a few electron orbits withh a fixed radius, and therefore the same tangential velocity, that move in the plane depictedd in the figure. The line ab represents the boundary of the container, its inte-riorr being on the right-hand side of the line. Bohr reasons that in the bulk, far from thee wall, there cannot be an anisotropy in the motion of electrons. At every point in thee region where the motion of electrons is restricted by the magnetic field such that theyy do not encounter the wall, their velocities are distributed equally in all directions.

1.1.1.1. Diamagnetism 3 3

Figuree 1.1: Electron orbits in the neighbourhood of a wall.

Fromm the picture it becomes clear that the paths of electrons reflecting from the wall formm continuations of the orbits they would follow if the wall were not present. One immediatelyy concludes that there cannot be an anisotropy in the electron motion close too the wall either. Now, since at any moment the magnetic effect of the motions of thee electrons in curved orbits is the same as if they were moving in straight lines with thee same velocity, it becomes clear that there is no diamagnetic response: the magnetic effectt of an electron is exactly cancelled by an electron moving at the same speed in thee opposite direction. Bohr then goes on by considering the paths of individual elec-trons.. The motion of a single bulk-electron in a circular orbit does produce a magnetic fieldfield in the opposite direction of the external magnetic field. However, if one follows thee motion of an electron that comes in contact with the wall, one sees that such an electronn "creeps" along the wall and describes an orbit in the opposite direction of the bulk-electrons.. This produces a magnetic field in the same direction as the external field andd the two contributions cancel exactly. In her Ph.D. thesis van Leeuwen gives a proof forr this cancellation using geometrical arguments.

Soo much for the classical case. How does quantum mechanics affect this picture? The electronn orbits become quantised. Landau [36] has shown that this leads to a diamag-neticc response in the free-electron gas. Landau's derivation involves calculating the free energyy from the (quantum-statistical) partition sum and employing ( 1 . 1 ) - much in the samee way as the reasoning given above for the classical case. Landau calculates the par-titionn sum by looking at the energy spectrum. If one only considers the motion of the

4 4

ChapterChapter 1. Introduction

electronss in a plane perpendicular to the magnetic field, one finds that the eigenvalues aree given by

Enn = (n + J)5^51, 11 = 0,1,2,... (1.4)

mc c

Inn an infinite plane there are infinitely many states belonging to each of these energy eigenvalues,, but per unit area the degeneracy is

(1.5) )

27irtc c

Nowadayss we refer to these states as Landau levels.

Thee motion in the direction parallel to the magnetic field adds a simple kinetic term to En.. Landau then directly calculates the free energy by means of Poissons summation

formula,, which is a good approximation when U.B |Bj <C (3_1 (where JIB = ^mc is the Bohrr magneton). But he does not provide an explicit expression for the partition sum itself.. For Maxwell-Boltzmann statistics the partition function per unit volume would read d

|e|BB I m 1_ 47iricc V 2n$h2 sinh(uB 0B]

Whenn |1B(3B is small, which is the limit considered by Landau, one can expand the sinhh in the denominator to arrive at the often quoted result of

zz = -EZ:J.22^,:. »,- (1-6)

- ^ uB2( 3 BB (1.7)

forr the magnetisation per electron.

Thee partition function (1.6) is calculated without taking into account a particular boundaryy condition. This suggests that the magnetisation is independent of, or at least insensitivee to, the boundary conditions. At the time when Landau published his results, manyy people doubted this insensitivity of the magnetisation to the exact nature of the boundaryy condition. Especially since in the classical case it seemed that the boundary wass responsible for the total absence of a magnetic response. Therefore it took quite a bitt of time for Landau's result to become generally accepted. The insensitivity of Lan-dau'ss result to the nature of the boundary has been (finally) proven by Angelescu et al [5]] for a broad class of boundary conditions, at least up to first order in the magnetic field. field.

Ass a side-mark it is worth noting that the concept of spin introduces another form of magnetismm in the free-electron gas: spin-paramagnetism. In fact the magnetisation due

1.2.1.2. Basic techniques 5 5

too Landau-diamagnetism in the free-electron gas given in (1.7) is exacdy one-third of thee magnetisation due to spin-paramagnerism, so the free-electron gas ultimately shows paramagneticc behaviour. In a real metal however the ratio of dia- and paramagnetism willl be different, because of the interactions of the electrons with the periodic potential off the lattice. This could lead to a net diamagnetism.

Evenn though the magnetisation is largely independent of the boundary conditions, it iss still interesting to investigate how the electron gas behaves in the neighbourhood of thee wall - behaviour that does depend strongly on the particular boundary conditions. Therefore,, a major part of this thesis is devoted to an investigation of the behaviour off the (charge) density and current density profiles close to the wall. In this investiga-tionn we will also incorporate the effects of quantum statistics. Landau's original deriva-tionn of the diamagnetic response does take into account the Fermi-Dirac statistics of thee electrons, but involves an approximation that breaks down for low temperatures and/orr strong magnetic fields. In fact, the final result would have been identical if he hadd started out with Maxwell-Boltzmann statistics. As we will see in this thesis, the behaviourr of a completely degenerate electron gas is considerably different from the Maxwell-Boltzmannn case.

1.22 Basic techniques

Throughoutt this thesis we will use some basic techniques to calculate density profiles andd correlation functions. Before we give an outline of the rest of this thesis, we will providee a short explanation of these techniques.

1.2.11 Green functions

Thee traditional approach to quantum mechanics has been to solve the Schrödinger equationn directly. And indeed in chapter 2 we will take this approach. Since the Schrö-dingerr equation is a partial differential equation a whole set of tools exist for solving it. Quitee a few of the techniques for constructing solutions for partial differential equa-tionss make use of Green functions.

Supposee that H is a differential operator, which in the context of the Schrödinger equa-tionn would be the Hamiltonian. Then the traditional definition of the Green function forr the eigenvalue equation Hi|>(r) = E\J>(r) is

6 6 ChapterChapter 1. Introduction

withh appropriate boundary conditions. More specifically, if we want to describe a parti-clee that is restricted to a certain domain D, with boundary 3D, the appropriate bound-aryy condition for \J)(r) is

i M r ) = 0 ,, r e 3D (1.9) whichh translates into

Gu( r , r ' )) = 0 r e 3D and/or r ' e 3D (1.10)

forr the boundary condition on the Green function. In terms of the eigenfunctions i^nn (T) we can write

G u ( r , r ' )) = Y. — i H > u ( r N > ; ( r ' ) ( l . H )

nn u tn

forr complex u.

Inn this thesis we will encounter a few other functions that are derived from this basic Greenn function. The first is the discontinuity of Gu( r , r ' ) at u = E

GE( r , r ' )) = ^ [ Gu = E + io { r , r ' ) - Gu = E- i o ( r , r ' ) 3 = X >n( r N > ; ( r ' ) 6(En - E).

(1.12) ) Heree iO is an infinitesimal imaginary number. From this energy Green function it is but aa small step towards the temperature or thermodynamic Green function

(

'OO O

d E e - PEG E ( r , r ' )) = 5 " e ^E" W r ) C ( r ' ) . (1.13)

00

n

Thiss G p ( r , r ' ) can be used to calculate physical quantities like charge and current densityy and correlation functions.

Noww suppose that G £ ( r , r ' ) is the Green function for an infinite domain. In order too construct the Green function for the boundary problem we must find a correction G u (r>r' )) t n a t satisfies

( H - u ) G < ( r , r ' ) = 00 (1.14) forr all r, r ' 6 D / 3 D (that is, inside D), with the boundary condition

G u ( r , r ' ) = - G j ( r , r/)) r e 3D and/or r ' e 3D. (1.15) Baliann and Bloch [8] give a way to construct such a correction for the wave equation

1.2.1.2. Basic techniques 7 7

formm of this multiple-reflection expansion depends on the nature of the boundary con-ditions.. John and Suttorp [31] looked a bit closer at this multiple-reflection expansion forr the Schrödinger equation and the Dirichlet boundary conditions (1.9) considered here.. Their version of the multiple-reflection expansion is given by

G£(r,T')) = - f d o - "wn " [ Vr" G £ ( r , r " ) ]r^r» w G i ( r "w, r ' ) JdD D

++ [ d < r "w[ d<j",wn"-[Vr»G°u(ry')]r^T„w JdDD J3D

x n ' " . [ V1- G ; ( r "w, r lr,T i r, w G i ( r ' "v v(r ' )) + . . . (1.16)

wheree n " denotes the normal vector, directed outwards perpendicular to the surface elementt dcr"w, at the point r " on the boundary. The symbol W is used to stress the factt that a symbol stands for a coordinate at the boundary, and r'" XT T ' "W stands for thee average (half the sum) of the limit r "' —» r "/ W from the inside and from the outside off the domain. John and Suttorp then use this multiple-reflection expansion to derive severall physical quantities for a weakly magnetised electron gas in the neighbourhood off a hard wall for Maxwell-Boltzmann statistics.

1.2.22 Path integrals

Thee Feynman path integral provides an alternative to direcdy solving the Schrödinger equation.. In a way it is closer to physical intuition. Indeed, we will see that the fact that itt can be interpreted as a sum over trajectories, makes path integrals a suitable tool for investigatingg inhomogeneous quantum systems. And if one would wish to somehow extendd the analysis given here to a full quantum plasma, by taking into account the Coulombb forces between charged particles, the use of path integrals seems to be almost unavoidable.. At least they have been widely used in recent calculations on homoge-neouss quantum plasmas [13, 14].

TheThe Feynman-Kac formula

Considerr a particle in an external potential V(r), i.e. with the Hamiltonian

HH = ^ + V(r) (1.17) wheree we have chosen units in such a way that the particle mass m drops out. The

8 8 ChapterChapter 1. Introduction

governedd by the temperature Green function G${r\ r ) , with 3 the inverse temperature. Itss path-integral representation is given by the Feynman-Kac formula

Gp( r ' , r )) = <r'|e-p H|r> =

.r',0 0

dnroP(cu)exp p

-I I

£ £

dxV(cu(T)) ) (1.18) )

wheree CU(T) describes the path and d u j f is the conditional Wiener measure. For sim-plicityy Tl has been set to 1.

Lett us take a closer look at this Wiener measure. The relevant underlying space is the sett CI of all paths cu with cu(0) = r and cu((3) = r ' . The most important property of thee Wiener measure is that for arbitrary (small) |r — r ' | and (3

»i(Q) )

L L

du{o>)) = (27t0)-3 / 2e Ï F (1.19) ) Too define the Wiener measure one has to assign a measure to certain subsets of CI, thee so-called cylinder sets. To construct these cylinder sets, we choose a sequence Ti,, T 2 , . . . , xn and for each Ti a "window" At (i.e. some Borel set) through which the

pathss in the cylinder set should pass. Denoting the cylinder set with C1A, the measure

iss then given by

ee 2 ( Ti + 1 -Ti >

H(QA)) = f dki(cü) = f d n f d rn T7[27t(Tl+1 - Ti)]"

J nAA J A , JAn f=Q

(1.20) ) wheree ro = r, rn +i = r ' , To = 0 and Tn +i = 3- By varying n, and the sequences

Ttt and Ai, we obtain sufficiently many cylinder sets to define the conditional Wiener measure.. Since both the underlying space CI and the measure \i depend on the points rr and r ' and the "time" interval [0, |3], we usually write |i.^ jf instead of u.. The Wiener measuree allows us to construct a (Lebesgue) integral

- ƒ ƒ

1(f)) = du(a>)f(o>) (1.21) )

off a function f (a>) that depends on the path tu. Such integrals are usually referred to as

pathpath integrals.

Thee Feynman-Kac formula is often interpreted as a sum over all possible paths from r too r' where each path gets an appropriate weight that consists of a factor that depends onlyy on its shape (the Wiener measure) and a factor that depends on the potential V(r).

1.2.1.2. Basic techniques 9 9

Notee that while the paths U>(T) should be continuous, there is no need for them to be differentiable.. In feet the class of differentiable paths has measure zero.

Evenn though a number of techniques to actually calculate path integrals exist, only aa limited set of path integrals can be solved exacdy. In most cases, we will have to be satisfiedd with an approximation of some kind. Fortunately, several useful approximation techniquess for path integrals do exist.

AA standard way of actually calculating path integrals is by using a process known as "timee slicing" (in fact this is often used as a constructional definition of a path integral, andd reminiscent of the definition of the Wiener measure). In using this method, one startss by dividing the "time" interval [0, |3] into n + 1 subintervals [Tm,Tm +i] of equal

lengthh en, that is

rmm = m en m = 0, . . . , r i en =

P P

nn + 1 (1.22) ) Uponn doing this, the exponential that incorporates the effect of the (external) potential

exp p Jo o dTV(cu(T)) ) becomes s exp p -- Y_ €nV(cu(Tm) .. m=0 (1.23) ) (1.24) )

Fromm (1.19) we see that the Wiener measure is essentially Gaussian and on the interval [Tm,, Tm +i ] it gives a factor

(27t€, , \ " 3 / 2 , , i i (1.25) ) Noww if we define cu(Tm) = rm, we can write the "time sliced" path integral as

Ge > n( r > )) = J d3rn. - J d3T ! ( 2 7 t en) -3^+ 1) / Vi I^ rLe -£"v^ ) e

lr2_-rirr e v, _ , rT- r | '

— 7 7 ^ —P- enV ( T i ) __ jT— 'n'n e

Thee full path integral can be found by taking the continuum limit: G p ( r ' , r ) == lim G ^ r » .

(1.26) )

10 0 ChapterChapter 1. Introduction

Too actually calculate the path integrals using this technique, we will have to be able to doo the integrations over rm in (1.26). When these integrations are Gaussian, that is if

thee potential is quadratic, this is straightforward. But for more complicated potentials itt is hardly ever possible to find an exact result.

Inn many cases (including the problems explored in this thesis) the region in space that iss accessible to a particle is restricted. This means that one should only sum over those pathss that stay inside that region. This can be accomplished by setting the potential to infinityy outside this region, but in practice one will most often restrict the integrations overr rm in (1.26) to the region accessible to the particle. In most cases one is forced

too make (further) approximations to be able to do this, even if the unrestricted path integrall is solvable exacdy since the integration over the components of rm no longer

extendss from minus infinity to plus infinity.

TheThe Feynman-Kac-Ito formula

Thuss far we have only considered particles moving in ordinary potentials. But in order too describe the magnetised electron gas using path integrals, we also need to incorporate thee effects of an external magnetic field. In terms of the vector potential (which we assumee to be time-independent), the path integral representation of G p ( r ' , r ) is given byy the Feynman-Kac-Itó formula [45]:

G p ( r ' , r }} = J d ^ ^ ( u ) ) e x p - J C!TV(CÜ(T)) + i d r "" A ( r " ) (1.28) )

wheree in addition to h and m, the electric charge e and the velocity of light c have beenn set to 1 as well. It is not a priori clear what is meant by the integral J ^ d r A ( r ) . Inn principle, this is the integral of the vector potential along the path CU(T). But since thee majority of the paths consists of paths that are not differentiable, this is not simply aa line integral. In the literature [47, 45] one can find basically two approaches to the definitionn of "stochastic integrals" such as J ^ d r - A ( r ) . These approaches are known by thee names of their inventors: Stratonovitch and Ito. Both approaches define the integral ass the continuum limit of a sum over finite intervals

TV V

d r - A ( r ) == lim Y Am Am (1.29) n—K»» *—

m = 0 0

wheree Am = tu(Tm +i) — cu(xm). But they differ in the definition of Am.

Itoo uses Am = A(cü(Tm)). This is the convention used by mathematicians, and

1.2.1.2. Basic techniques 11 1

off the vector potential "in the future". Unfortunately it has one "unphysicaT aspect: it doess not transform in a correct way under a gauge transformation.

Considerr the gauge transformation A —> A + Vf. Under such a gauge transformation thee Green function Gp{r',r) should transform as G p ( r ' , r ) —> e, [ f { r ) _ f { r ) 1G p ( r '(r ) .

Inn other words A[iv) = ƒ ^ d r - A f r ) should transform as A{iv) —»A{tv)+1[r') — f{r). Itóss Lemma [47] states that for a function f (r) and a path cu with cu(0) = r, tu(3) = r ' wee have

[[ d r " (Vf )(r") = f (r') - f (r) - \ \ dT V Vf (cu(t)). (1.30)

Ja>> 2 Jo

Soo in order to restore gauge-independence when interpreting A{iv) as an Ito integral, onee needs to add the term ^ Jo dT V A(CU(T)).

Thee Stratonovitch approach uses Am = j[A(cü(Tm +i)) + A(cu(Tm))] and has

gauge-independencee built-in, since the difference between Itö and Stratonovitch is exacdy thee divergence term \ ƒ£ dT V A(CU(T)) mentioned above. For this reason it is more popularr among physicists. Fortunately, the difference between Ito and Stratonovitch disappearss when V A = 0. Therefore we are free to choose how to interpret the stochasticc integral in (1.28) as long as the vector potential is divergence-less. This may bee convenient since the mathematical framework for Itö integrals (Itó calculus) is much moree developed.

Notee that (1.29) provides the recipe for "slicing" the vector potential term in (1.28) that resembless the "time-slicing" described in (1.22). It is a convenient method to use when actuallyy calculating path integrals that involve the interaction with an electromagnetic field. field.

1.2.33 Degeneracy

Inn the previous sections we have treated some techniques that make it possible to cal-culatee properties of a free-electron gas for Maxwell-Boltzmann statistics. But by doing soo we would ignore the fermionic (or bosonic) character of the particles, and its con-sequencess for the statistics. This is fine for high temperatures and/or weak magnetic fields,fields, where the average occupancy of the states is small, but it does not provide a cor-rectt description for a cold and dense electron gas in a strong magnetic field. Some ad-hocc attempts at treating the (completely) degenerate electron gas have been made [43], butt a more elegant method exists: it turns out to be possible to derive the properties off a degenerate gas of non-interacting particles by starting from results for Maxwell-Boltzmannn statistics. In particular for T = 0 — the completely degenerate case — the two

12 2 ChapterChapter 1, Introduction

statisticss are related by a Laplace transform, as was shown by Sondheimer and Wilson [48].. In their paper they make a more thorough re-examination of the free-electron gas inn a magnetic field (with Fermi-Dirac statistics), in order to put the older results on a firmerr theoretical basis. The essence of their approach is explained in this section. Considerr a single-particle operator ai(r), where i is the label of a particle. We are interestedd in the expectation value for the operator A(r) = 2^i=i ai (r) in a g35 °f* N noninteractingg particles. Since all particles are identical we can drop the particle label i and,, if we ignore the quantum statistics, we can write the expectation value of A(r) at inversee temperature (3 in the canonical ensemble as

<A(r))pp = ^ T r [ e -p Ha ( r ) ] . (1.31)

Heree V is the volume occupied by the gas and Z is the partition function per unit volume.. In other words

(A(r))pp = Np E nX e - PE" ( n | a ( r ) ) n ) (1.32)

2 _ nee n n

wheree En is the energy level corresponding to the single-particle state n .

Iff we would take into account the quantum statistics for spin- j fermions we would get

( A ( T ) ) » , , = 2 £e > [ EJ| 00 + 1( n | a ( r ) | n ) (1.33)

insteadd (in the grand canonical ensemble). Here we assumed that the hamiltonian is independentt of the spin, hence the overall factor 2. The chemical potential (x is deter-minedd by 2 ^ 1n[ e| 3 ( E n - > i' + 1]_T = N (or rather the other way around: the chemical

potentiall u determines the number of particles). In particular at T = 0, i.e. |3 = oo the factorr [ e ^E n _^ + I ]- 1 becomes a step-function which yields

{ A ( r ) ^ = 2 ^ 0 ( u . - En) < n | a ( r ) | n )) (1.34) n n

wheree n o w 2 ^n0 ( | x — En) = N.

Sincee the Laplace transform of the step-function is given by

i i

d u e - ^ e ( ^ -- En) = d n e - < ^ = - e ~p E" (1.35) 'o o

1.3.1.3. Outline 13 3

wee see that

1

000 2 7

d n e - ^ < A ( r ) ) ^^ = — <A(r))p (1.36)

wheree p = N / V is the particle density. In other words, (A(r))^ is the inverse Laplace transformm of |§{A(r)) p with respect to 0

ii fc+oo 2 7

<

A(r)

>"-5dL

d|le

"

l,

'

l

^<

Alr

»»--

( u 7 )

AA similar, though somewhat more complicated relation holds for T ^ 0:

(AfrJV.uu =

-r

a , i

'^(i^)Tï)

< A ( r ) >

^--

(1

'

38)

Inn this thesis however, we will restrict ourselves to the case T = 0.

1.33 Outline

Noww that we have explained these basic techniques, we can give an outline of the rest off this thesis. In chapters 2 and 3 we will calculate charge and current density profiles forr the completely degenerate magnetised free-electron gas near a wall. The expressions forr these density profiles are asymptotic expansions in terms of the distance from the walll and give information on how the profiles decay towards their bulk values.

Chapterr 2 uses the Green function approach outlined in section 1.2.1. In chapter 3 wee will use the path integral techniques from section 1.2.2. The final result, the afore-mentionedd asymptotic expressions for charge and current density, will be very similar, butt nevertheless, both methods are of interest. The Green function approach makes it straightforwardd to calculate more terms in the asymptotic expansions for the densities. Itt also yields an interesting integral relation for parabolic cylinder functions. The path integrall approach from chapter 3 on the other hand, makes it possible to give an intu-itivee derivation of the multiple-reflection expansion. We will see that the higher-order termss in this expansion correspond to paths reflecting from the wall a number of times. Usingg the multiple-reflection expansion, we will then derive the density profiles for Maxwell-Boltzmannn statistics, and via the inverse Laplace transform technique from sectionn 1.2.3, the profiles for the completely degenerate case.

Thee density profiles derived in chapter 2 and 3 have the form of a sum over Landau levels.. For weak magnetic fields the number of Landau levels will become very large,

14 4 ChapterChapter 1. Introduction

andd the region where the asymptotic expressions for the profiles are valid, shirts further andd further away from the wall. In chapter 4 we will therefore investigate the weak field limit,, by looking first at the much simpler case where the magnetic field is perpendicular too the wall. After a comparison of our results with numerical data we will treat the originall geometry, where the magnetic field is parallel to the wall, in a similar way. Finally,, in the last chapter, we will derive correlation functions for the inhomogeneous electronn gas. Before we calculate those correlation functions in the magnetised case we willl derive the correlation functions for the field-free case in order to be able to com-paree the magnetised case with the field-free case. When the external magnetic field is present,, the full two-point correlation function, i.e. the correlation function where the componentss of both coordinates are all different, is difficult to evaluate. However, the correlationn function for points at the same distance from the wall, with the difference vectorr either parallel or perpendicular to the magnetic field, is much more manageable. Wee will derive asymptotic expansions for these correlation functions both for small and largee distances from the wall.

Chapterr 2

Densityy profiles

Everr since Landaus original derivation [36] of diamagnetism in a magnetised free-electronn gas, there has been interest in boundary effects. This is not surprising, since thee diamagnetism of a finite sample is caused by currents flowing near the boundary. Too get a deeper insight in the diamagnetic effect one needs to investigate the behaviour off these currents in the neighbourhood of a wall parallel to the external magnetic field. Off particular interest is the question how the current density decays in the bulk. Forr high temperatures it is adequate to use Maxwell-Boltzmann statistics. In that ap-proximationn the precise form of the current profile in the neighbourhood of a hard wall hass been studied by Ohtaka & Moriya [42] and by Jancovici [27] within the framework off linear response, and, more recently, by John & Suttorp [31] with the use of a Green functionn method. In both approaches a Gaussian decay of the current density in the bulkk has been found: asymptotically the decay is proportional to exp{— x2), with x the

distancee from the wall in suitable units. A similar decay has been found [31, 30] for the excesss charge density and the excess (kinetic) pressure.

Forr lower temperatures the effects of quantum statistics have to be taken into ac-count.. In that regime Macris, Martin and Pulé [40] have derived an exponential bound (~~ exp(—x)) on the decay of the current density in the bulk, at least for non-zero temperature.. For the strongly-degenerate case of vanishing T, Ohtaka & Moriya [42] andd Jancovici [27] have obtained a closed expression for the current density, via an inversee Laplace transform of the expression for Maxwell-Boltzmann statistics. Remark-ablyy enough, their results exhibit a much slower algebraic decay proportional to x_ 1. Usingg the same method one easily derives similar expressions for the excess charge den-sityy and the excess pressure at T = 0. However, the expression for the excess pressure obtainedd along these lines shows the unphysical feature of an oscillatory behaviour that iss no longer damped in the bulk.

16 6 ChapterChapter 2. Density profiles

Thee various findings for the asymptotic behaviour of physical quantities near the bulk, ass described above, justify a closer look at the problem. In this chapter we will derive systematicc asymptotic expansions for the charge and the current density near the bulk, byy starting from exact integral expressions valid at T = 0, which will be established onn the basis of a Green function formulation. The validity of these asymptotic expan-sionss will be assessed by a comparison with the results of a numerical evaluation of the integrall expressions.

2.11 Green functions; charge and current density

Considerr the half-space x > 0, with a hard wall at x = 0. Choose the magnetic field inn the z-direction, with vector potential A = (0, Bx,0). The transverse part of the Hamiltoniann for a particle with charge e and mass m in this field is given by

Hj__ = —z— Aj_ + ïï\xvcx— + - m t uc2x2 (2.1)

2mm oi| 2

wheree wc — eB/mc is the cyclotron frequency associated with the particle.

Inn order to simplify the notation we will choose units such that e = 1, m = 1, c = 1 andd h = 1 (which implies cuc = B), such that the Hamiltonian becomes

H_LL = + i B x ^ - + 1 B2_X2_{. (2.2)}

22 9-y 2

Thee Green function for the eigenvalue equation Hiii)n(r) = Enipn(r) (r = (x,y)) is

definedd by

(Hj.. - u ) G ( r , r ' ) = - 6 ( r - r ' ) (2.3) withh u a complex energy variable and with boundary condition Gj_>u(r, r') = 0 for

xx = 0 and/or x' = 0. This means we can express the Green function as

| U(r,T')) = £ T|>n(r)a|>;(r')—L-. (2.4) n n

Thee discontinuity of G j .u at u — E

E( r , r ' )) = ^ [ G x , u = E+i o ( rtr ' ) - G ,u = E_i 0( r , r ' ) ] (2.5)

2.. /. Green functions; charge and current density 17 7

Duee to the translation invariance in the y-direction of both the Hamiltonian and the boundaryy condition, a Fourier transform is appropriate. If we define the transform by G-L,u(i*>r')) = (27t)_1 J Ü ^ d k exp[ik(y—y')]Gj.,u(k,x,x')) the Hamiltonian becomes

)) = -j~ + \

k2

~

Bkx

+ l*

2

*

2

<

2

'

6 )

andd the equivalent of (2.3) is

[H_L(k}-u]G_L > u(k)x,x')) = - 6 ( x - x ' ) . (2.7)

Thiss means that we can write the Fourier transform of the energy Green function as

Gx,E(k,x,x,)) = X ^ ( k)x ) 4 ) ; ( k , x/) 6 ( En( k ) - E ) (2.8) n n

wheree the \l)n(k,x) are eigenfunctions of Hx(k), with eigenvalues En(k), normalised

suchh that J*U° dx !ii>n (k, x) | = 1.

Forr high temperatures, we can use Maxwell-Boltzmann statistics to calculate the prop-ertiess of the electron gas. The charge density for a gas of charged particles without mutuall interaction at inverse temperature |3 is given by

^M^M = j-^n(r)\2e-^ (2.9)

n n

wheree p is the bulk density and is the part of the one-particle bulk partition function (perr unit volume) corresponding to the degrees of freedom in the directions perpendic-ularr to the magnetic field. With the help of the Fourier transform of the energy Green functionn (2.8) we can write this as

pOOO poo

P p MM = ^ H d E e "p E d k G ( k , x , x ) . (2.10)

Jo J-oo

Att lower temperatures we will have to take into account the effects of quantum statis-tics.. For fermions at T = 0 we can use the inverse Laplace transform technique from thee introduction:

11 fc+ioo 7 7 i i*c+ioo -yj

27tiJc_iooo pP 2mJc_i o o P ^

-(2.11) ) wheree the constant c can be chosen arbitrarily as long as it is positive. Note the factor 22 that takes into account the spin degeneracy; we ignore Zeeman splitting here.

18 8 ChapterChapter 2. Density profiles

Sincee Zy = (27t3) 1 / 2, the charge density for a gas of spin- j fermions without mutual interactionn at temperature T = 0 and chemical potential u. is given by

23/2 2

P Ü MM T l^n(r)|2 ( u - En)1 / 2.

7TT z

—

(2.12) ) Again,, by using the Fourier transform of the energy Green function (2.8) we can write thiss as

-y\/2-y\/2 r\i. POO

Pn(x)) = — d E O i - E )1'2 d k G ( k , x , x ) . (2.13)

7tZZ Jo J-oo

Likewise,, the current density in the y-direction is given by

ww=—L{4 4

a

y y 3y y

-BxhMrjfJ^-EJ

1

/

2 2 (2.14) ) or r W WW = 21 / 2 2 7T^ ^ .. rOO d E ( u - E )1 / 22 d k ( k - B x ) G ( k , x , x ) . (2.15) 00 J-oo

2.22 Explicit form of Green function; parabolic cylinder functions

Wee now define dimensionless quantities by expressing the position x in units 1 / \ / B (orr [h/fmiüc)]1/2 in the original units) and the wavenumber k in units y/B (or in the origionall units (mtuc/ri)1 / 2)- The relevant variables become1 £, — \/Bx, K = k / \ / B .

Wee also express all energies in units B (or Tuuc). Therefore we will use e = E/B and

"vv = u/B. Using these new variables, we get the following dimensionless Hamiltonian

Thee corresponding eigenfunctions are the parabolic cylinder functions [41]

(2.16) )

*M*>£)) =

-1/2 2

dlDidlDin{K]n{K]__y2y2(Vl(l-K))(Vl(l-K)) De n ( K )_1 / 2( v/2 ( £ , - K ) ) (2.17)

wheree we have applied a similar normalisation as before (but now with dimensionless £,, and K). The function en(K)> which gives the eigenvalues, is defined by the boundary

conditionn at £, = 0

D D £ „ { K ) - 1 / 2 2 - \ / 2 K ) = 0 . . (2.18) )

Thee function is plotted in figure 2.1. It has been studied before by MacDonald and

1.. We use £, (with a bar) here to avoid confusion when we will use £,, TJ and C for differences berween coordinatess later on in chapter 5.

2.2.2.2. Explicit form of Green function; parabolic cylinder functions 19 9

n=4 4

n=3 3

n=2 2

n=0 0

Figuree 2.1: The function en(i<), for n = 0 up to 4.

Stredaa [39] and by Kunz [35]. As can be seen in figure 2.1, en(ic) has the property that

limK^ooo en« j =n+ \ a n d en( 0 ) = 2 n + §.

Iff we substitute (2.17) into (2.8) we get the following expression for the energy Green function n -II r rOO Gi,E(k,x,x)) = - f ^ dl'D2en(K]_y2(V2(l'-K)) VBB - Uo x D ^( K )_1 / 2( V 2 ( ^ - K ) ) 6 ( e - en( K ) ) . . (2.19) )

Insertingg this expression into (2.13) and (2.15) we can carry out the integration over E. Definingg Kn(-v) by en ( KnM ) = "v we arrive at the following expressions for the charge

density y P|x(Xj j v/2B3/2 2 7IZ Z ,, poo Y_Y_ d K [ ^ - en( K ) ]1 / 2 d t ' D2n ( K )_1 / 2( V 2 ( ^ --Jo o - i - i i •0) ) n2 2 (Vl(l-K))(Vl(l-K)) (2.20)

20 0 _{ChapterChapter 2. Density profiles}

andd the current density v/2B22 ' f00 ) v » == - - ^ 2 ~ Z dK[w-en{K)}^2{l-K) (2.21)

x

[T

d t

'

D D

L K > - , / 2 ( ^ ( E ' - K » » - 1 1 D LK, -1 / 2( ^ - K ) ) . .

Thee summations are over all n < -v — j , which we indicate by the prime. From these expressionss it is fairly easy to see that in the bulk the charge density is given by

p»=p»= lim p^{x) = ^ — V" f v - f n + J ) ]1/2. (2.22)

x—>ooo 7 tz _{* —}

n n

Thee current density in the bulk can be calculated in a very similar way:

j „ , n == lim j (x) = - ^ - = - 5 " [ ^ - ( n + 1 ) ]1/2^ ^ - dAAD^v^A). (2.23)

Sincee the functions Dn[y/2\) and ADn{\/2A) are orthogonal, this means that there is

noo current in the bulk.

Alternativee expressions for the charge and the current density can be found by writing Gj_iU(k,, x, x') as the sum of the Green function for an infinite domain and a correction

duee to the boundaries. The infinite-domain Green function is given by [21]

G l ,u0 c , x , x ' ) == - ^ = r ( - u / B + l ) Du / B_1 / 2( vy2 ( v/B x -K) )

yno yno

xx Du / B- ,/ 2( - v ^ t V B x ' - <)) (2.24)

forr x > x'y and an analogous expression for x < x ' . The correction for the chosen

geometryy is [31]

G l , J l e , x , x ' )) = 7L r ( - u / B + l ) " " " - " f ^ y

V7tBB DU / B- I /2( - V 2 K )

xx Du / B- i /2{ y / ï [ v ^ x - K ) ) DU / B_1 / 2{ V 2 ( ^ X ' - K)) (2.25)

forr all x > 0 and x ' > 0.

Thee energy Green function is determined by the poles of G° u + Gj_ u. Since the

2.2.2.2. Explicit form of Green function; parabolic cylinder functions 21 1

thee denominator in (2.25) contribute. They give a residue proportional to the derivative [9D€_i/2(—\/2K)/9e]_ 11 in e = en(K), which results in

Gx.EdC.X.x)) = ~j= Y, n - € u ( K ) + \)D2

en[K)_}/2(V2(l- <))

3 De_1 / 2( - v/2 K ) )

(2.26) )

xx De n ( K )_1 / 2( \ / 2 K )

Fromm (2.18) we see that

a De_1 / 2( - y 2 K ) ) 3e e dD D e - 1 / 2 2 de e \-y/l* \-y/l* -i-i - i ££ = £ n ( K ) 6 ( e - en( K ) ) . . e = €n( K ) ) ÖK K e = en( K ) ) den(K) ) i - 1 1 dK K (2.27) )

Withh the help of the Wronskian W [DA(z),DA[-z)] = V ^ t / H - A ) [41] we derive

)) = - ^ = £ r2( - en( K ) + 1 ) D ^K )_1 / 2( V 2 K )

xx D2e n ( K )_1 / 2( V 2 ( ^ - K ) ) ^ ^ 6(e - en(K)). (2.28)

dK K

Byy comparing this with (2.19) we find

dl'Dldl'Dln(K)n(K)__]/2]/2(V2(ï-(V2(ï- K)]\ = ^ r2( - en( K ) + i ) D Ïn ( K )_1 / 2( > ^

wheree we made use of the fact that den(K)/dK < 0.

den{<) )

dK K

(2.29) )

Pluggingg (2.29) into (2.20) and (2.21) gives alternative expressions for the charge den-sityy and the current density. Unfortunately neither these nor (2.20) and (2.21) allow uss to evaluate the integrals over K analytically. Both sets of formulas can be used for a numericall evaluation, although the expressions based on (2.28) are more convenient, sincee they involve a single integration only. Numerical results obtained along these lines aree presented in figure 2.2. Both the charge and the current density decay to their bulk valuess within a distance of a few times the typical length scale of the system 1 / \ / B . Near thee boundary, the current density exhibits a layered structure of currents flowing in al-ternatee directions. The number of layers increases with the number of filled Landau levels. .

222 Chapter 2. Density profiles

Figuree 2.2: The charge density ( ), in units VlB3/2/n2) and the current density

(( , in units \/2B2/7t2) for y = 2.0.

2 . 33 Asymptotic expansions

Thee expressions (2.20) and (2.21) are a suitable starting-point to derive the asymptotic behaviourr of the charge density and the current density for large £,. We will start out withh the latter since it is somewhat simpler; the current density vanishes in the bulk. Fromm (2.21) we see that for the current density, we need to determine asymptotic ex-pansionss of the integrals

llnn(l)=(l)= dK[v-en(K)V/2(l-<)

poo o

11

dl

'

D

£ n ( K ) - 1 / 2 2 {V2{1'-K)) {V2{1'-K))

- 1 1

n

2 2 (V2{l-K)).(V2{l-K)). (2.30)

Itt will turn out that In( t ) decays as exp(—£,2/2), so we can discard any terms that decay

fasterr than that.

2.3-2.3- Asymptotic expansions 23 3

In(£,)) from the interval [KnhO, K'] can be estimated. Consider the normalisation factor

1

000 roo poo _{dfdf Di{y/l{l' -K))=\ dV Dlirfl') > d£' Dl{V2l') = c}

n(A)

00 J - K J - Kn( - V )

(2.31) ) withh A = en{K) - j , which implies that A € [n,-v — £]. Since cn(A) is finite in the

closedd interval [n.-v — j]t we conclude that cn(A) is bounded from below by a certain

cnn independent of K. NOW we use the following asymptotic series [41]

Dx(z)) « e-z2/4zxAx(z/V2) (2.32)

whichh is valid for large and positive z. Here we introduced

== £ ( - A / 2 )m( ( 1 - A ) / 2 )m (_g 2 )_m ( 2 3 3 )

*—— m!

m=0 0

wheree {a)n is Pochhammer's symbol a(a + 1)---(a + n—1). Note that An(z) with n

integerr has a finite number of terms only; it is related to the Hermite polynomials by Hn(z)) = (2z)nAn(z). From en(K) < "v we conclude that

^ ( ^ V Z I ^ - K ) ) ^ - 1 ^ - ^ - ' ^ - ^ - 11

[1 +0({e-K)-

2

)] (2.34)

forr large positive £, — K. This means that the contribution of the interval [KU(*V), K'] to In(£,)) is smaller than

2 ^ - 1 / 2 ^ 1 / 22 TK'

ff dKe-( l-K ) 2(£,-K)2^ [1 + 0 ( ( l - K ) -2) ] . (2.35)

Sincee we have chosen K' < (1 — jy/l)t, this decays faster than exp(—£,2/2), so that it cann be discarded.

Forr K > K' we can use the asymptotic expansions of en(K)

[^-(n+.^^e-'V-'^gll (2.36)

(seee appendix 2.A for a derivation) and of the normalisation factor

dÊ'D2n ( K )_1 / 2(V2(£'-K)) )

« - T - ii - -Aü

2 n + 1 e - K 2 K 2 n + l c

-w

-v/Tm!! Tiln!)^

24 4 ChapterChapter 2. Density profiles

(seee appendix 2.B), both of which are valid for large K. Since [en(K) - ( n + \)] is

small,, we can write

D2e n( K ) - i /2{ v ^ - K ) )) =

[e

n

(K)-(n+J)]+h.o.t.. (2.38)

DD

22

jV2{l-K))+^Di(Vl(l-K)) jV2{l-K))+^Di(Vl(l-K))

A=n n

Withh the help of these expressions we find

r°°° 1

In( £ ) «« dKfr-fn + J ) ] ' /2- = - ( £ - K ) D * { V 5 ( E - K ) ) JK'' V7™-1

_£

d

4,

v

.

( u +

.

r

V

2

_l_

2 V

,

K 2

n

t

,g)

( l

_

K ) D

^

( l

_

K | ) )

-- d K t - v - f n + l ) ]1/2— -I2n + 1e -K 2K2 n + 1Cn( K ) ( ^ - K ) D i ( ^ 2 ( £ , -K) ) J K '' 7t(Tl!jz

+ +

++ h.o.t. (2.39) Thee first term can be discarded. This can be seen by writing Dn in terms of the Hermite

polynomiall Hn [41] Dn(z)) = 2 -n / 2 e -z 2 / 4Hn( z / V / 2 ) . (2.40) Ass H2 (z) is even in z, we have

pOOO POO

d K ( ^ - K ) D2( v/2 ( ^ - K ) ) = 2 -nn d K e -( £-K ) 2( l - K ) H2( l -K) . (2.41)

J K '' J 2 £ , - K '

Sincee K' is less than (1 — \y/l%> this decays faster than exp(—£,2/2).

Inn the remaining terms of (2.39) we split the integration interval once more, now at K"" = a " L with a " > \\fl. The contribution from < > K" in the second and the third termm is negligible. This can be shown in the same way as we did for the first term. For thee fourth term we use the following integral representation of the parabolic cylinder functionn [41]

DA(z)) = J^ez2/4 f ° ° d t e ~t 2 / 2c o s ( A 7 t / 2 - z t ) tA (2.42)

too show that AD2(V 2 ( É - K ) ) ) 7 - n / 2 + 3 / 2 2

== ~r— Hn(£-ic) (2.43)

roo o X X

II

oo o

2.3.2.3. Asymptotic expansions 25 5

Thee absolute value of the part between curly brackets is smaller than |ln t[ + 7t/2, which impliess that

I

f000 _{f tr}

d t e "t 2 / 2tnn |cos[n7T/2 - y/l{l- K)t] l n t - - sin[n7t/2 - y/i{l - ic)t]}

wheree c^ is independent of K and £. As a consequence we find that

(2.44) )

0) )

A=n n

ff d K e - ^ K ^ ' ^ r t t - K ) A

D

^(V2(£-

K

)

| J K "" D n l K j ÖA

f

2 d K e- KK K2n+1 H Ê - K J H ^ - K ) ! (2.45) K " "

wheree c^ is independent of K and £, as well. The right-hand side decays faster than exp(—£,2/2)) since we have taken K" > \yfll,.

Wee now collect all remaining terms, evaluating the quotient An(K)/Bn(ic) as a single

series,, writing Dn in terms of Hermite polynomials, and using

d d 3A A Dx(z) ) _{A=n n} , - ^ 7 4 , n n An( z / v / 2 ) l n z ++ — AA(z/V5) A=nJ J (2.46) ) Thiss asymptotic relation is valid for large and positive z and follows by differentiating (2.32)) with respect to A. The result for In(£) as defined in (2.30) is

US)) « ^ j ï f v - <

n

+ W

/Z

J

K

dKe-

K2

e-

(t

-

K

'

2

K

2n+1

(^- K)

2n+1

P

n

(*,Ê- K)

(2.47) ) with h

PPnn{K,l-K)={K,l-K)= -

{^-(nn + l^^i^-^-^

2

^-^}^^-^

++ Ln( K , £ - K )

containingg the asymptotic series

Kn(K,£,, - K) = 1 -—^ K-* + (2.48) ) 1+nn + n2 - 2 . n - n2( l_K )_2_ 4 + 9 n - n ^ .4 ^ nn — n 8 8 3nn - 6n2 + 4 n3 - n* 44 K -2( ^ - K ) -2- " ' " mv y* '^-{l- K)-4 + TT / F Ï l + 2 n _2 1 — 2 n ,P . ? 9 - 4 n3 4 Ln( K , £ , - K } == — K 2+ _ - _ ( £ , - K ) -2 — K"4 l - 4 n33 _2,F . , 3 - 1 2 n + 1 2 n2- 4 n3 , _4 (2.50) ) ( l ~ K ) -44 + ...

26 6 ChapterChapter 2. Density profiles

Wee can now integrate over K. In order to do so we expand the integrand around K = Ë./2.. If we choose K' and K" symmetrically around £,/2 we can make use of

*£ / 2 +d K e -2< « - ^2( K - l / i rr * i ^ - ü l l . / f + 0 ( e -2 Q 2a2- -1) (2.51)

k/2-ak/2-a 2 V 2

whichh is valid for n even. The same integral yields 0 for n odd. Note that if we choose K'' and K" as indicated before the 0 ( e_ 2 a a2 n _ 1) can be discarded when £,» 1. Term byy term integration then gives us the following asymptotic expansion for ln{t)

i - 2 n - 3 / 22 __

In(l)) » r . u2[y-(n+\)V/2e-^ /2i4*+2Rn(l) (2.52)

V7t(n!}z z wheree the series Rn(£) is given by

^(D^(D = ~ I Tr r^TTTTÏÏ + L - - Y - l n ( ë2/ 2 ) l Mn(l) + NnÜ) (2.53) ^ 4 [ - V - ( T I + 1 / 2 ) 11 ^—, m J

withh the asymptotic series

Mn(£)) = 1 - (3 + 2n + 4 n2) r2 - (12 + 21n - 10n2 - Sn4)r4 + . . . (2.54)

N n ( l )) =

_

(1

+

4n)

£-

2

-

2 1

-

2

° ^ -

3 2 n 3

r

4

+ ... (2.55)

Thiss result is independent of the particular choice of K' and K" as it should be. Finally,, substitution of (2.52) in (2.21) yields the asymptotic expansion for the current densityy that we set out to establish. It has the form

B22 « - ' 2 -2 n

WWW » - ^ T ï Z ' S i ï * -

( n +

i))

V 2

e-

£ V 2

£

4 n t 2 R

-(£) Ö-56)

TV V

withh the asymptotic series Rn(£) as given in (2.53).

Thee asymptotic behaviour of the charge density can be determined in a similar fashion. Insteadd of (2.30) we now have the integral

I

oo o

d K [ - V - €n( K ) ]1 / 2 2

KnhO O

rr°°° l~1

23.23. Asymptotic expansions 27 7

Contraryy to the current, the charge density has a finite bulk value, so it is appropriate too subtract the bulk density

(

OOO 1

d K[v_( n + 1 /2 ) ] 1 / 22 ' D2(_ ^K ) ( 2 5 g )

fromm I^(£,). Splitting the integral in the same way as before we arrive at:

I ; ( X ) - I ; ( O O ) « « - r d K f v - ( n + i ) ]1/2-7l rI2T V + 1e -, s 2K2 n + 1Cn( K ) D i { > / 5 ( E - K ) ) ) JK'' n[n\r ++ rdK [ v - ( n + i ) ] V 2 ^2ne- ^K2 n+l A n ( K ) 9 ^ ^ ^ JK'' 7i{n!)2 BU{K) 3A x=n - ff d K N - ( n + ^ ) ]1/2-7L - D2i( v/2 ( ^ - K ) ) ++ h.o.t. (2.59) Thee last term decays faster than exp(—£2/2). This can be shown by expressing Dn in

termss of the Hermite polynomial Hn

d K D2( v/2 ( ^ - K ) ) = 2 -nn dKe-{l~K) H i ( £ , - K )

J—ooo J—OO

(2.60) ) andd using that K' is less than (1 — jy/2)E,.

Thee remaining terms in (2.59) can be handled in a similar way as we did for the current densityy (see (2.39)). The only difference is the absence of the factor (£,— <). One finds

2-2n-1/2 2

y/n[n\) y/n[n\)

wheree we introduced the abbreviation

i;(x)) - i;(oo) « <= , _n 2[ - v - ( n + | ) ]1 / 2e - ^2E4 n + 1R ; ( E ) (2.61)

R

n(£>> = - {4 t v- ( n+1 / 2 ) ] + £ i - V - l n ( £ V 2 ) J Mi(E) + N ; ( E ) (2.62)

withh the asymptotic series

M.M.ffnn(l)(l) = 1 - ( 2 + 2n + 4 n2) r2- ( l 0 + 1 9 n - 6 n2- 8 n4) r4 + ... (2.63)

28 8 ChapterChapter 2. Density profiles 0.4 4 0.0 0 iwj~ ~ -0.4 4 -0.8 8

Figuree 2.3: Comparison between numerical results ( ) and the asymptotic ex-pansionn ( ) of the current density for -v = 1.0. The plotted function is f(£,) = exp(£,2/2)£,-2I0(£,),, with I0(£) as defined in (2.30).

Substitutionn of (2.61) in (2.20) gives us P n M - P n ( o o ) ) B 3 /2^ ' 2 2 T t V2 2 y-'2~y-'2~2n 2n ^ -- (nT)2 n n 11 , i 1 / 2ü- E V 2 t 4 n + 1

IB B

i«i«n+]n+]K(l).K(l). (2.65) 2.44 Discussion

Too check the validity of our asymptotic expansions we have compare them with nu-mericall results for the charge and the current density. In figure 2.3 we have plotted Io(x)) for y = 1.0. For this value of -v there is only one (partially) filled Landau level, soo I o M represents the complete current density. Because of its fast decay the pre-factor exp(—£,2/2)£,4n+22 has been divided out. The solid line corresponds to the numerical results,, the dotted line to the asymptotic expansion (2.52). As can be seen, the conver-gencee is quite good.

Ass (2.56) and (2.65) show, the contribution of each Landau level n to both the cur-rentt density and the charge density has a Gaussian decay for large x (in leading order

2.4.2.4. Discussion 29 9

proportionall to e x p ( - £2/ 2 ) £4 n + 2l n ( £2/ 2 ) and e x p ( - £2/ 2 ) £4 n + 1 ln(£,2/2), respec-tively).. Hence, when only a limited number of Landau levels is rilled, in other words forr every finite magnetic field, both densities decay with a tail proportional to a Gaus-sian. .

Thee decay found here is consistent with the bound derived by Macris, Martin and Pulé [401.. However, it disagrees with the results of Ohtaka & Moriya [421 and of Jancovici [27].. In the latter paper the current density at T = 0 is given as

J

^( X )) " 1 5 ? J8* "2 [ ï - S i ( 23 /V/ 2x ) ] + ( J L j - l ) s i n ( 2 ^ V/ 2x )

-- (

2V2

^

1/2x

+ 2"V

/ 2

*)

C O S ^ V '2* ) }

(2.66)

withh Si(z) the sine integral. The right-hand side decays algebraically, with a tail propor-tionall to x~1 for large x. It is obtained via an inverse Laplace transform of the current densityy jy,p (x) for a magnetised free-electron gas with Maxwell-Boltzmann statistics as

explainedd in section 1.2.3. The Maxwell-Boltzmann form of the current density em-ployedd in [27] is obtained by a linear-response method valid for small magnetic field. Inn fact, the dimensionless parameter that has to be small is fJB. The integration in the inversee laplace transform is taken over all values of fi, and thus in particular over all valuesvalues of |3B. Hence, it is not justified a priori to insert the linear-response expression forr jy,p(x) and to carry out the integration subsequently. As a consequence, the

ex-pressionn (2.66), and the ensuing algebraic decay is not guaranteed to be correct. As hass been remarked already in the introductory section, the procedure of taking inverse Laplacee transforms of Maxwell-Boltzmann expressions for small fields may even lead too weird effects like undamped oscillations, if it is applied to other physical quantities. Questionss about the validity of (2.66) in the limit x —> oo have been raised before by Shishidoo [46], who argues that the expression is not uniformly convergent, and is valid onlyy for small x (and small B).

Itt should be noted here that our asymptotic expansions (2.56) and (2.65) are rather awkwardd when it comes to studying the limit B —» 0. In that limit the number of filled Landauu levels goes to infinity. The coefficients in the expansion rapidly grow with the labell n of the Landau level, as is clear from (2.54), (2.55), (2.63) and (2.64). Hence, thee asymptotic region moves further and further away from the wall, as B goes to 0. Thiss weak-field limit will be investigated in chapter 4.

Ourr approach to determine the asymptotic behaviour of profiles for finite magnetic fieldsfields can easily be generalised to other physical quantities, for instance the kinetic pres-sure.. In general, the leading term is proportional to exp(—£,2/2)f,m ln(£2/2), where m

30 0 ChapterChapter 2. Density profiles

increasess with the number of filled Landau levels and with the number of particle mo-mentaa occurring as factors in the expression for the physical quantity being calculated.

22 .A Appendix: Asymptotics o f en( K)

Inn section 2.2 we introduced the function en(K), which defines the eigenvalues of the

Fourier-transformedd Hamiltonian (2.16). It is defined by

De n(K) - i / 2 ( - \ / 2 K ) = 0 .. (2.67)

Thee asymptotic expansion of D\{—\/ÏK) for large and positive K is given by [41]

\/27tt 1 Dxt-y/it Dxt-y/it eeinXinXee-K-K / 2( V/ 2K )AA A ( K ) +

r(-A) )

e~~ ^ ( V ^ K J - ^ - ' B X I K )

withh A\ as defined in (2.33) and with B* given by

BX( K ) == f ( d + A ) / 2 )m( ( 2 + A ) / 2 U( K2rm

-tn=0 0 TTL! !

(2.68) )

(2.69) )

Settingg (2.68) to zero and expanding around A = TL we arrive at

[en(K)-(nn + J)] 11 2

n

e~K 2 K2 n + 1 ^ n ^

y/ïva\y/ïva\ " " BU( K .

forr large positive K. This is a generalisation of the expression given by Kunz [35]

(2.70) )

2.BB Appendix: Asymptotics o f the normalisation factor

Inn section 2.3 we needed the asymptotic expansion of the normalisation factor

J

'' 'OO

dl'Dldl'Dl

n[K)n[K)

__

W2W2

(V2(i'-K)) (V2(i'-K))

__ 0

(2.71) ) forr large K. In the previous appendix we have seen that for large K the function [en( K) —

(nn + j)] is small. Therefore we can write

oo poo didi//UU22£n{K)£n{K)_,_,/2/2(V2(ï-K))=\(V2(ï-K))=\ d ^ D ^ V ^ ' - K ) } oo Jo ++ ^-\ dïDl[V2(ï-K))\ [ e_{Ö A} n( K ) - ( n + i ) l J oo U=n »22 roo - [ en{K) - ( nn + i ) ]2+ h . o . t . (2.72)

2.B.2.B. Appendix: Asymptotics of the normalisation factor 31 1 Thee first term in this expansion is given by

(

ooo poo r—K

d£'Di(V2(£'-K))== dl

f

Di(V2l')-\ di'DiiVll')

00 J—oo J—oo _{y/iïn\-2y/iïn\-2}nn_--}}_e-e-K2K2_KK2n+2n+-- 3n + n_{'' (<~}2 ~-2

K"44 +

) ) (2.73) )

ass can be derived by expressing Dn in terms of the Hermite polynomial Hn, followed

byy term by term integration of the resulting series.

Withh the help of the integral representation (2.42) of D* the coefficient of the second termm in (2.72) becomes 3A A

((

ooo ?\/2 f°° ?

d ï / D ^ v ^ ' - K ) )) = ~ \ dse

s /2

D

n

(V2s)

00 A=n V7Ï J - K

J

ooo „

dl'e-dl'e-ll''Z/2Z/2llmm[cos{nn/2-[cos{nn/2- V2sl')]nl' - - sin(n7t/2- v/2sl')].(2.74) oo 2 Repeatedd partial integration yields

I

oo o d£'Dj(V5(S'-K)) ) 0 0

aa

ÖAJO O n / 2 + m / 2 + 1 1 {nn —m)

I

oo o d l '' e"1' / 2^/ n"m-1 {cos[(n + m + 1 )TT/2 - Vlsf] In I' o o -- j sin[(n + m + 1 )TU/2 - Vïsl']}

Forr all m < n we write the contribution at s = — K of the sine term as

ll /2/2 7tn-m-1ei\/2Kl' ——Im m 2 2 : n + m + +

II

oo o d £ ' e -1'2' ' o o • Hn-m( s ) ) (275) ) (2.76) ) Wee now use a theorem [17] stating that for large x the Fourier integral

dt(|>(t)e,xt t

Joe Joe

hass an asymptotic expansion to which the end point a contributes as

A =

£ .

n + 1

o - c M a }

x

_

n

_

1 e i x a a

(2.77) )

n = 0 0 d a

32 2 ChapterChapter 2. Density profiles

iff (J>(t) has no singularity in [ex, (3]. With the help of this theorem we can show that

££ Jo

»» -f (-1H^)--"" f '

2l +

^ 7 -

1

'

! K

-". (2.79)

1=0 0

Thee contribution of the cosine term at s = — K in (2.75) can be written as

Re e

ff

oo o _{d £ / e}-_^^ / 2_{£ ,}m _ m _ 1_ei v /_^K_{^ ' l n £ '}

o o

(2.80) )

Becausee of the logarithm we need a generalisation of the previous theorem to Fourier integralss with logarithmic singularities. This generalisation can also be found in [17]. Itt states that for (f)(t) = 4>i (t) ln(t — a) the asymptotic expansion of (2.77) contains a contributionn from the lower end point which reads

A = L L

;TI+ + n ^ O O id"(Mtx) ) d an n .n .n i | > ( r n - l ) - l n xx + i - . X~n-1eixa_ _ _{(2.81) )}

Withh the help of this theorem we see that the contribution from s = — K of the cosine termm is identical to the contribution of the sine term. Using the same method, we find thatt for s —> oo the two terms in (2.75) cancel, at least for m < n .

Thee contributions for m = n can be calculated in a similar fashion, although they need somee extra attention because of the additional f~} singularity. They add up to

55 _e~l_'2/2 _n

d£// - icos[(2n + 1 )n/2 - Vise'] In I' - - sm[(2n + 1 )n/2 - y/lsl']}

7 t f - 1 1

_

( 2 l - 1 ) ! „ _ 2 i i

Y

_

m (

v2

K ) +

x:if^K--wheree y is Euler's constant. Collecting all these terms we get

(2.82) ) 33 ro dX X dl'T>l(>/2(l'-K)) dl'T>l(>/2(l'-K)) 2v / 7TTl! ! X=n n UU 1 VV - - y - l n f v ^ K ] *—.*—. m. .m=1 1 11 + 2n _2 3 + 6n + 6n2 ++ — A —K + T2 44 16 K"44 + .. .. (2.83)

2.B.2.B. Appendix: Asymptotic; of the normalisation factor 33 3

Finally,, we have for the third term in (2.72)

11 d2 r

23A2 2 d Ê ' D ^ V ^ ' - K ) ) ) d£// DA( V5(£' - K)) -^D^y/iil' - K))

1

00 0

0 0

iD»(v5(l' '

(2.84) )

Ann asymptotic expansion of the first integral can be derived along the same route as above.. The only new ingredient is a straightforward extension of the theorem by Erdélyi too Fourier integrals with squared logarithmic singularities. It states that if in (2.77) one takess (f)(t) = (J)2 (t) In (t — a), the contribution of the lower boundary is given by the asymptoticc expansion

A = f ; in + 1^ ^ ^ { [ T K nn + l ) - I n x ]2 + C ( 2 , n + 1 )

++ i7t[i|>(n + 1 ) - I n x ] - - j - >

n = 0 0

l n x ] - ^ U "n - 1ei x a. . (2.85) )

Whilee it is not very difficult to derive this extension of Erdélyi s theorem, a slighdy differentt way to calculate the integral can be found in [50].

Ass a consequence of the results mentioned above, the asymptotics of the first integral in (2.84)) is found to be of order 1TI2(\/2K). This implies that for large K the first integral iss negligible with respect to the second, as we shall see.

Thee asymptotic behaviour of the second integral in (2.84) follows by noting that for largee K the dominant contribution comes from the lower end of the integration interval. Withh the help of (2.42), (2.78) and (2.81) we can derive the following asymptotic expansionn for the integrand

ii

DM DM

v ^ n ï e ^ z - " -11 Bn( z / \ / 2 ) (2.86) )

\=n \=n

forr large and negative z. Term by term integration leads to

\2\2 roo '00 A=n 11 ^ 2 roo )) )) 7 t ( n ! )22 -n-1eK 2K -2 n" " • • ( ' ' 22 5 + 5 n + n2 _4 22 + K 4 + (2.87) )

Thee right-hand side grows exponentially as K —> oo, but this is compensated by the factorr exp(—2K2) in [en(ic} — (n + \)}2t resulting in an overall exp(—K2) behaviour.