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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Weakk magnetic fields

Inn the previous two chapters, we have investigated the density profiles for a magnetised electronn gas in the presence of a hard wall. We have seen that for the completely degen-eratee case, i.e. at zero temperature, the ultimate decay of the profiles towards their bulk valuess is Gaussian with algebraic and logarithmic pre-factors. At the end of chapter 3 wee concluded that because of these pre-factors, this ultimate Gaussian decay shifts fur-therr away from the wall, towards the bulk, when the number of Landau levels increases. Sincee the number of Landau levels becomes large for weak magnetic fields, it would be interestingg to know how the profiles behave before the ultimate Gaussian decay sets in. Inn terms of the variables £, = \/B X and *v = u/B for the (dimensionless) distance from thee wall in magnetic lengths and dimensionless chemical potential, we have established thatt the Gaussian decay sets in for 1} » v. However, in the case where t2 « "V, with v,, t large, we can still use the multiple-reflection expansion that we used in the previous chapter.. Incidentally v » 1 corresponds precisely to the weak field limit that attracted ourr attention in the first place.

Inn order to investigate the situation where t2 «*v (for large v), we will first look at the casee where the magnetic field is perpendicular to the wall instead of parallel to it. This simplifiess matters considerably, and it may be helpful in tackling the more difficult case wheree the magnetic field is parallel to the wall.

4.11 Magnetic field perpendicular to the wall

Thee situation with the magnetic field perpendicular to the wall has been investigated beforee by Horing and Yildiz [24, 25]. However, as we will see below, the analysis in 57 7

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thee first part of this chapter gives access to a slightly different regime, and provides uss with more insight under which circumstances the approximations made in [25] are applicable. .

Wee keep the magnetic field in the z-direction, which means that the wall is now in the xx — y-plane, and the p(x) of chapter 3 gets replaced by p(z), and the (charge) density profilee is given by

P3(Z)) = £ G | |I P( Z , Z ) (4.1)

wheree p is the bulk density. Calculation of this density profile for the non-degenerate casee is almost trivial now since the magnetic field only couples directly to the x- and u-directions.. The bulk Green function for the z-direction is simply

G

S.e

(z

'

z,))

= 7 ^

e x p _{23 3} (4.2) )

soo if we use the reflection principle (see (3.10) in section 3.1) we see that

5P p(z)--pe-2 z 2/P.. (4.3)

Forr the completely degenerate case we employ the inverse Laplace transform (3.46). Since e

zz =

f M

3 / 2

_ J 5 B

2n$)2n$) 2sinh(BB/2)

thiss means that we have to evaluate the integral

-JJ rC+iOO Be0 ^ l - 2 Z2/ P

8 ( V , Z ))

= " 5 5 Li o o d P2 3 / 2 ^ / 2 p 3 / 2s i l l h, pB / 2 r <4'5)

Heree the integration is in principle over a straight line Re B = c (with c > 0) parallel too the imaginary axis. However, we can always deform the contour, as long we do not pulll it through any singularities.

Inn contrast to the case where the magnetic field is parallel to the wall, we can derive ann exact expression (exact in the sense that it does not contain any integrals) for the densityy profile in the completely degenerate case. Using

-- oo

'' = 2£e-( n + 1 / 2 ) ( 5 B (4.6)

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inn (4.5) gives us

ii fc+ioo ny00 p3 u - ( n + 1 / 2 ) 0 B - 2 z2/ p

6 M z )) =

"2^ri J

d

P

n =

° ^ „-»/,,,„,

(4-7)

'c—lOO O 21/2^3/2^3/2 2

Iff we set s = 6B and use the dimensionless variables *v = u/B and C = B z / y ' p instead off p. and z we get

R 3 / 22 °° i fc'+ioo

6pH(z)) = - - ^ — £ - L d s es [ , - ( n+, / 2 ) ]s- 3 / 2e- 2 , c V s ,

^ / 2n3 / 2 ^ - 2 7 n Jc, _i o o o

(4.8) )

Thee inverse Laplace transform in (4,8) is given by1

- f f

27tiJc c

c+iooo I

d s es tt s "3 / 2 e "a / s = -^= sin (2>/at) 0(t) (4.9) forr a > 0 and c > 0. Use of this identity in (4.8) results in the following formula for thee exact density profile:

R 3 / 22 ,

SPtU)) =-ln2yv2~JL sin(2V2^^y-{n + 1/2)t) (4.10)

TV V

wheree the prime indicates that the summation is over all n with n < -v — 1 / 2 . This expressionn has the form of a sum over Landau levels. Therefore it is not very practical iff the number of Landau levels becomes very large, i.e. for large v. Even numerical evaluationn becomes difficult since there can be quite a bit of cancellation between terms inn the sum, which leads to loss of precision.

4.1.11 Density profiles for large "v

Inn order to derive an expression for the density profile that behaves better for v » 1, wee go back to (4.5). Taking 0 = i t / B , we have

6

P , WW = " 2 5 7 1 ^ 7 2 Jcd t (i t) 3 / 2s i n h(i t / 2, <4-n>

withh C a suitable contour. The integrand has singularities at t = 27m for all integers nn (simple poles for rt ^ 0, TL = 0 is the endpoint of a branch cut). This allows us to choosee a contour as pictured in figure 4.1. Note that the contribution from t < 0 is the 1.. [19], p. 245.

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\rr r

—Ann —In Im m

XT" "

27rr An

"Cr r

Re e

Figuree 4.1: Contour of integration for (4.11). The dots represent the poles at t = Inn.

complexx conjugate of the contribution from t > 0. Therefore we can focus on t > 0 andd easily recover the full result afterwards.

Too investigate the behaviour for large "v, we have to look closer at the points where the phasee f(t) = t + 2C2/t is stationary, that is, where f'(t) = 0, thus t = >/!£. In addition, itt is useful to know that f'(t) < 0 when t < A/2£ and f'(t) > 0 when t > y/2l. The contributionn arising from the point t = \/2C to the integral can be calculated using the methodd of steepest descent.

Inn order to apply this method, we have to find a path through t = ylt where the reall part of f(t) is constant. Setting Re f(t) — f(\/2£) = 0 we see that for t close to

\flt\flt we get R e t = ) + %/2C The minus sign corresponds to a path of steepest ascent,, the plus sign to a path of steepest descent. Since we are looking for steepest

descentdescent we have to choose the plus sign, which is convenient, since that means that the

pathh stays away from the cut along the positive imaginary axis (see figure 4.2). In the neighbourhoodd of t = y/ÏZ, we can now parametrise the path as t = yïï, + em / 4s . Thiss means that the contribution arising from the point at t = V2C is given by the asymptoticc expansion of the integral

"25/2^5/2 2 in/4in/4 i2V2-vl ds s

exp(---m* exp(---m*

i V ^ C F2!! sin(C/\/2 + ei 7 t/4s/2)' (4.12) ) Itt is necessary to keep the s-dependence in the argument of the sine in the denominator, evenn if s is small. The reason is that when C/\/2 is close to a multiple of n, small changes inn s have a large impact on the overall value of the integrand. The s-dependence in the factorr ( i t )3 / 2 in the denominator on the other hand can be dropped immediately, since

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ss will be small in comparison to \/2C It will prove to be convenient to write (4.12) as g3/22 e2i-vV2l poo

213/4^5/22 £3/2 J ds s

expf-- ^ ^ s2l l

sin{l/y/l)sin{l/y/l) cos(ei7t/4s/2) + c o s ^ / v7! ) sinfe*7 1/4*^)' (4.13) ) Sincee -v » 1, the exponential in the numerator of the integrand suppresses contribu-tionss from s outside a small region around zero. In other words, we can indeed assume ss to be small and write

ds s

ex

P

(

-7fc

s2

--

_4isin(C/v/2) )

sin{l/V2)+cos(l/V2)ë^sin{l/V2)+cos(l/V2)ë^44s/2s/2 ~ cos^C/v7!] forr the integral in (4.13). With the help of the relation2

ds s exp(-- ~ ^ s 2 1 1 s22 + 4itan2(C/V^) (4.14) ) d s ^ -T T I= E r f c ( a b ) ^:ea 2 b 22 | a r g a | < ^ , R e b > 0 s22 + b2 2b b for r b = 2 ei 7 t / 44 tan[l/Vl)

wee can evaluate this integral. The result is .. /A tan(C/V2) 2 7 t em / 4 4 (4.15) ) (4.16) ) il/y/2) il/y/2) ^ i W t c / v ^ E r f cc n^01ein/4 |t a n (£/ v^ | ) . (4.17)

Becausee of the factor exp(i7t/4), the argument of the error function is complex. Ex-pressingg the error function in terms of the Fresnel integrals S(x) and C(x) [41]

Erf(ei7r/4x)) = Vlein/4 -x-x I - i S I \l-x

Jk Jk

(4.18) ) allowss us to write (4.17) in terms of functions with a real argument. The Fresnel inte-gralss S(x) and C(x) are defined by

S ( x ) == f d t s i n ^ t2) C(x) = d t c o s s

m m

(4.19) ) Thee arguments of these are still rather complicated. Therefore we introduce the short-hand d

25/4^1/22 _

xx

= Si72p7lHc/V5)|. (4.20)

2.. Eq. 3.466.1 in [23].

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— 4 T I \ ^ ^

-V2?s s

-In -In Im m 27t t y SS AlX

v% %

Re e

Figuree 4.2: Deformed contour of integration. The crosses represent the stationary pointss at t = In this example nm i n = 1.

Thiss gives the following result for the contribution of both stationary points to the densityy profile

B3/2X X 47 n, i / 2 £s i n( £ /v/ 2 ) )

C(X)) - - sinn (2V2-VC + ^ X2)

S(X) ) _{cos s}

(iVivi (iVivi

+?x; ;

(4.21) ) However,, as we will see now, this is not the complete result.

Inn order to follow the path of steepest descent, we had to deform C in (4.11) towards aa contour as depicted in figure 4.2. In doing so, we have pulled it through the poles att T for all n > —k~. Therefore we will have to add the residue in those poles too (4.21) in order to find the correct asymptotic expression. The sum of the relevant residuess is given by B3/2 2

L L

-1)n --coss f 27mrv + vlvl1 1 " ) ) n3 / 2 2 (4.22) )

wheree nm;n is the smallest integer larger than —k-.

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-- S ( X ) - - cos i ] c o s ( 2 V 2 ^ + ^ X2) } }

R 3 / 22 °° COS (27CTYV + ^ ITT )

T V = T lmi n n

forr "v » 1 and £ not too close to zero. The dependence of the sum over n on rtmin seemss to suggest that there is a jump in the density when I goes through V2.7tm. This wouldd be rather unphysical, and therefore we can expect a cancellation of this jump by otherr terms in this expression. We will see below that this is indeed the case.

Off course the Fresnel integrals S(x) and C(x) are still special functions. It is therefore interestingg to see if there are special cases in which (4.23) can be simplified. This is indeedd possible if X » 1 or if X <C 1.

Thee strong inequality X » 1 holds for -v » 1 and finite C if | tan(C/\/22) | is not close too zero, which is true if C is not close to yflnvn, (for m integer). In other words, the followingg approximation is valid if C is chosen such that the stationary point is not close too the poles. In that case

S(X)

-I«-JLcos(^)) CPO-I-JL*,^) (4.24)

whichh leads to

R3/2 2

cos(2\/2vC)) (4.25) 47t2-v1/2tsin(t/\/2) )

forr the term containing the Fresnel integrals in (4.23). We would have obtained the samee result by neglecting the s-dependence in the sine in the denominator of the inte-grandd in (4.12).

Whereass X » 1 corresponded to I away from y/lnvn^ the case X <C 1 corresponds too C close to y/lnrx. So this case gives us information about the situation where the stationaryy point and a pole coalesce. We can now approximate the Fresnel integrals by

S(X)) « ^ X3 C(X) « X. (4.26)

6 6

Iff we look at the full expression for hp^iz) in this approximation

6 f v ( 2 ))

* " 47r3m3/2 Wit ~ ^ m ) cos \2V2vl - -n)

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wee see that there are now two jumps when C reaches a multiple of y/2n: one in the firstfirst term, and the jump that we already observed in the last term of (4.23). Indeed thesee discontinuities compensate each other exactly, and the density is continuous in thee points I = x/2nm.

Horingg and Yildiz [25] give their approximation for weak magnetic fields as the sum of twoo terms: pr [z) and PdHvA(z)- The first term comes from the branch cut (which in our casee lies along the imaginary axis) and is described by Horing and Yildiz as "monotonie inn magnetic field dependence". The second term comes from the poles (which in our casee lie along the real axis) and is "oscillatory in the de Haas-van Alphen (dHvA) sense". Wee will give a comparison between their result in the limit T —> 0 and the results derivedd here. The T —> 0 limit of the contribution of the term PÖHVA(Z) to 6p^(z) is givenn by

B3 / 22 ~ COS (27HW + 2 £ i - f 7t)

6p

dHV

A(z)) = - f - r L

V n 3 /

;

<

4

-

28

>

n=1 1

Iff we compare this with (4.22) one can spot two differences. The first is the absence of thee factor (—1 )n. This can be attributed to the fact that Horing and Yildiz do explicitly takee into account the Zeeman-splitting, which we preferred not to do for consistency withh the rest of this thesis. Incorporating the Zeeman-splitting in our result would amountt to replacing the sinh(BB/2) with tanh(6B/2) in the denominator in (4.4). Indeedd this does account for the absence of the factor (—l)n from (4.28).

Thee second difference is that in (4.28) the summation over n starts at 1 instead of Umin-Thiss can be explained by the fact that the pr [z) term in the result of Horing and Yildiz cann only be valid for C <C Vln (although they do not mention this explicitly). So in a sensee their result is complementary to the results derived here.

Thee contribution to öp^Jz) from the branch cut (which is called P2r(z) in [25]) is givenn by B3 / 2 2 ópr(z)) = _{2 3 / 2} n2 ^ 1 / 2 2 sin(2\/5-vC)) eos(2\/2vC) 23/2-^33 J2 (4.29) )

Thiss result does not change if one ignores the Zeeman-splitting; its derivation is based onn the approximation of l/tanh{(3B/2) by the first term in its Laurent expansion, whichh is identical to the first term in of the Laurent expansion of 1/sinh(6B/2). But inn spite of this, (4.29) differs considerably from our result (4.21).

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0.03 3

CO O Q. .

-0.02 2

-0.03 3

8 8

Figuree 4.3: Numerical evaluation of 6p^(z)/B3 / 2 for v = 5: exact (4.10) ( ), proximationn according to Horing and Yildiz (4.30) ( ) and our asymptotic ap-proximationn (4.23) ( ). The latter is almost indistinguishable from the exact ex-pression. .

4.1.22 Numerical results

Inn order to check the quality of our asymptotic expression (4.23) for the density profile wee would like to compare it numerically with the approximation given by Horing and Yildiz,, and the exact result (4.10), for moderate values of "v.

Inn the absence of Zeeman coupling the result of Horing and Yildiz would read

6pH(z) ) B3 / 2 2 23/2^2^1/2 2 B3 / 22 ~

2^2J->: :

n = l l sin(2\/2vC)) cos(2v/2"vC) 2 3 / 2 ^ 33 Z2 coss [2mw + ^ - f 7i) i UU \ / _{(4.30) )}

Figuree 4.3 shows that for large I, our approximation is clearly better than the one given byy Horing and Yildiz. In fact, the curve corresponding to (4.23) is almost indistin-guishablee from the exact result, even though -v = 5 is not particularly large. We also see

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Figuree 4.4: Numerical evaluation of 6p^(z)/B3 / 2 for *v = 5: exact (4.10) ( ), proximationn according to Horing and Yildiz (4.30) (— • —) and our asymptotic ap-proximationn (4.23) ( ). Here the result of Horing and Yildiz is almost indistin-guishablee from the exact result.

thatt for C smaller than s/lm., (4.30) still seems to work reasonable well. But closer to

VïnVïn and beyond, it makes no sense.

Off course it is not fair, not to discuss the situation with C, closer to zero. Here our approximationn is not supposed to work. In fact, in the limit C —> 0 (4.23) diverges. Figuree 4.4 shows that (4.30) behaves much better for smaller values of C, illustrating thatt our approximation and the approximation given by Horing and Yildiz are com-plementary. .

4.22 Magnetic field parallel to the wall

Afterr having treated the case where the magnetic field is perpendicular to the wall, let us returnn to the original geometry with the magnetic field parallel to the wall. As indicated before,, we are interested in the regime where £, 3> 1 and 7 > 1 , with £,2 « v. In order too investigate this regime, it is useful to introduce the new variable £, = i,/yfv. We will

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startt from (3.29) in the previous chapter, which, in terms of the new variable £,, reads pp ByV2£ 1 - q o2 / ^ f°° dp

Zj.8V57tï/22 q^2 e X PV 2q<Jjo ,/p(1 + p)

xx ^/l + d - q o ^ p e x p ( - " "2q f) P^2) • (4.31)

Wee will use once more the inverse Laplace transform (3.46). Choosing a contour slightlyy different from that in (3.47), by setting 0 = (it + £)/B, we arrive at

(4.32) ) Heree J("w,u) is defined as

**^ - r * ^ ^ --**

<4

-

33)

Inn the new variables t and £,, the parameter q0 is given by tanh[(it + £,)/4]. Since I is

O(l),, qo is complex (and not in the vicinity of 1, as in (3.47)). Hence, the argument beloww (3.29), which justified rewriting the integral J in terms of a modified Bessel function,, cannot be used here. Instead, we will have to determine the properties of J(w,u)) itself, in particular for large values of Rew. A suitable asymptotic expression forr J(w, u) is derived in appendix 4.A.

Becausee of the asymptotic form (4.49) of J(w,u), it is useful to split the integration in (4.32)) in intervals of length 27t. We do this by setting t = T + 27m, with T e [0,27t]. Thiss yields

B3 / 2e^ ££ oo ,2TT 1

VV

**VV ] &V2n5/2 ntrj Jo (iT + 27Tin+£,)3/2

xx (n +1) cosech ( —-— J exp < rv T + — cotanh f —-— J >

++ i a s e c h

(^+l)

e X

p|

i v

[

t +

. (4.34)

Sincee -v is large, the integrand is in general a rapidly fluctuating function of T, which allowss us to employ the saddle-point method to calculate the integral over T. The in-tegrall over the first term between the curly brackets is dominated by the saddle point

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thatt follows by setting the derivative of the function

ff (T) = IT - y cotanh (^j^\ (4.35)

equall to zero. This leads to T = 2\\> + i£, with TJ> = arccosfl — £,2/4). Clearly, only valuess of I in the interval 0 < i, < l\fl make sense here. They correspond to saddle pointss T with a real part between 0 and In. If £ approaches its lower or upper bound, thee asymptotic behaviour for Sp^fx) as derived here will no longer be valid, as (4.49) thenn no longer holds.

Treatingg the second integral in (4.34) in a similar way, we arrive at the following asymp-toticc result for the excess particle density in the regime of large "V and £, € (0,2\/2):

6 p n MM « exp[2rv(\|> + sin op}] 8v27t2-v1/2sinn ' ip ^-^- lrv4-'\ IpZraiw TO fn — 1 1P-2TOUT

VV (_i)nl!LLLi?

+ i

y r-i^lH: L^

^ - ;; J (mi + i|))3/2 + lZ _l , J ( 7 m_ , | , ) 3 / 2 .n=00 n=2 ++ C.C. (4.36)

Forr small values of t the variable \|> is close to zero. As a consequence the n = 0 term inn the first sum in (4.36) dominates. Hence, the asymptotic expression simplifies to:

B3/2 2

óPn(x)) » . n 2 1 / 2 . 1 / 2 , , ,/ 2 COS[2-V(I1J + sini|>)]. (4.37)

4v27t2'v1/2sinn 7 \[> ij>3/2

Numerically,, it is found that this approximation is quite useful even if t is of order 1, ass can be seen in figure 4.5.

Thee approximation (4.37) ceases to be accurate for values of *v that are close to a half-oddd integer. This can best be seen by writing the sums in (4.36) in terms of Lerch functionss Q(z, s, a) that are defined as [18]

00 0

<P(z,s,a)) = 2 j a + n ) -szn (4.38)

n.=0 0

forr \z\ < 1, s > 0 and a non-integer. An analytical continuation as a function of zz in the complex plane can be carried out, except for a cut from 1 to oo along the positivee real axis. Setting apart the first term in the first sum in (4.36) and defining VV = -v + 1/2 — [-v + 1/2] (that is, V is the non-integer part of -v + 1 /2), we get

B3/2 2

6 f V WW W

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CO O Q . . 0.02 2 0.00 0 -0.02 2 -0.04 4

Figuree 4.5: Comparison between results from (4.36) ( ) and the approximations (4.37)) ( ) an d (4.44) ( ) for 5 p ^ x ) / B3 / 2 at -v = 10 as a function of £,. I ^ ïï i 2nrv'' 1 1 . * 7t t ++ i e- 2 ™ ^ '0 |e -27ri-v'' 1 1 _ i ' 2 '' 71

++ 1

^ ^ i e e 7t t - 2 7 t i - v ' ' "'"'0>(e0>(e22™™vv,?,1,?,1 +

2 -

2 7 t i V

, f , 1 -- -

_{!)]} }}

7T T ++ C.C. (4.39) )

IfIf -v' is close to an integer, the Lerch functions with s = 1/2 dominate. In fact, for • v ' C ll one may write [18]

\e\e22™',j,a ™',j,a jiTi/4 4

/ 2 V V (4.40) )

whereass for 1 — V <?C 1 one has:

<S>[e<S>[e22™'™' ,-,OL

-in/4 -in/4

y/itt-y') y/itt-y')

(4.41) ) Thee sum of the Lerch functions with s = 1 / 2 in (4.39) becomes large for V small, while itt stays finite for v ' near 1. Hence, a better approximation for small V is obtained by

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«1 1 CO O

2 2

a. a. 0.10 0 0.05 5 0.00 0 - 0 . 0 5 5 - 0 . 1 0 0 rr h I''' i\ I' r**-'i i **f 1***7 \ " VV , 1 . , , , 1 0.5 5 1.0 0 1.5 5 2.0 0

Figuree 4.6: Comparison between results from (4.36) ( ) and the approximations (4.37)) ( ) and (4.42) ( ) for 6 p ^ x ) / B3 / 2 at v = 10.501 as a function of £,.

incorporatingg the dominant contribution of the Lerch functions in (4.37), so that one getss for small V :

B3/22 r i

óp^(x) )

4 \ / 2 7 r2V / 2s i nV 2 ^^ | ^ 3 3/2 2

cos[2-v(4>> + sinij>)]

++ n3/2*n/2 cos[2Mit> + sintl>) + re/4] I . (4.42)

AA numerical comparison of this expression with (4.36) shows that it indeed outper-formss (4.37) for "v just above a half-odd integer, as is seen in figure 4.6. As remarked at thee end of section 3.4, the singular behaviour near half-odd integer -v is associated with thee de Haas-van Alphen effect.

Returningg to the expression (4.37), which holds for u not in the vicinity of a Landau level,, we see that it can be simplified further if £, is so small that the expressions for \\> andd sin i|) can be replaced by the first few terms in their power series expansion around ££ = 0. In this way, one finds

spuM M

V~" "

Isfln Isfln

2v2 2 cos s

2 ^ x 1 1 1

2^2 2

B^x x

48u u

++

....

(4.43) )

Thiss approximation is valid for x in the range l / \ / B <C X <C y/\i-/B, so that x is large comparedd to the Fermi wavelength 1/^/jL The cosine can be expanded if x is limited

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stilll further to the range 1/VË « X « H1 / 6

/ B2 / 3 «C yfii/B. In that case one finds:

a a

6 f v WW W

ijhzH* cos{2^L x) + ^ s i n(2V ^ x) + . . . . (4.44) Accordingg to (3.52) the current density in this regime is

jWi|l(x)) « - - ^ - c o s ( 2 V ^ x ) (4.45)

4v27t/x x

inn leading order of B. These formulas describe the behaviour of the excess particle densityy and the current density for l / \ / B < x < u V6/ B2 / 3, in systems with \i » B. Ass we see, the dependence on the distance to the wall shows an algebraic decay in this regime,, with an oscillatory pre-factor. This type of decay, which agrees with that found fromm perturbation theory for small B [42, 27], is much slower than the Gaussian decay inn the far region x » y i l / B , which cannot be obtained from perturbation theory. The validityy of an algebraic decay in the far region was questioned before [46]. A numerical comparisonn of (4.44) with (4.36) and (4.37) is made in figure 4.5, which shows that (4.44)) is useful for small t, only, as expected.

Finallyy we make a comparison of the asymptotic expansion in terms of Lerch functions forr the density profile (4.39) with numerical results based on (2.20). The results are plottedd in figure 4.7. It is clear that (4.39) reproduces the essential features of the exact resultt such as the oscillation frequency and overall (algebraic) decay. Nevertheless the differencee between the two curves is considerably larger than in the case where the magneticc field is perpendicular to the wall (see figure 4.3).

44 A Appendix: Properties o f J f vv, u)

Thee function J(w,u) is defined as

i ( w

'

u )

=r

d y

v^)

e _ w u ï

--

(4

-

46)

Thiss integral representation is valid for |argwu| < TT/2 and |argu| < n. In (4.32), bothh the variables w u and u encircle the origin repeatedly as t varies. In fact, for t e

(-27T,2TI)) the arguments of both these variables are in the interval (-7t,7t). When t passess through {2m + 1 )27i, for integer m, both w u and u cross the negative real axis in thee clockwise direction. In contrast, the variable w in (4.32) has a simpler behaviour:

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CO O

£ £

Q. .

Figuree 4.7: Numerical evaluation of 6 p ^ ( x ) / B3 / 2 for y = 10: exact ( ) and our asymptoticc approximation (4.36) ( ).

itt stays in the right half-plane for all values oft. Hence, it is more convenient to switch too an integral representation for J(w,u) in which the variables w and u are separated. Thiss representation is found by considering (4.46) for real and positive u and w and rescalingg y by a factor u. In this way one gets the representation

J(w,u) )

r r

dy y // i + y

y(yy + u )

-wy -wy _{(4.47) )}

forr which an analytical continuation to any w and u with |arg w| < 7t/2 and |arg u| < n iss obtained trivially. Hence, in this new form further analytical continuation is neces-saryy for the variable u only. That continuation is easily carried out by evaluating the discontinuityy across the cut for negative u. Writing J ( w , um) for the analytical

contin-uationn of J(w, u) that follows by letting u encircle the origin m times in the clockwise directionn (corresponding to t e ((2m — 1 )27t, (2m + 1 )2n) in (4.32)), one finds:

I ( w , um) ) dyy A i + y y e"

w yy

+ 2im dy y 1 1

yd d

_Uj y _gw u y y _{(4.48) )}

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thee contour followed by u in (4.32) avoids this cut, as it crosses the positive real axis betweenn 0 and 1.

Havingg performed the analytical continuation of the integral J(w,u), we can now de-terminee its asymptotic behaviour for large -v (and finite £). For these values of v and

tt the real part of w is large, whereas |u| is finite. Hence, the first integral in (4.48) is

approximatelyy equal to b t / ( w u ) ]1 / 2. In the second integral in (4.48), the dominant contributionss come from the end points of the integration interval, since |wu| is large. Evaluatingg the end-point contributions one finds an asymptotic expression that de-pendss on the location of w u in the complex plane (with a cut at the negative real axis). Inn the right half-plane one finds VI —u exp(wu) y/n/{w\i). In the left half-plane thee result depends on the quadrant: in the second quadrant, with a r g w u 6 (7t/2,7t), thee result is iy/n/{wu), and in the third quadrant, with a r g w u e [nt3n/2) one gets

—iy/n/{wu).. Finally, the asymptotic expressions along the imaginary axis follow by addingg the limits of the expressions valid in the neighbouring quadrants. Collecting thee results, we have found the following asymptotic expression for the analytic contin-uationn of J(w,u) for large Rew:

J ( w , um)) « J^£ [(1 T 2m) + 2imVT^u" ew u] (4.49)

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