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.

1. 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
23
10 , the London penetration depth as
2
c
4π
Λ would vary accordingly, from as small as
8
10 m−
∼ at 23
10sn = to as big as 2
10 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)
c
π
∇× = −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]:
2
2* *
i i i i i i( )
2mi mc
e e
φ 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 mc
e e
φ 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 z
i ie (x,y)uα
φ −
= [5], and for non-surface state electrons there is
2 2
i 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 z
i i j t
i i j
e (x,y) (x,y,z) e U U (5 3)u uα α
φ
−
− −
→ + = + −∑ ∑ ∑ ,
where
N '
2
i
i
(x,y) Uu =∑ is independent of z, and
N N ' 2
j t
j
(x,y,z) Uu
−
=∑ is of non-surface state
electrons.

2. 2
Multiplying the two sides of
4
c
π
∇ × =B j by 2 z
e α
and taking curl, the left side becomes:
2 z 2 z 2 z 2 2 z 2 z 2
3(e ) ( e ) ( ) e (e 2 ) ( ) eα α α α α
α∇ × ∇ × = ∇ × ∇ × − ∇ = × ∇ × − ∇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 i
N'
2 * *
i i i i i i
i
2 2
2 z 2 z
t 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 2
t 3
2N '
2 * * 2 z
i i i i i i t
i
c c
(U e U ) (e 2 ) ( ) e
mc 4 4
{ [( ) 2i ( )]} [ (U e U )] (5 10).
2mi mc
e
+
e e
u + u u u u
α α α
α
α
π 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 B
B e (2 B) 0 (5 11)
mc 4 z
e α
α
π
∂
− + ∇ = −
∂
,
where 2 z
e α
acts as a conversion factor, so that when the solution has the form of:
2n z
n
n 1
B B (0)e (5 12)α
∞
−
=
= −∑ ,
the
2 z
e α
factor converts the power factor 2(n 1) z
e α− +
of the last two terms of (5 11)− to 2n z
e α−
,
leading to:
2
2 2
n n 1
U c
B (0) B (0)( 2 2 (n 1) 4(n 1) ) 0 (5 13)
mc 4
e
α α α
π
+− − + + + = − , or
2
n 1 n2 2
U
B (0) B (0) (5 14)
mc n(n 1)
eπ
α
+ = −
+
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 zn
x
n 1
B (0)
A A e (5 15)
2n
α
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
of
2
2
i
mc
e
φ A, the situation should be considered where the summations in (5 10)− include all

3. 3
electrons in the system. We still assume solutions as:
2n z
y y,n yn
n 1
B Y (y)b e (5 20)α−
=
= −∑ ,
y,n 2n z
x yn
n 1
Y (y)
A b e (5 20')
2n
α
α
−
=
= − −∑ ,
y zA A 0= = ,
2n z
y,n2n z x
z z,n z,n yn
n 1 n 1
Y (y)A e
B Y (y)b e b (5 21)
y 2n y
α
α
α
−
−
= =
∂∂
= = − = −
∂ ∂
∑ ∑ , and
yn y,n
z,n
z,n
b Y (y)1
b (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:
2
y2 z 2 z 2 z 2z
t y 3 1 y
2
y,n2 z 2n zt
t yn
n 1
Bc B c
(U e U )B (e 2 ) ( ) e B
mc 4 y z 4
Y (y)U
e [2 U ] b e (5 33 1)
mc z 2n
e
+
e
+
α α α
α α
α
π π
α
α
−
=
∂∂
+ × − − ∇
∂ ∂
∂
= − − − −
∂
∑
e e
and
2
2 z 2 z 2
t z z
N ' N'
2 * *
i i i i i i 3
i i
2
2 z
t x
c
(U e U )B e B
mc 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 0
e α− ×
terms in (5 33 2)− − becomes:
2N' N'
2 y,1* * t
i i i i i i 3 y1
i i
Y (y) U
{ [( ) 2i ] [ ( )]} b (5 34),
2mi mc 2 y
e e
u + u u u u
α
∂
− ∇ × ∇× ∇ − ∇ = −
∂
∑ ∑ ik e
which gives
N' N'
2 * *
t i i i i i i 3
i iy1 y,1
c
U { [( ) 2i ] [ ( )]} dy (5 35)
ib Y
u + u u u u
e
α
= − ∇ × ∇× ∇ − ∇ −∑ ∑∫ ik e .
Substituting (5 21)− into (5 33 2)− − , the coefficient relation of 2 z
e α−
terms is:
2 4 z 4 z
y,2 y,22 z 2 z 2
t y2 y2
2
y,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 α
α α
α α α
α π α
α α
− −
− −
∂ ∂
− ∇
∂ ∂
∂ ∂
= − +
∂ ∂
which leads to (by integrating with respect to y):
2 2
2t
y2 y,2 y1 y,1 y2 y,22 2
U mc 1
b Y (y) b Y (y)U b ( 8 )Y (y) C (5 33 2')
2 4 2 ye
α
π
∂
+ − + = − −
∂
In (5 33 1)− − , the coefficient relation of 2 z 0
e α− ×
terms is:

4. 4
2
2 z 2 z 2 z 2 2 z 2 z 2 z 2
t y,1 y1 y,1 y1 y1 y,1
2
y,12 z 2 z
t t y1
c c
[ (e U )Y b e (e 4 Y b e ) e b e (4 )Y
mc 4 4
Y
e [2 U (U )]( b )e ,
mc z 2
e
+
e
+
α α α α α α
α α
α α
π π
α
α
− − −
−
−
∂
=
∂
which gives:
N 2
t
j
j
U
( ) 0 (5 36)
z z
u
∂ ∂
= = −
∂ ∂
∑ .
Generally, the coefficient relations of
2n z
e 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
2
2
y,n 1 y,n 12
n
[b UY (y) b U Y (y)]
mc n 1
c n
b [ 4 n(n 1)]Y (y) 0 (5 40 1)
4 n 1 y
e
α
π
+ +
+ +
+ −
+
∂
+ + = − −
+ ∂
,
and
2
y,n y,n y,n 1 t y,n 1
2
2
y,n 1 y,n 12
n
[b UY (y) b U Y (y)]
mc n 1
c n
b [ 4 n(n 1)]Y (y) C (5 40 2).
4 n 1 y
e
α
π
+ +
+ +
+ −
+
∂
+ + = − −
+ ∂
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. 5