Transformation of Thernal Energy in Electric Energy in an Inhomogeneous Superconducting Ring

10 Pages·0.11 MB·English

by  A. V. Nikulov

9
9
9 TRANSFORMATION OF THERMAL ENERGY IN ELECTRIC
1 ENERGYINANINHOMOGENEOUSSUPERCONDUCTINGRING

n
a A.V.Nikulov
J
InstituteofMicroelectronicsTechnologyandHighPurityMaterials,Russian

2
Academy of Sciences, 142432 Chernogolovka,Moscow Region, Russia.
1
Abstract
]
n
Aninhomogeneoussuperconductingring(hollowcylinder)placedina
o
magnetic field is considered. It is shown that a direct voltage appears on
c
section with lowest critical temperature when it is switched periodically
-
r fromthenormalstateinthesuperconductingstateandbackwards,ifthe
p
magnetic fluxcontained within thering is not divisible by thefluxquan-
u
tum. The superconducting transition can be first order in this case. In
s
. thevicinityofthistransition,thermalfluctuationscaninducethevoltage
t
a in thering with rather small sizes.
m
-
d
n
o
1 Introduction
c
[

Superconductivity is a macroscopic quantum phenomenon. One of the conse-
1
quencesofthisisthe periodicaldependence ofenergyofasuperconductingring
v
3 onamagneticfluxwithinthisring. Thisdependenceiscausedbyaquantization
0 of the velocity of the superconducting electrons v . According to (See ref.[1])
s
1
1 1 dφ 2e 2e Φ0dφ
0 vs = (h¯ − A)= ( −A) (1)
m dr c mc 2π dr
9
9 the velocity along the ring (or tube) circumference must have fixed values de-
/ pendent on the magnetic flux because
t
a
m 2e
- Zldlvs = mc(Φ0n−Φ) (2)
d
n and n = dl(1/2π)dφ/dr must be an integer number since the wave function
l
o Ψ = |Ψ|eRxp(iφ) of the superconducting electrons must be a simple function.
c Where φ is the phase of the wave function; Φ0 = π¯hc/e = 2.07 10−7G cm2 is
:
v the flux quantum; A is the vector potential; m is the electron mass and e is
i the electron charge; l = 2πR is the ring circumference; R is the ring radius;
X
Φ= dlA is the magnetic flux contained within the ring.
r l
a TRhe energy of the superconductor increases with the superconducting elec-
tron velocity increasing. Therefore the |v | tends towards a minimum possible
s
value. IfΦ/Φ0 isanintegernumber,thevelocityisequaltozero. Butvs cannot
be equal to zero if Φ/Φ0 is not an integer number. Consequently, the energy of
1
thesuperconductingstateoftheringdependsinaperiodicmanneronthemag-
netic fieldvalue. The energyofthe supeconducting ringchangesby tworeason:
2
because the kinetic energyof the superconducting electronsn slmv /2 changes
s s
and because the energy of a magnetic field LI2/2 induced by superconducting
s
current I = sj = s2en changes. Here L is the inductance of the ring; n is
s s s s
the superconducting pair density; s is the area of cross-sectionof the ring wall.
The first is the cause of the Little-Parks effect [2]. Little and Parks discov-
ered that the critical temperature, T , of a superconducting tube with narrow
c
wall depends in a periodic way on a magnetic flux value within the tube. This
effect has been explained by M.Tinkham [3]. According to ref.[3], the criti-
cal temperature of the homogeneous ring (which we considered now) is shifted
periodically in the magnetic field:
2 2
Tc(Φ)=Tc[1−(ξ(0)/R) (n−Φ/Φ0) ] (3)
because the kinetic energy changes periodically with magnetic field. Here ξ(0)
is the coherence lengthat T = 0. The value of (n−Φ/Φ0) changesfrom-0.5 to
0.5. The T shift is visible if the tube radius is small enough (if R≃ξ(0)).
c
The magnetic field energy F = LI2/2 = Ls22e2v2n2 does not influence
L s s s
on the critical temperature value because it is proportional to n2. A value of
s
this energy depends on the temperature because the n value depends on the
s
temperature. The n value change causes the superconducting current change
s
andavoltageinconsequencewithelectromagneticinductionlaw. Consequently
thesuperconductingringcanbeusedfortransformationthethermalenergyinto
the electric energy if the velocity v 6= 0. A temperature change will induce a
s
voltage in the ring if v 6= 0. In the present paper a most interesting case - a
s
induction of direct voltage in an inhomogeneous ring - is considered.
A ring (tube) with the narrow wall (the wall thickness w ≪ R,λ) is con-
sidered. In this case Φ = BS ≃ HS, because the magnetic field induced by
the superconducting current in the ring is small. Here H is the magnetic field
induced by an external magnet, S =πR2 is the ring area and λ is the penetra-
tion depth of the magnetic field. We consider a ring whose criticaltemperature
varies along the circumference l = 2πR, but is constant along the height h. In
sucha ring,the magnetic flux shifts the criticaltemperature ofa sectionwith a
lowest T value only. When the superconducting state is closed in the ring, the
c
current of the superconducting electrons must appear as a consequence of the
relation(2)ifΦ/Φ0 isnotanintegernumber. ThereforethelowestTc valuewill
be shifted periodically in the magnetic field as well as T of the homogeneous
c
ring.
2
2 Inhomogeneous superconducting ring as a thermal-
electric machine of direct current
It is obvious that the current value must be constant along the circumference,
becausethecapacitanceissmall. Thevalueofthesuperconductorcurrentmust
be constant if the current of the normal electrons is absent. Therefore the
velocity of the superconducting electrons can not be a constant value along the
circumference of a inhomogeneous ring if the superconducting pair density is
notconstant. (I considerthe ringwithidenticalareasalongthe circumference.)
Let us consider a ring consisting of two sections l and l (l +l = l = 2πR)
a b a b
with different values of the critical temperature T > T . According to the
ca cb
relation (2) the superconducting current along the ring circumference, I , must
s
appear below Tcb if Φ/Φ0 is not an integer number. Then
I =I =s j =s 2en v =I =s j =s 2en v (4)
s sa a sa a sa sa sb b sb b sb sb
if the normal current is absent. Here n and n are the densities of the
sa sb
superconducting pair in the sections l and l ; v and v are the velocities of
a b sa sb
the superconducting pairs in the sections l and l and s and s are the areas
a b a b
ofwallsectionof l and l . s=s =s =wh. dlv =v l +v l . Therefore
a b a b l s sa a sb b
according to (2) and (4) R
2e n 2e n
sb sa
v = (Φ n−Φ); v = (Φ n−Φ) (5)
sa o sb o
mc(l n +l n ) mc(l n +l n )
a sb b sa a sb b sa
s4e2 n n
sa sb
I = (Φ n−Φ) (5a)
s o
mc (l n +l n )
a sb b sa
According to (5a) a change of the superconducting pair density induces a
change of the superconducting current if Φ n − Φ 6= 0. The change of the
o
superconducting current induces the change of the magnetic flux Φ=HπR2+
L(I +I ) and, as a consequence, induces the voltage and the current of the
s n
normalelectrons(thenormalcurrent,I ). ThetotalcurrentI =I +I mustbe
n s n
equalinbothsections,becausethecapacitanceissmall. ButI canbenoequal
sa
to I . Then I 6= I and consequently, the potential difference dU/dl exists
sb na nb
along the ring circumference. The electric field along the ring circumference
E(r) is equal to
dU 1dΦ dU LdI ρ
n
E(r)=− − =− − = I (6)
n
dl l dt dl l dt s
where ρ is the normal resistivity.
n
Because the normal current exists the relations (5) becomes no valid. The
velocity v and the current I can be not equal zero even at n = 0. The
sa ca sb
3
current decreases during the decay time of the normal current L/R . R =
nb bn
ρ l /s is the resistance of the section l in the normal state.
bn b b
The voltage, the normal current, and superconducting current change peri-
odically in time if the n value changes periodically. But in addition a direct
sb
potential difference U can appear if the l section is switched from the normal
b b
state in the superconducting state and backwards i.e. some times (t ) n = 0
n sb
and some times (t ) n 6=0. Let us consider two limit cases: t ≪L/R and
s sb n nb
t ≫L/R . t ≫L/R and l ≫l in the both cases.
n nb s nb a b
In the first case the total current I is approximately constant in time and
because t ≫L/R
s bn
2
4e n <n >
sa sb
I ≃s2ensa <vsa >≃s (Φ0n−Φ) (7)
mcl n +l <n >
b sa a sb
Here < n > is a average value of n . The average value of the resistivity of
sb sb
the l section, ρ ≃ρ t /(t +t ). Consequently,
b b bn n s n
lb <nsb > (Φ0n−Φ)
U =R I ≃ ρ (8)
b b l n +l <n > λ2 b
b sa a sb La
whereλ =(mc/4e2n )1/2 istheLondonpenetrationdepth. Accordingto(8)
La sa
U 6=0 at <n >=6 0 and ρ 6=0. Consequently, the direct potential difference
b sb b
can be observed if we change the temperature inside the region of the resistive
transition of the section l (where 0<ρ <ρ ).
b b bn
In the second case
s<nsb > (Φ0n−Φ)
U ≃ Lf (9)
b l n +l <n > λ2
b sa a sb La
where f is the frequency of the switching from the normal into the supercon-
ducting state.
Thus, the inhomogeneous superconducting ring can be used as a thermal-
electric machine of direct current. The power of this machine is small. It
decreaseswiththeincreasingoftheringradius. Letustoevaluatethemaximum
power at l n ≪ l < n >. This condition means that the temperature of
b sa a sb
l changes enough strongly. The power will be maximum at t ≃ t ≃ L/R .
b n s bn
The frequency of the switching f =R /2L in this case. The power of the ring
bn
can not exceed
sl ρ Φ2 sl ρ Φ2
b bn 0 2 b bn 0
W =IUb = 2l2λ4 (n−Φ/Φ0) < 8l2λ4 (10)
a La a La
For example, at s = 0.1µm, l = 0.1µm, l = 1µm, λ = 0.1µm, ρ =
b a La bn
100µΩcm the power is smallerthan W =10−4Vt. The power canbe increased
by the ring height h increasing.
Aboveweconsideredthecasewhenthechangesofthetemperatureisenough
strong. In the opposite case, when the temperature change is small, l n ≫
b sa
4
l < n > and therefore the current and the voltage are proportional to the
a sb
< n > /n value. Because n must be equal zero some times, this means
sb sa sb
that the voltage is proportional to the amplitude of the temperature change.
Consequently, the inhomogeneous ring is a classical thermal machine with a
maximum efficiency in the Carno cycle [4].
Atasmalllb/la valueandaenoughbig|(n−Φ/Φ0)|valuethesuperconduct-
ingtransitionofl isfirstorder. Inthiscaseinordertoswitchthel sectionfrom
b b
the normal state in the superconducting state and backwards the temperature
must be change on a finite value, because the hysteresis of the superconducting
transition exists.
3 First order superconducting phase transition
Accordingto(5)thev valuedecreaseswiththen valueincreasing. Therefore
sb sb
the dependence ofthe energyof the superconducting state onthe n value can
sb
have a maximum in some temperature region at T ≃ T (Φ). The presence
cb
of such a maximum means that the superconducting transition is a first order
phase transition.
The existence of the maximum and the width of the temperature region
where the maximum exists depends on the n − Φ/Φ0 value and on the ring
parameters: l , l , w, h and T /T . These dependencies can be reduced to
a b ca cb
two parameters,B and L , which are introduced below. It is obvious that the
f I
maximumcanexistatonlyn−Φ/Φ0 6=0. Thereforeonlythiscaseisconsidered
below.
The Ginsburg-Landau free energy of the ring can be written as
mv2 β mv2 β LI2
F =s[l ((α + sa)n + an2 )+l ((α + sb)n + bn2 )]+ s (11)
GL a a 2 sa 2 sa b b 2 sb 2 sb 2
Here αa = αa0(T/Tca−1), βa, αb =αb0(T/Tcb−1) and βb are the coefficients
of the Ginsburg-Landautheory. We do not consider the energyconnected with
thedensitygradientofthesuperconductingpair. Itcanbeshownthatthisdoes
not influence essentially the results obtained below.
The Ginsburg-Landau free energy (11) consists of F (the energy of the
GL,la
section l ), F (the energy of the section l ) and F (the energy of the
a GL,lb a L
magnetic field induced by the superconducting current):
F =F +F +F (12)
GL GL,la GL,lb L
Substitutingtherelation(4)forthesuperconductingcurrentandtherelation
(5) for the velocity of the superconducting electrons into the relation (11), we
obtain
β
a 2
F =sl (α (Φ,n ,n )n + n ) (12a)
GL,la a a sa sb sa 2 sa
5
β
b 2
F =sl (α (Φ,n ,n )n + n ) (12b)
GL,lb b b sa sb sb 2 sb
2 2 2 2 2
F = 2Ls e (Φ0n−Φ) nsansb (12c)
L mc (l n +l n )2
a sb b sa
Here
αa(Φ,nsa,nsb)=αa0(TT −1+(2πξa(0))2((nl n−Φ+/Φl0n)2n)2s2b)
ca a sb b sa
αb(Φ,nsa,nsb)=αb0(TT −1+(2πξb(0))2((nl −n Φ+/Φl0n)2n)2s2a)
cb a sb b sa
ξa(0) = (h¯2/2mαa0)1/2; ξb(0) = (h¯2/2mαb0)1/2 are the coherence lengths at
T=0.
Accordingtothe meanfieldapproximationthetransitionintothesupercon-
ducting state ofthe sectionl occurs atα (Φ,n ,n )=0. Because n 6=0 at
b b sa sb sa
T =T thepositionofthesuperconductingtransitionofthel sectiondepends
cb b
on the magnetic flux value:
T (Φ)=T [1−(2πξ (0))2(n−Φ/Φ0)2n2sa] (12d)
cb cb b (l n +l n )2
a sb b sa
.
At l = 0 the relation (12d) coincides with the relation (3) for a homoge-
a
neous ring. A similar result ought be expected at l ≫ l . But at l ≪ l
b a b a
the T (Φ) value depends strongly on the n value. At n = 0 T (Φ) =
cb sb sb cb
Tcb[1−(2πξb(0)/lb)2(n−Φ/Φ0)2] whereas at lansb ≫ lbnsa Tcb(Φ) = Tcb[1−
(2πξb(0)/la)2(n−Φ/Φ0)2]. Consequently a hysteresis of the superconducting
transition ought be expected in a ring for l ≪l .
b a
To estimate the dependence of the hysteresis value on the ring parameters,
wetransformtherelation(12)usingtherelationsforthethermodynamiccritical
fieldHc =Φ0/23/2πλLξ;α2/2β =Hc2/8πandfortheLondonpenetrationdepth
λ =(cm/4e2n )1/2. Weconsideraringwithl ≫ξ (T)=ξ (0)(1−T/T )0.5.
L s a a a ca
n ≃−α /β in this case. Then
sa a a
1 1
′ ′
F =F +Fn (τ + +n (B+ (2+L ))) (13)
GL GLa sb (n′ +1)2 sb (n′ +1)2 I
sb sb
′
Here n =l n /l n ;
sb a sb b sa
2 4 ′
F =−sl Hca(1+ (2πξa(T)) ((n−Φ/Φ0)nsb)4)
GLa a 8π l4 (n′ +1)
a sb
Because l ≫2πξ , F ≃−sl H2 /8π.
a a GLa a ca
sξ (T)H2 2πξ (T)
F = a ca a (n−Φ/Φ0)2
2 l
a
6
T l2
τ =(T −1)(n−Φ/Φ0)−2(2πξb(0))2
cb b
β l l2
Bf =0.5βb lb (2πξb(0))2(n−Φ/Φ0)−2
a a b
s L
L =4π
I λ2 l
La a
For h > R, L = k4πR2/h where k = 1 at h ≫ R. Consequently, L =
I
4π(l/l )(lw/λ2 (T)) in this case. At h,w ≪ R, L ≃ 4lln(2R/w), therefore
a La
L =16π(l/l )(s/λ2 (T))ln(2R/w) in this case.
I a La
′
The numerical calculations show that the F (n ) dependence (13) has a
GL sb
maximum at small enough values of B and L in some region of the τ values.
f I
′
The width of the τ region with the F (n ) maximum depends on the B
GL sb f
value first of all. At L ≪ 2 the maximum exists at B < 0.4. For example
I f
at B = 0.2 and L ≪ 2 the maximum takes place at −1.02 < τ < −0.89.
f I
This means that the transition into the superconducting state of the section l
b
occurs at τ ≃ −1.02, (that is at Tcs = Tcb(1−1.02(n−Φ/Φ0)2(2πξb(0)/lb)2)
and the transition in the normal state occurs at τ ≃ −0.89, (that is at T =
cn
Tcb(1−0.89(n−Φ/Φ0)2(2πξb(0)/lb)2)if thermalfluctuations arenottakeninto
account.
2
TheinequalityL ≪2isvalidforatube(whenh>R)with2πlw≪λ (T)
I La
and for a ring (when h < R) with 8πhw ≪ λ2 (T). The hysteresis value
La
increases with decreasing B value and decreases with increasing B value.
f f
The Bf value is proportional to (n−Φ/Φ0)−2. Consequently, the hysteresis
value depends on the magnetic field value. Because the hysteresis is absent at
B > 0.4, it can be observed in the regions of the magnetic field values, where
f
Φ/Φ0 differs essentially from an integer number. The width of these regions
depends on the 0.5(β /β )(l3/(2πξ (0))2l ) value (see above the relation for
b a b b a
2
Bf). Since (n−Φ/Φ0) <0.25 and βb ≃βa in the real case, the hysteresis can
beobservedintheringwithl3 <0.2(2πξ (0))2l ). Forexampleintheringwith
b b a
lb =2πξb(0) and la =10lb, the hysteresis can be observedat |n−Φ/Φ0|>0.35
(if βa = βb). At |n− Φ/Φ0| = 0.5 Bf = 0.2 and the hysteresis is equal to
T −T ≃0.03T in this ring.
cn cs cb
4 Transformation of thermal fluctuation energy
into electric energy
The hysteresis of the superconducting transition can be observed if the maxi-
mum is high enough. The maximum height is determined by a parameter F,
which is introduced below. The hysteresis will be observed if the maximum
height is much greater than the energy of the thermal fluctuation, k T. In the
B
7
opposite case the thermal fluctuation switches the l section from the normal
b
state into the superconducting one and backwards at T ≃T (Φ).
cb
Above we have used the mean field approximation which is valid when
the thermal fluctuation is small. In our case the mean field approximation
is valid if the height of the F −F maximum, F , is much greater
GL GLa GL,max
than k T. This height depends on the F, τ, B and L values: F =
B f I GL,max
FH(τ,B ,L ). The F parameter is determined above. The H(τ,B ,L ) de-
f I f I
pendence can be calculated numerically from the relation (8). To estimate the
validity of the mean field approximationwe oughtto know the maximum value
ofthe H(τ) dependence: H (B ,L ). We canuse the meanfield approxima-
max f I
tionif FH (B ,L )≫k T. This is possible if the heightofthe ring is large
max f I B
enough, namely
1 l Gi1/2 Φ
a −2
h≫ξ (0) (n− )
a
πHmax(Bf,LI)wTca/Tcb−1 Φ0
Here Gi = (k T /ξ (0)3H2 )2 is the Ginsburg number of a three-dimensional
B ca a ca
superconductor. We have used the relation for the F parameter (see above).
For conventionalsuperconductors Gi=10−11−10−5. H ≃10−2 for typical
max
B andL values. Forexampleinthe ringwithB =0.2andL ≪2the H(τ)
f I f I
dependence has a maximum H (B = 0.2,L ≪ 2) = 0.024 at τ ≃ −0.94.
max f I
Consequently, the value of h cannot be very large. As an example for a ring
with parameter value B = 0.2, L ≪ 2, l /w = 20, T /T −1 = 0.2, and
f I a ca cb
fabricated from an extremely dirty superconductor with Gi = 10−5, the mean
field approximation is valid at h≫20ξa(0) if |n−Φ/Φ0|≃0.5.
If the mean field approximation is not valid, we must take into account the
thermalfluctuationswhichdecreasethe valueofthehysteresis. Theprobability
ofthe transitionfromnormalintosuperconductingstateandthatofthetransi-
tionfromsuperconductingintonormalstatearelargewhenthemaximumvalue
of F −F is no much more than k T. Therefore the hysteresis can not
GL GLa B
be observed at FH (B ,L ) < k T. This inequality can be valid for a ring
max f I B
made by lithography and etching methods from a thin superconducting film,
where h is the film thickness in such a ring.
As a consequence of the thermal fluctuations, the density n (r,t) changes
sb
with time. We can consider n (r,t) as a function of the time only if h,w,l ≃
sb b
or < ξ (T). At T ≃ T (Φ) (at the resistive transition) l is switched by the
b cb b
fluctuations from the normalstate in the superconducting state and backwards
i.e. some times (≃ t ) n = 0 and some times (≃ t ) n 6= 0. Consequently,
n sb s sb
accordingto the relations (8) and (9) the direct potential difference canappear
in the region of the resistive transition of the section l . Thus, the energy of
b
the thermal fluctuations can be transformed into the electric energy of direct
currentintheinhomogeneousringatthe temperatureoftheresistivetransition
of the section with the lowest critical temperature.
In order to evaluate the power of this transformation one ought take into
8
accountthatintheconsequenceofthethermalfluctuationtheGinsburg-Landau
freeenergyF changesintimewithamplitudek T. Accordingtotherelations
GL B
(7), (9) and (12c) U I/f ≃ F . Because F is a part of F (see (12)) the
b L L GL
power can not exceed k Tf. The maximum value of the switching frequency
B
f is determined by the characteristic relaxation time of the superconducting
fluctuation τ : f ≃1/τ . In the linear approximationregion [5]
GL max GL
¯h
τ = (14)
GL
8k (T −T )
B c
Thewidthoftheresistivetransitionofthesectionl canbeestimatedbythe
b
value T Gi . Gi = (k T/H2(0)l s)1/2 is the Ginsburg number of the section
cb b b B c b
l . Consequently the power value can not be larger than
b
8Gi
b 2
W = (k T ) (15)
B cb
¯h
and the U value can not exceed
b
8R Gi
U =( b b)1/2k T (16)
b,max B cb
¯h
The U value is large enough to be measured experimentally. Even
b,max
at T = 1 K and for real values R = 10Ω and Gi = 0.05, the maximum
c b b
voltage is equal to U ≃3µV. In a ring made of a high-Tc superconductor,
b,max
U can exceed 100µV. One ought to expect that the real U value will
b,max b
be appreciably smaller than U . This voltage can be determined by the
b,max
periodical dependence on the magnetic field value (see the relations (8) and
(9)).
Transformation of thermal fluctuation energy into electric energy does not
contradict the second thermodynamic law, because it is valid to within the
thermal fluctuations [6].
ACKNOWLEDGMENTS
I thank the National Scientific Council on ”Superconductivity” of SSTP ”AD-
PCM” (Project 95040) and the International Association for the Promotion of
Co-operationwithScientists fromNew Independent States(ProjectINTAS-96-
0452) for financial support.
References
[1] M.Tinkham, Introduction to Superconductivity. McGraw -Hill Book Com-
pany (1975).
9
[2] W.A.Little andR.D.Parks,Phys.Rev.Lett9,9(1962)andPhys.Rev.133,
A97 (1964).
[3] M.Tinkham, Phys. Rev. 129, 2413 (1963).
[4] C.Kittel, Thermal Physics. John Wiley and Sons, Inc. New York (1970)
[5] W.J.Skocpol and M.Tinkham, Rep.Prog.Phys.38, 1049 (1975)
[6] L.D.Landau and E.M.Lifshitz, Statistical Physics Part 1, Pergamon Press,
New York, (1989).
10