A phenomenological theory of phase transitions in high T_{c} superconductors

0.13 MB·English

by  D. M. Sedrakian

Europhysics Letters PREPRINT
5 A phenomenological theory of phase transitions in high T su-
0 c
0 perconductors
2

n D. M. Sedrakian
a
J Yerevan State University, 1 Alex Manoogian Str., Yerevan, 375025, Armenia

1
3
PACS.nn.mm.xx – 74.25.-q.
]
n
o
c
r- Abstract. – Aphenomenologicaltheoryissuggestedtoexplaintheexperimentallydiscovered
p “paramagnetic peculiarity” and theunconventionalchangeof theenergetic state of a layersof
u highTc ofY-Ba-Cu-Ointhevicinityofthephasetransition. Thephysicalconditionsarefound
s
underwhichthesepeculiaritiesarerevealed. Itisshownthatthesuggestedtheoryqualitatively
.
at describes theexperimental data.
m
-
d
n
Despite the substantial success in the investigations of the high T superconductors, the
o c
microscopic nature of the superconducting state in these materials has not been understood
c
[ to date [1]. In recent years promising attempts of experimental investigations of the su-

3 perconducting phase transition were made in high Tc Y-Ba-Cu-O [2,3]. In this work a
new “paramagnetic” peculiarity and unconventional change in the energetic state of layer of
v
2 high Tc superconductor in the vicinity of the phase transition is discovered. This success
1 was achieved due to the construction of a highly sensitive magnetometer, which permits to
3 measure verysmallchanges∆λ ofthe penetrationdepth ofthe magneticfield with frequency
1
of the order several MHz in a sample of a flat high T superconductor. The experimental
0 c
set-up permits to measure changes ∆λ of absolute magnitude of the order ∆λ 1 3 A
5
0 with relative accuracy ∆λ/λ 10−6 [2]. The first of these peculiarities shows a∼n in−crease
∼
/ of the penetration depth of the order of few micrometers when the temperature is decreased
t
a in the vicinity of the phase transition, prior to its decrease from the value of δ of the order
m
of hundreds of micrometers up to the London penetration depth λ of the order of a few mi-
- crometer [2]. The second peculiarity is manifested in an increase of the energy of the sample
d
with decreasing temperature prior to the well-known decrease due to the transition to the
n
o superconducting state [3].
c The aim of this work is to show within the framework of a phenomenological theory of
v: superconductivity that peculiar dependence of the the penetration depth on the temperature
i can be explained within an approach, which suggests a specific behavior of the Cooper pairs.
X
We will show as well, that the account of the Coulomb interaction in a solid-state plasma,
r which is composed of normal ions a and normal and superconducting electrons, can explain
a
the observedabsorptionoftheenergyofthe electromagneticfieldduringthe superconducting
phase transition [3].
Section 2 derives an expression for the penetration depth λ and the conditions are obtain
under which the maximum of the λ can be at T =85.4 K, as observed in ref. [2]. In Section
(cid:13)c EDPSciences
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EUROPHYSICSLETTERS
3the electrostaticpotentialis foundfor achargedparticle Zewith accountofthe screeningof
the surrounding plasma. In the closing Section 4 the change in the free energy of the sample
with decreasing temperature is computed. Here we show that the energy of the sample as a
function of temperature shows a maximum and only beyond the maximum one observes the
familiar temperature dependence of the difference in the free-energies of the superconducting
and normal phases.
ThepenetrationdepthoftheelectromagneticfieldinahighT superconductor. – Suppose
c
that in the superconductor the electric current is a sum of the normal and superconducting
currents,~j and~j . These currents are relatedto an external electric field E exp( iωt) by
n s
∼ −
the following relations
e2τ n
~j = n E~, (1)
n
m 1 iωτ
e2n −
~j = i sαE~, (2)
s
mω
where n and n are the densities of the normal and superconducting electrons, respectively,
n s
τ is the mean-free-flight time of a normal electron, e and m are the charge and the mass of
electron and α is the fraction of the Cooper pairs which are involved in the supercurrent.
Our main assumption is that when T < T not all the Cooper pairs participate in the
c
current, i.e. α < 1, and with a decrease of the temperature, α increases and, already for
the temperatures which correspond to the fast decrease of the penetration depth, tends to
unity. Sucha behaviorofα canbe understoodin the frame ofnonlocaltheoryofPippard[4],
according which in the well-known London’s connection Eq.(2) the coefficient
ξ 1
α= =
ξ ξ /l+1
0 0
haveto appear. Here ξ is the coherentlenghtofCooperpairsandl is the lenghtoffree path
0
of electrons. Near the phase transition T . T ξ l, hence α l/ξ 1. With decrease
c 0 0
≫ ≈ ≪
oftemperaturethe superconductingcorrelationsbetweenelectronsbringtothe decreaseofξ
0
and increase of l, hence ξ /l will become less than unity and α will tend to one. When α=1
0
the dependence of λ on T corresponds to the familiar dependence λ = λ(T), which follows
from the theory of the superconductivity.The experimental measurement of the temperature
dependence of the penetration depth λ(T) would allow to find form of the function α(T).
Upon substituting the net current~j =~j +~j in the Maxwell equation
s n
4π
curlB~ = ~j (3)
c
andactingwiththecurloperatoronbothsidesofEq. (3)andusinganotherMaxwellequation
1∂B~
curlE~ = , (4)
−c ∂t
we finally find
4πe2n iωτ n n
curl curlB~ = n α s B~. (5)
m2 1 iωτ n − n
c (cid:18) − (cid:19)
If we assume that the magnetic field depends only on the distance z from the surface of the
superconductor, then Eq. (5) within the bulk of a superconductor z >0 assumes the form
d2B~(z)
+k2B~(z)=0, (6)
dz2
D.M.Sedrakian:AphenomenologicaltheoryofphasetransitionsinhighT superconductors3
c
where
4πe2n iωτ n n
2 n s
k = α . (7)
m2 1 iωτ n − n
c (cid:18) − (cid:19)
The solutions of Eq. (6) are of the from
B~(z)=B~0eikz−iωt, (8)
i.e. the penetration depth of an electromagnetic field is
1
λ(T)= . (9)
Imk
Introducing the short hand notations
2 2 2 2 2 2 2
2 mc 2 mc 1+ω τ δ0 1+ω τ
λ0 = 4πe2n, δ0 = 2πe2n ωτ , Z0 = 2λ2 = ωτ , (10)
0
we find
n 2 n ω2τ2 n 2 n ω2τ2 n −1/2
n s n s n
λ=δ0"s(cid:16) n (cid:17) +Z0(cid:18)α n + 1+ω2τ2 n (cid:19) +Z0(cid:18)α n + 1+ω2τ2 n (cid:19)# . (11)
A compact form for the penetration depth (11) is obtain by introducing further abbreviation
n n ωτ
n s
=x, =1 x, αZ0 =Z, β =1 ; (12)
n n − − Z
we find
−1/2
λ(T)=δ0 x2+Z2(1 βx)2+Z(1 βx) . (13)
− −
hp i
Since ωτ 1 and the minimal value of Z is of the order of unity, we replace with high
≪
accuracy β by 1. Then Eq. (13) simplifies to the from
−1/2
λ(T)=δ0 x2+Z2(1 x)2+Z(1 x) . (14)
− −
hp i
As can be seem from Eq. (14), for T > T , when x = 1, the penetration depth λ is equal
c
the depth of the skin effect δ0. With the decreasing temperature, when Z 1, that λ(T)
≃
increases and passes through a maximum at T = T0 and becomes equal δ0 at temperature
T = T1. As follows from ref. [2], T0 =85.4 K, T1 =85 K, whereas Tc = 88.7 K. The relative
−3
change of the penetration depth at the maximum is ∆λ/δ0 5 10 . Assuming that in the
≃
temperature range T1 T Tc Z is constant, we find the maximal value of λ(T) from Eq.
≤ ≤
(14). Asimple calculationshowsthatifthe maximumofλ(T)islocatedatthe pointx0,then
this point is determined by the formula
1
x0(2 x0)= (15)
− 1+∆λ/δ0
(cid:20) (cid:21)
At the point were λ(x) is again equal δ0 the following relations hold
2
x1 =2Z 1, (16)
−
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EUROPHYSICSLETTERS
where
2 x0
Z = . (17)
2 x0
−
Substituting in Eq. (15) the value of ∆λ/δ0, we obtain from (15) and (17) x0 = 0.8586 and
Z = 0.8673. If, following the Gorter-Casimir theory, one assumes x0 = (T0/Tc)4, then from
the requirement that the maximum is at T0 = 85.4 K, we obtain the value of Tc = 88.7
K, which coincides with the value of Tc quoted in ref. [2]. If we require that T1 = 85 K,
then one needs to take the value Z = 0.92; this shows that there is an initial increase of Z
which accompanies the drastic decrease λ(T). The further decrease of λ(T) with decreasing
temperature,seenintheexperiment[2],canbeexplainedbytheincreaseinZ,i.e. anincrease
of α up to unity. A more detailed comparison between Eq. (14) and the results of ref. [2]
will be given in future.Of great interest is the derivation of the function α = α(T) from the
microscopic theory.
Electrostatic potential in the solid-state plasma at T 0. – As is well known from
→
the plasma theory, the electrostatic potential of a charged particle in plasma is screened
by the distribution of charged particles of opposite sign [5]. The Fourier-component of the
longitudinal component of the dielectric constant ε(0,k) when ω 0 for mixture of ions,
→
normal and superconducting electrons has the following form
1 1 1
ε (k,0)=1+ + + , (18)
l k2a2 k2a2 k2a2
i e s
where a , a and a are the Debye radii of ions, normal and superconducting electrons, re-
i e s
spectively. The radius a can be found from the formulae [6]
i
ν E
i
a = , ν = (19)
i i
Ω M
i r
where E is the mean kinetic energy,M is the mass ofthe ion,Ω is the ion plasma frequency,
i
4πn Z2e2 1/2
Ω = i i . (20)
i
M
(cid:18) (cid:19)
Here eZ andn arethe chargeandthe density ofions,respectively. FromEqs. (19) and(20)
i i
we obtain for a the following expression
i
E
i
a = (21)
i 4πe2n
r
where
n
E =E . (22)
i n Z2
i i
Consideringthe sample ofthe highT superconductoras asolid, atlowtemperatures forthe
c
mean kinetic energy of ion vibrations one can use the following expression
k T θ
B
E = D( ), (23)
2 T
whereD(θ)is the Debyefunction, θ is the Debye temperature. Inanalogousmanner onecan
T
find the Debye radius of the normal electrons
ε
F
a = , (24)
e 4πe2n
r n
D.M.Sedrakian:AphenomenologicaltheoryofphasetransitionsinhighT superconductors5
c
where ε is the electron Fermi-energy. The role of the Debye radius of the superconducting
F
electrons, as shown in [7], plays the London penetration depth λ, i.e.
mc2
a λ= , (25)
s ≡ s4πe2nsα
where n is the density of the superconducting electrons. If we define
s
1 1 1
−1
a = + + , (26)
sa2i a2e a2s
then the Fourier-componentof the electric statical potential φ will take the form [6]
k
eZi/ǫ0 eZi/ǫ0
φ = = , (27)
k k2ε (0,k) k2+a−2
l
where ǫ0 is the dielectric constant of the sample under consideration. Finally, for the electro-
statical potential of a charge particle eZ , we can write the following expression
i
φ (r)= φ ei~k·~r d~k = eZie−r/a, (28)
i k (2π)3 ǫ0r
Z
where r is the distance to the charge eZ , while a is determined from the Eqs. (21)-(26).
i
The Coulomb interaction energy of solid-state plasma. – The energy of the Coulomb
interaction between the charged particles in plasma is of the form [5]
1 ′
W = V Z en φ . (29)
2 i i i
i
X
′
Here n is the density of the charged particles, φ is the potential created by all charged
i i
particles at the point r = 0, except the field of the particle with the charge eZ , located at
i
that point and V is the volume. Expand the potential φ (r), Eq. (28), in the vicinity of the
i
point r =0:
eZ r eZ eZ
i i i
φ (r)= 1 = . (30)
i ǫ0r − a ǫ0r − ǫ0a
′ (cid:16) (cid:17)
Consequently,φ canbeidentifiedwiththesecondtermontheleft-hand-sideoftheexpression
i
(30). Substituting this expression in Eq. (29), we obtain the energy per unit volume
W e2
2
ε = = Z n /a. (31)
c V −2ǫ0 i i
i
X
Using the Eqs. (21)-(26) we can compute the quantity a. Taking into account the smallness
of the quantities E′/ǫ and E′/mc2 compared to unity, we rewrite ε as
F c
e2 4πe2n 1/2 E n E α n
2 i n i s
ε = Z n 1+ + . (32)
c −2ǫ0 i (cid:18) Ei (cid:19) i i(cid:20) 2εF n 2mc2 n (cid:21)
X
The energy density of the Coulomb interaction ε in the case where electrons are normal
c
follows from (32) when n /n=1 and n =0, i.e.
n s
2 2 1/2
e 4πe n E
2 i
ε = Z n 1+ . (33)
c −2ǫ0 i (cid:18) Ei (cid:19) i i(cid:20) 2εF(cid:21)
X
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EUROPHYSICSLETTERS
The gain in the energy density of the sample due to the coulomb interaction is
e2n Z2n αε n
∆ε =ε εn = (4πe2nE )1/2 i i 1 F s. (34)
c c− c 4ǫ0εF i n − mc2 n
Xi (cid:16) (cid:17)
If we take into accountthat α<1 and ε /mc2 1 and substitute for E the expression(22)
F i
≪
in (34) then we obtain
1 πZ2n 1/2 (e2n)3/2 E 1/2 n (T)
∆ε = i i s . (35)
c ǫ0 i (cid:18) 4n (cid:19) ε1F/2 (cid:18)εF(cid:19) n
X
IfwesubstituteforE theexpression(23)andusetheasymptoticalexpressionforthefunction
D(T/θ) in the limit T/θ 1, we obtain
≪
2
T n (T)
s
∆ε =A , (36)
c
T n
(cid:18) c(cid:19)
where
π5 1/2 (e2n)3/2(k θ)1/2 T 2 Z2n 1/2
A= B c i i , (37)
(cid:18)40(cid:19) ǫ0 εF (cid:18) θ (cid:19) i (cid:18) n (cid:19)
X
and T is the temperature of the superconducting phase transition. According to the phe-
c
nomenologicalGorter-Casimir theory
4
n (T) T
s
=1 , (38)
n − T
(cid:18) c(cid:19)
which can be used in Eq. (36) to eliminate n . The correction to the free energy density of
s
the normal state ∆f can be obtained from ∆ε by integratingthe thermodynamicalrelation
c c
∆ε/T2= (∂/∂T)(∆f/T)
−
∆ε
∆f =f fn = T cdt+C. (39)
c c− c − t2
Z
The integration constant C is determined from the requirement ∆f (T ) = 0. Substituting
c c
Eqs. (36) and (37) into Eq. (39) and carrying out the integration, we finally find
2 4
T T
∆f =0.2A 4 5 . (40)
c
( −(cid:18)Tc(cid:19) " −(cid:18)Tc(cid:19) #)
To obtain the total change in the free energy density due to the superconducing phase tran-
sition within the sample, we need to add to ∆f the change of free energy density due to the
c
correlationinteraction of the Cooper pairs
∆f = Hc2(0)[1 ( T 2 2, (41)
s
− 8π − T
c
(cid:1) (cid:3)
where H (0) is the criticalmagnetic field at T =0. The net change in the free energy density
c
is
2 2
∆f =∆f +∆f =0.2A 4 y(5 y ) b(1 y) , (42)
c s
− − − −
(cid:2) (cid:3)
D.M.Sedrakian:AphenomenologicaltheoryofphasetransitionsinhighT superconductors7
c
where b=H (0)2/8π and y =(T/T )2.
c c
One shouldnote that the change in the free energy in superconductorsis commonly given
by the secondtermin Eq. (42), consequently,asthe temperatureis decreasedit decrasesand
reaches the value b at T = 0. As shown in the exprimental work [3], such a behaviour of
−
the variationof the free energy density is not realized in the experiment. The experimnetally
deduceddependenceof∆f ontemperatruequalitativelyconincideswiththethatpredictedby
(42). Withdecreasingtemperature∆f increasesfirstreachingamaximumaty0 anddcreases
subseqently goingthroughzeroatthe pointy1. As the experimnet showsthe deviationsofy0
from the y1 is small compared to unity. To guarantee that this is the case it is sufficient to
require A b. An estimate of the values of A and b shows that this condition can be easily
≪
met. In the frameworkofthe Ginzburg-Landautheoryαn /n is the squareofthe modulus of
s
the equilibriumvalue ofthe orderparameter ψ 2. If the changeinthe free-energydensity in
| e|
the Ginzburg-Landau theory is written in the form
d(T)
2 4
∆f(ψ,T)=c(T)ψ + ψ , (43)
| | 2 | |
then in the framework of the present theory
c(T) 1c2(T)
2
ψ = =α(1 x), ∆f = . (44)
| e| −d(T) − −2 d(T)
Thus,thefunctionsc(T)andd(T)canbedeterminedfromthemeasurementsλ(T)and∆f(T)
according the the following relations
2∆f 2∆f
c(T)= d(T)= (45)
α(1 x) −α2(1 x)2
− −
It is sopposed that the function x=x(T) defineds by the Eq.(38).
As we see from Eq.(42) ∆f is not only negative, as it have to be for ordinary supercon-
ductors. Indeed for T > Tc1, where Tc1 = √y1Tc, ∆f is positive and according to Eq.(45)
c(t) andd(T)will change theirs signes. For ordinarysuperconductorsit is impossible, but for
high-Tc superconductors it takes place for temperatures T > Tc1. It seems that for high-Tc
superconductorsthere aremore thanone criticaltemperature. More workneed to be done to
understand thess new results. The quanititative comparison of Eq. (42) with the experiment
will be given in our next paper..
In closing I would like to thank S. G. Gevorgyan for discussions and presentations of the
experimatalworkobtainedinrefs.[2,3]whichstimulatedthiswork. Theauthorisalsogreatful
to A. Mouradyan and M. Hayrapetyanfor the discussion of the results of this work.
REFERENCES
[1] 1. The Applied Superconductivity Conference (ASC’ 2000, VA, USA, September 2000), IEEE
Trans. on Applied Supercond.,11 (2001).
[2] 2.S.G.Gevorgyan,T.Kiss, T.Ohyama,M.Inoue,A.A.Movsisyan, H.G.Shirinyan,V.S.Gevorgyan,
T.Matsushita,M.Takeo.Supercond.Sci.Technol.,14,1009-1013(2001)[seealso: S.G.Gevorgyan.
J.Contemporary Physics (Armenian Academy of Sciences), v.38, N1, pp 41-53 and v.38, N2,
pp.45-50 (2003)].
[3] 3. S.G.Gevorgyan, T.Kiss, A.A.Movsisyan, H.G.Shirinyan, T.Ohyama, M.Inoue, T.Matsushita,
M.Takeo. Physica C., 363, 113-118 (2001).
[4] Pippard A.B. Proc. Roy.Soc., A 216, 547, (1953)
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EUROPHYSICSLETTERS
[5] L. D. Landau,E.M. Lifshitz, Statistical Physics, M. Nauka,1964(in Russian).
[6] E.M. Lifshitz, L.P.Pitaevski Physical Kinetics. M. Nauka,1979(In Russian).
[7] D. M. Sedrakyan and R.A. Krikoryan,Astrofizika,47, 237-240 (2004).