1

Solutions of Maxwell Equation for a Lattice System with Meissner

Effect

Qiang LI Jinheng Law Firm, Beijing, China

(Revised and posted on Slideshare on April 10, 2015. ) We show that Maxwell equation of a lattice system may have Meissner effect solutions when all carriers are surface state electrons. Some limitations on the wave function distributions of electrons in the system are identified.

PACS numbers: 73.20.-r, 75.10.-b, 05.50.+q

Introduction

London Theory is an explanation to Meissner effect [1] [2]. But the derivation of London Theory had its problems, namely:

1) In the current derivations of London Theory and penetration depth, Λ was treated as a constant independent of spatial coordinates r . Such treatment is unreasonable. And

2) Problem with the value of Λ . Since sn (number density of carriers) might vary from 1 to

such as 2310 , the London penetration depth as2c

4πΛ would vary accordingly, from as small as

810 m−∼ at 2310sn = to as big as 210 m∼ when sn corresponds to a few carriers. Core Equation

Here, we propose an alternative approach to derive London penetration depth. Our derivation starts from Maxwell equation:

4 (5 1)cπ∇ × = −B j ,

and we assume the limitation “all carriers are electrons of surface states”; on the other hand, we do not use any limitation like that of London equation; instead, we use existing current expression of probability flow, according to which the ith electron, with its wave function iφ , has a current contribution of [3]:

22* *

i i i i i i( )2mi mc

e eφ φ φ φ φ= − ∇ − ∇ −j A .

Then, the sum of current of all electrons is the total current: 2

2* *i i i i i

i[ ( ) ] (5 2)

2mi mce eφ φ φ φ φ= − ∇ − ∇ − −∑j A

We consider a half-infinite superconductor, as Kittel did [4], with boundary being at z 0= and the superconductor is at the positive side of the z axis. For surface state electrons there is

2 22 zi ie (x, y)uαφ −= [5], and for non-surface state electrons there is

2 2i i (x, y,z)uφ = . In a

sample there are N electrons, of which N’ are surface state electrons. For reasons that would later become clear, the summary of surface and non-surface state electrons are separated, as

N ' N N ' 22 22 z 2 zi i j t

i i j

e (x, y) (x, y,z) e U U (5 3)u uα αφ−

− −→ + = + −∑ ∑ ∑ ,

whereN '

2i

i(x, y) Uu =∑ is independent of z, and

N N ' 2

j tj

(x, y,z) Uu−

=∑ is of non-surface state

electrons.

2

Multiplying the two sides of4cπ∇ × =B j by 2 ze α and taking curl, the left side becomes:

2 z 2 z 2 z 2 2 z 2 z 23(e ) ( e ) ( ) e (e 2 ) ( ) eα α α α αα∇ × ∇ × = ∇ × ∇ × − ∇ = × ∇ × − ∇B B B e B B .

When all carriers are surface state electrons, the right side becomes: 2

22 z 2 z * *i i i i i i

i iN '

2 * *i i i i i i

i2 2

2 z 2 zt t

e e ( ( ) )2mi mc

[2i( ) ( )]2mi

[ (U e U )] (U e U )( )mc mc

e e

e u + u u u u

e e

α α

α α

φ φ φ φ φ∇× = ∇ × − ∇ − ∇ −

= − ∇ × ∇ × ∇ − ∇ −

∇ + × − + ∇ ×

∑ ∑

∑

j A

k

A A

.

Thus, we obtain the core equation: 2

2 z 2 z 2 z 2t 3

2N '2 * * 2 z

i i i i i i ti

c c(U e U ) (e 2 ) ( ) emc 4 4

{ [( ) 2i ( )]} [ (U e U )] (5 10).2mi mc

e +

e eu + u u u u

α α α

α

απ π

+ × ∇ × − ∇

= − ∇ × ∇ × ∇ − ∇ − ∇ + × −∑

B e B B

k A

( 1e , 2e and 3e are unit vectors in x, y, and z directions respectively.) As such, we correlate Schrödinger's wave functions (5 3)− of electrons in a lattice to Maxwell equation (5 1)− using the relation of (5 2)− .

Series solution

With 2 yB=B e , if we only consider the surface state electrons, the y component equation of

(5 10)− would become: 2

2 z 2U c BB e (2 B) 0 (5 11)mc 4 ze α α

π∂− + ∇ = −∂

,

where 2 ze α acts as a conversion factor, so that when the solution has the form of:

2n zn

n 1B B (0)e (5 12)α

∞−

== −∑ ,

the 2 ze α factor converts the power factor 2(n 1) ze α− + of the last two terms of (5 11)− to 2n ze α− , leading to:

22 2

n n 1U cB (0) B (0)( 2 2 (n 1) 4(n 1) ) 0 (5 13)

mc 4e α α α

π +− − + + + = − , or

2

n 1 n2 2

UB (0) B (0) (5 14)mc n(n 1)

eπα+ = −

+

Unless U is too big, solution (5 14)− always converges very quickly (if non-surface state electrons are considered, convergence of solution would not be affected, as can be seen later.) Accordingly, there would be:

2n znx

n 1

B (0)A A e (5 15)

2nα

α

∞−

== = − −∑ .

Solutions of the core equation with respect to all electrons in the system Since in the current expression of (5 2)− all electrons would have their current contribution

of2

2imc

e φ A , the situation should be considered where the summations in (5 10)− include all

3

electrons in the system. We still assume solutions as: 2n z

y y,n ynn 1

B Y (y)b e (5 20)α−

=

= −∑ ,

y,n 2n zx yn

n 1

Y (y)A b e (5 20 ')

2nα

α−

== − −∑ ,

y zA A 0= = , 2n z

y,n2n z xz z,n z,n yn

n 1 n 1

Y (y)A eB Y (y)b e b (5 21)y 2n y

αα

α

−−

= =

∂∂= = − = −

∂ ∂∑ ∑ , and

yn y,nz,n

z,n

b Y (y)1b (5 21')2n Y (y) yα

∂= −

∂.

( ynb and z,nb may seem unnecessary, as they are constants; but they can have corresponding signs, so they are retained.) When all carrier electrons are surface state electrons, the component equations of (5 10)− in y and z directions respectively become:

2y2 z 2 z 2 z 2z

t y 3 1 y

2y,n2 z 2n zt

t ynn 1

Bc B c(U e U )B (e 2 ) ( ) e Bmc 4 y z 4

Y (y)Ue [2 U ] b e (5 33 1)

mc z 2n

e +

e +

α α α

α α

απ π

αα

−

=

∂∂+ × − − ∇∂ ∂

∂= − − − −

∂ ∑

e e

and 2

2 z 2 z 2t z z

N ' N '2 * *

i i i i i i 3i i

22 z

t x

c(U e U )B e Bmc 4

{ [( ) 2i ] [ ( )]}2mi

[ (U e U )]A (5 33 2).mc y

e

e u + u u u u

e

α α

α

π+ − ∇

= − ∇ × ∇ × ∇ − ∇ −

∂− + − −∂

∑ ∑k ei

With (5 21)− , the relation of all 2 z 0e α− × terms in (5 33 2)− − becomes: 2N ' N'

2 y,1* * ti i i i i i 3 y1

i i

Y (y) U{ [( ) 2i ] [ ( )]} b (5 34),2mi mc 2 y

e eu + u u u uα

∂− ∇ × ∇× ∇ − ∇ = −∂∑ ∑ ik e

which gives N' N'

2 * *t i i i i i i 3

i iy1 y,1

cU { [( ) 2i ] [ ( )]} dy (5 35)ib Y

u + u u u ue

α= − ∇ × ∇× ∇ − ∇ −∑ ∑∫ ik e .

Substituting (5 21)− into (5 33 2)− − , the coefficient relation of 2 ze α− terms is: 2 4 z 4 z

y,2 y,22 z 2 z 2t y2 y2

2y,1 y,22 z 2 z 4 z t

y1 y2

Y (y) Y (y)e c e[ (e U b ) e b ]mc 4 y 4 4 y

Y (y) Y (y)U U[( )b e e ( b )e ],mc 2 y 4 y

e

e

α αα α

α α α

α π α

α α

− −

− −

∂ ∂− ∇

∂ ∂∂ ∂= − +∂ ∂

which leads to (by integrating with respect to y): 2 2

2ty2 y,2 y1 y,1 y2 y,22 2

U mc 1b Y (y) b Y (y)U b ( 8 )Y (y) C (5 33 2')2 4 2 ye

απ

∂+ − + = − −∂

In (5 33 1)− − , the coefficient relation of 2 z 0e α− × terms is:

4

22 z 2 z 2 z 2 2 z 2 z 2 z 2

t y,1 y1 y,1 y1 y1 y,1

2y,12 z 2 z

t t y1

c c[ (e U )Y b e (e 4 Y b e ) e b e (4 )Ymc 4 4

Ye [2 U (U )]( b )e ,

mc z 2

e +

e +

α α α α α α

α α

α απ π

αα

− − −

−

−

∂=∂

which gives: N 2 t

jj

U( ) 0 (5 36)z z

u∂ ∂= = −∂ ∂∑ .

Generally, the coefficient relations of 2n ze α− terms as determined by (5 33 1)− − and (5 33 2)− − respectively are:

2

y,n y,n y,n 1 t y,n 1

22

y,n 1 y,n 12

n[b UY (y) b U Y (y)]mc n 1c nb [ 4 n(n 1)]Y (y) 0 (5 40 1)

4 n 1 y

e

απ

+ +

+ +

+ −+

∂ + + = − −+ ∂

,

and 2

y,n y,n y,n 1 t y,n 1

22

y,n 1 y,n 12

n[b UY (y) b U Y (y)]mc n 1c nb [ 4 n(n 1)]Y (y) C (5 40 2).

4 n 1 y

e

απ

+ +

+ +

+ −+

∂ + + = − −+ ∂

Obviously, (5 40 2)− − and (5 40 1)− − are exactly the same when integral constant C 0= ; thus, y,n 1Y (y)+ can be determined from y,nY (y) according to (5 40 1)− − , and solution of

Meissner effect in the form of (5 20)− is ensured. As such, we have shown that, when all carriers are surface state electrons, the core equation (5 10)− (and Maxwell equation (5 1)− ) may have Meissner effect solutions (5 20)− , (5 20')− , (5 21)− and (5 40 1)− − if the electron wave functions are as limited by (5 35)− , (5 36)− and (5 39)− . The limitation “all carriers are surface state electrons” is somewhat surprising. On the other hand, if the current expression (5 2)− is true, the wave function of a carrier electron has to be of the type of that of a surface state electron in order to form a current suitable for Meissner effect. As such, we have provided a possible explanation to the mechanism of Meissner effect. In addition, the result of (5 40 1)− − may relate to some possible explanation to the origin of stripe phase [6]. [1] Meissner, W.; R. Ochsenfeld (1933). "Ein neuer Effekt bei Eintritt der Supraleitfähigkeit".

Naturwissenschaften 21 (44): 787–788. [2] London, F.; London, H. (1935). "The Electromagnetic Equations of the Supraconductor".

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 149 (866): 71.

[3] Michael Tinkham: Introduction to Superconductivity, Second Edition, McGraw-Hill, Inc., 1996, sec. 1.5.

[4] Kittel, Charles (2004). Introduction to Solid State Physics. John Wiley & Sons. pp. 275. [5] Sidney G. Davison, Maria Steslicka (1992). Basic Theory of Surface States. Clarendon Press.

ISBN 0-19-851990-7. [6] Emery, V. J.; Kivelson, S. A.; Tranquada, J. M., Stripe Phases in High-Temperature

Superconductors, Proceedings of the National Academy of Sciences of the United States of America, Volume 96, Issue 16, pp. 8814-8817.

5