Can Anyon statistics explain high temperature superconductivity?

0.11 MB·

by  Ahmad Adel Abutaleb

7 Can Anyon statistics explain high temperature
1
0 superconductivity?
2

n
a Ahmad Adel Abutaleb
J
Department of Mathematics, Faculty of science,

7
University of Mansoura, Elmansoura 35516, Egypt
2

] February 3, 2017
h
p
-
n Abstract
e
g In this paper, we find a reasonable explanation of high temperature
. superconductivity phenomenausing Anyonstatistics.
s
c
i
s 1 Introduction
y
h
p Superconductivity is a phenomena occurring in certain materials when cooled
[ below some characteristic temperature called a critical temperature. In a su-

1 perconductor,the resistancedropssuddenlytozerowhenthe materialiscooled
v belowitscriticaltemperature. Thisphenomenawasfirstdiscoveredbythedutch
2 physicistHeike KamerlinghOnnes,whichreceivedthe Nobelprize ofphysicsin
9
1913 for discovering this phenomena and the related work regarding liquefac-
6
tionofhelium[1]. Thereisanotherphenomenarelatedtosuperconductorcalled
0
0 (Meissner effect), which is the expulsion of a magnetic field from a supercon-
. ductor during its transitionto the superconducting state. This phenomena was
2
discovered in 1933 by Walther Meissner and Robert Ochsenfeld [2]. Although
0
7 themicroscopictheoryofsuperconductivitywasdevelopedbyBardeen,Cooper
1 andSchrieffer (BCStheory)in 1957[3], other descriptionsofthe phenomena of
: the superconductivity were developed [4-6].
v
i Whereasordinarysuperconductorsusuallyhavetransitiontemperaturesbe-
X
low 30 Kelvin, It is well known that some materials behave as superconductors
r at unusually high critical temperatures and this materials called high tempera-
a
turesuperconductorsor(HTS). The firstHTS wasdiscoveredin1986byGeorg
BednorzandK.AlexMuller,whowereawardedthe1987NobelPrizeinPhysics
[7].
Although the standard BCS theory (weak-coupling electron–phonon inter-
action) can explain all known low temperature superconductors and even some
high temperature superconductors, some of high temperature superconductors
cannot be explained by BCS theory and some physicist believe that they obey
a strong interaction.
1
Inthispaper,wefindthatifweuseaunifiedstatistics[8]insteadFermi-Dirac
statistics, then we can find a reasonable explanation of all high temperature
superconductivity phenomena using a standard BCS theory.
2 The model
We follow the derivation of Hamiltonian model using the mean field approxi-
mationandBogoliubovunitarytransformationasinthe reference[9],wherewe
can write the following relation,
∆
∆ = V | | [f( E ) f(E )], (1)
K K
| | | | 2 (ǫ(k) µ)2+∆2 − −
Xk −
p
where f(E ) determined from the unified statistics as follow [8],
K
(ǫ(k) µ)2+∆2
1 +1 1−p 1 1 p =ep k−BT . (2)
(cid:18)(1 p)f(E ) (cid:19) (cid:18)pf(E ) − (cid:19)
k k
−
3 BCS Theory ( p 1)
→
For p 1, we recover the usual BCS Theory.
→
When p 1,we have the following usualform ofthe Fermi function f(E ),
K
→
1
f(E )= , (3)
K
(ǫ µ)2+∆2
−
1+ep kBT
so, we have,
1  1 1 
1= V .
| |Xk Z 2 (ǫ(k)−µ)2+∆2  − (ǫ(k)−µ)2+∆2 − (ǫ(k)−µ)2+∆2 
p 1+e p kBT 1+ep kBT 
(4)
Replacing the summation by integration over the energy states, we have,
µ+ǫD N(0)dǫ  1 1 
1= V .
| |Zµ−ǫD 2 (ǫ−µ)2+∆2  − (ǫ−µ)2+∆2 − (ǫ−µ)2+∆2 
p 1+e p kBT 1+ep kBT 
(5)
2
Let ζ =ǫ µ, then we have the master equation for BCS theory as,
−
1 ǫD dζ  1 1 
= . (6)
|V|N(0) Z0 ζ2+∆2  − ζ2+∆2 − ζ2+∆2 
p 1+e pkBT 1+epkBT 
Determine T .
C
At ∆=0, we have T =T , so we can write,
C
1 ǫD dζ  1 1 
= . (7)
V N(0) Z ζ ζ − ζ
| | 0  − 
1+ekBTC 1+ekBTC 
ζ
Define =x, we can write,
k T
B C
ǫ
D
1 K T dx
= B C φ(x), (8)
V N(0) Z x
| | 0
1 1
where φ(x)= .
(cid:20)1+e−x − 1+ex(cid:21)
Using integration by parts we have,
ǫ ǫ ǫ
D D D
K T dx K T K T dφ(x)
B C φ(x)=[φ(x)lnx] B C B C lnxdx. (9)
Z x 0 −Z dx
0 0
Foraweakcouplinglimit,i.e. 1 ⋗⋗1,k T ⋖⋖ǫ , ∆ ⋖⋖k T ,we
n|V|N0 B C D | 0| B Co
have,
ǫ
D
[φ(x)lnx]KBTC = ln ǫD , (10)
0 K T
B C
ǫ
D
K T dφ(x) ∞ dφ(x)
B C lnxdx lnxdx 0,12563. (11)
Z dx ≈ Z dx ≈−
0 0
So, we have
1 ǫ
=ln 1,134 D . (12)
V N(0) (cid:18) K T (cid:19)
B C
| |
Wecanrewritethelastequationtoobtainthewellknownformulaforcritical
temperature for BCS Theory as,
3
1
1,134ǫD V −N(0)
T = e . (13)
C | |
K
B
Determine ∆ .
0
At T =0, we have ∆=∆ , so from the equation we have,
0
1 ǫD dζ ǫ
= =sinh−1 D , (14)
|V|N(0) Z0 ζ2+∆20 (cid:18)∆0(cid:19)
q
so we have,
ǫ 1
D
=sinh . (15)
∆ (cid:18) V N(0)(cid:19)
0
| |
1
Again, for a weak coupling limit, we have sinh 1 1e V N(0), so
(cid:16)|V|N(0)(cid:17) ≈ 2 | |
we can write the BCS formula of ∆ as follow,
0
ǫ 1
D 1
= e|V|N(0). (16)
∆ 2
0
From equations (13) and (16), we arrive at the universal BCS formula,
∆ =1.76K T . (17)
0 B C
Using equations (16) and (17) in the master equation (6), we can plot the
T ∆
universal curve y =y(x), where x= ,y = as follow,
T ∆
C 0
sinh 1
|V|N(0)
1 y dz  1 1 
= .
V N(0) Z √z2+1 1.76y√z2+1 − 1.76y√z2+1
| | 0  − 
1+e x 1+e x 
(18)
4 Anyon superconductivity
Fromequations(1)and(2),wecanwritethegeneralformofthemasterequation
for any value of p as follow,
ǫD
1 dζ 1
= φ( ) φ(r(ζ)) , (19)
V N(0) Z ζ2+∆2 (cid:20) r(ζ) − (cid:21)
| |
0
p
4
where,
ζ2+∆2
pK T
r(ζ)=e B , (20)
and φ(r) determined from the following equation,
1−p p
1 1
+1 1 =r. (21)
(cid:18)(1 p)φ(r) (cid:19) (cid:18)pφ(r) − (cid:19)
−
4.1 Determine TC for any value of p
At ∆=0, we have T =T and then from equations (19-21) we can write,
C
ǫ
D
K T
B C
1 dx 1
= φ( ) φ(r(x)) , (22)
V N(0) Z x (cid:20) r(x) − (cid:21)
| |
0
r(x) = ex, (23)
and again, φ(r) determined from the equation (21).
4.2 Determine ∆0 for any value of p
AtT =0,wehave∆=∆ andfromequation(20),wehaver = ,soequation
0
∞
(21) takes the form,
1 1−p 1 p
+1 1 = . (24)
(cid:18)(1 p)φ(r) (cid:19) (cid:18)pφ(r) − (cid:19) ∞
−
Then for any value of p,we have
φ(r)=0 at T =0. (25)
Similarly, we can evaluate φ 1 by solving the following equation,
r
(cid:0) (cid:1)
1−p p
 1   1  1
+1 1 = . (26)
1 1 − r
(1 p)φ  pφ 
 − (cid:18)r(cid:19)   (cid:18)r(cid:19) 
   
1
So at T =0, We have r = , =0 and then equation (26) takes the form,
∞ r
1−p p
 1   1 
+1 1 =0 (27)
1 1 −
(1 p)φ  pφ 
 − (cid:18)r(cid:19)   (cid:18)r(cid:19) 
   
5
The solution of (26) (for positive φ(1)) takes the form,
r
1 1
φ( )= at T =0. (28)
r p
Using (25) and (28) in the master equation (19), we have at T = 0 the
following equation,
ǫD
p dζ ǫ
= =sinh−1 D, (29)
|V|N(0) Z0 ζ2+∆20 ∆0
q
or in the equivalent form,
p
ǫ =∆ sinh . (30)
D 0 V N(0)
| |
At weak coupling limit, from the equation (30), we have the following uni-
versal relation for ∆ at any value of p as,
0
p
−
V N(0)
∆ =2ǫ e . (31)
0 D | |
In the next section,we will study some specialcases ofAnyonsuperconduc-
tivity for some special values of p.
5 Special cases of Anyon superconductivity
5.1 p = 1 Anyon
2
From the unified statistics (equation(2)), we can find the formula of f(E ) as
k
follow,
2
f(E )= . (32)
k 1
2 (ǫ µ)2+∆2 2
−
1+e p kBT 
 
 
So, after some calculations, we arrive at the master equation for the case
p= 1 as follow,
2
6
 
 
 
1 ǫD dζ  1 1 
=  .
2|V|N(0) Z0 ζ2+∆2  1 − 1
p  2 ζ2+∆2 2 2 ζ2+∆2 2
 − 
1+e pkBT  1+e pkBT  
 
    
    
    
(33)
Determine T
C
At ∆=0, we have T =T , so from equations (32,33) we can write,
C
ǫ
D
1 K T dx 1 1 
= B C . (34)
2 V N(0) Z x 1 − 1
| | 0  
(1+e−2x)2 (1+e2x)2
 
 1 1 
Define φ(x)= , we can calculate the last inte-
1 − 1
 
(1+e−2x)2 (1+e2x)2
 
gral as,
ǫ ǫ
D D
KBTC dxφ(x) [φ(x)lnx]KBTC ∞dxdφ(x)lnx=ln ǫD +0.4228=ln1.535ǫD.
Z0 x ≈ 0 −Z0 x KBTC KBTC
(35)
So,we can write the equation(34) as,
1 1.535ǫ
D
=ln , (36)
2 V N(0) K T
B C
| |
or on the equivalent form,
1
1.535ǫD 2 V−N(0)
T = e . (37)
C | |
K
B
Comparing equation (37), the criticaltemperature for the Anyon case (P =
1) and equation (13), the critical temperature for BCS theory (at the same
2
value of Debye energy) we have,
7
1
T
Cp=21 =1.35e2 V N(0). (38)
| |
T
CBCS
1
So, for some weak coupling limit, say =5, we have
V N(0)
| |
T 16,446T . (39)
Cp=21 ≃ CBCS
So, if the critical temperature in the standard BSC is equal to 1 Kelvin
1
(with = 5), then using a unified statistics with P = 1, the critical
V N(0) 2
temper|at|ure jumps to 16,446 Kelvin.
Notice that although the weak coupling is the reason of why we cannot
explainsome HTSusing the standardBCStheory,inthe Anyonicstatistics the
critical temperature is increased whenever the coupling limit is decreased!!
1
Asanexample,considerthatwehavemoreweakinteraction(say =
V N(0)
| |
10), then we have,
T 201T !! (40)
Cp=12 ≃ CBCS
So at the same condition, if the critical temperature in the standardBSC is
1
eual to 1 Kelvin then for a weak coupling =5 we see that the critical
V N(0)
| | 1
temperaturejumpsto16,446Kelvinandformoreweakcoupling =10
V N(0)
| |
the critical temperature jumps to 201 Kelvin!
Determine ∆
0
At T =0, we have ∆=∆ , so from the equation (29) we have,
0
1 ǫD dζ ǫ
= =sinh−1 D , (41)
2|V|N(0) Z0 ζ2+∆20 (cid:18)∆0(cid:19)
q
so we have,
1
−
2 V N(0)
∆ =2ǫ e .. (42)
0 D | |
From (24) and (28), we have
∆ =1.3K T . (43)
0 B C
T ∆
Finally, we can plot the universal curve y = y(x), where x = ,y =
T ∆
C 0
for p= 1 as follow,
2
8
 
sinh2|V|1N(0)  
 
1 y dz  1 1 
=  .
2 V N(0) Z √z2+1 1 − 1
| | 0  
 2.6y√z2+1 2 2.6y√z2+1 2
 − 
1+e x  1+e x  
 
 
    
    
(44)
5.2 p = 1 Anyon superconductor
3
For p= 1, we can determine φ(r) from equation (21) as,
3
3
φ(r)= . (45)
1 1
−
(2r3+2√r6+r3+1)3 +(2r3+2√r6+r3+1) 3 1
−
Fromequations(22-23)andaftersomecalculations,wecanfindtheformula
of T for the Anyon p= 1 as,
C 3
1
−
32.627 3 V N(0)
T = ǫ e . (46)
C D | |
K
B
Also, from the general equation (31) we have,
1
−
3 V N(0)
∆ =2ǫ e . (47)
0 D | |
Conclusion 1 Inthispaper,wecalculatedthecriticaltemperatureofsomemet-
als using the standard BCS theory but assuming that the particles inside this
metals obey Anyonic statistics instead Fermi-Dirac statistics. We find that the
lower of the interaction we have, the greater of the critical temperature which
maybeareasonableexplanationofsomehightemperaturesuperconductorswhich
the standard BCS theory cannot explain it.
Acknowledgement 2 I want tothank E. Ahmad for his advice and very useful
discussion.
6 References
[1] Simon Reif-Acherman, ”Heike Kamerlingh Onnes: Master of Experimental
TechniqueandQuantitativeResearch”,Physics in Perspective, vol. 6,pp. 197–
223, (2004).
9
[2]W.Meissner,R.Ochsenfeld,”EinneuerEffektbeiEintrittderSupraleit-
fahigkeit”, Naturwissenschaften, vol. 21, pp. 787–788,(1933).
[3] J. Bardeen, L. N. Cooper, J. R. Schrieffer, ”Microscopic Theory of Su-
perconductivity”, Physical Review, vol. 106, (1957).
[4] C. S. Gorter, H.Casimir, ”On superconductivity”, Zeitschrift fu¨r Tech-
nische Physik , vol. 15, (1934).
[5]H.LondonandF.London,”TheElectromagneticEquationsoftheSupra-
conductor”, Proceedings of the Royal Society A, vol.149, (1935).
[6] V. L. Ginzburg, L. D. Landau, Soviet Physics JETP 20, vol. 1064,
(1950).
[7]J.G.Bednorz,K.A.Muller, ”PossiblehighTC superconductivityin the
Ba-La-Cu-O system”, Zeitschrift fu¨r Physik B, vol. 64, pp. 189–193,(1986).
[8]A.A.Abutaleb,”Unifiedstatisticaldistributionofquantumparticlesand
Symmetry”, Int. J. Theor. Phys, vol. 53, pp. 3893-3900,(2014).
[9]http://www-personal.umich.edu/˜sunkai/teaching/Fall 2012/phys620.html.
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