On the inhomogeneous magnetised electron gas

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Kettenis, M.M.

Publication date

2001

Link to publication

Citation for published version (APA):

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

offsetdrukkerij b.v.

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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.

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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)

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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.

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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.

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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 ) ) n2 2 (Vl(l-K))(Vl(l-K)) (2.20)

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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

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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. .

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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.

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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!)^

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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

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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 ) AD^(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 + ...

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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

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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)

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

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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

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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

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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)

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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 n-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

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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)

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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.

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Thatt is the reason why we had to expand (2.72) up to second order in [ €U( K ) — ( n + j ) ] .

Higher-orderr derivatives of J^° dï.' D2(\/2(£/ — K)) are also of order exp(«2) or less, so thatt we do not have to go beyond second order.

1 1 1 1

Substitutionn of (2.70), (2.73), (2.83) and (2.87) in (2.72) gives

[

oo i — 1

di'Didi'Din{K)n{K)__]/2]/2[V2(i'-K)) [V2(i'-K))

Jo o

1/701!! 7t(n!)2 2n + le- KK K2 n + 1C n ( K) Cn(K)) = yy — \n[y/l* Lm=1 1 1 -- 11 + n + n 2 _2 4 + 9n - n4 _, •KK — 8 8 , l + 2 nn 2 9 - 4 n3 4 ++ | - 4 - K + ^ 6 - K + (2.88) )

++

...

(2.89) )

On the inhomogeneous magnetised electron gas - Chapter 2 Density profiles

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