CHAPTER I INTRODUCTION TO SUPERCONDUCTIVITY 1 ... - …shodhganga.inflibnet.ac.in/bitstream/10603/17418/7/07_chapter 1.pdf · 3 can be thought of as infinite: a superconductor. The

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CHAPTER I

INTRODUCTION TO SUPERCONDUCTIVITY

1.1 Introduction

Superconductivity is a fascinating and challenging field of Physics. Today,

superconductivity is being applied to many diverse areas such as: theoretical and

experimental science, military, transportation, power production, electronics, medicine

as well as many other areas. Scientists and engineers throughout the world have been

striving to understand this remarkable phenomenon for many years.

In 1911, Kamerlingh Onnes began to investigate the electrical properties of

metals in extremely cold temperatures. It had been known for many years that the

resistance of metals fell when cooled below room temperature, but it was not known

what limiting value the resistance would approach, if the temperature were reduced

very close to 0 K. Some scientists, such as William Kelvin, believed that electrons

flowing through a conductor would come to a complete halt as the temperature

approached absolute zero. Other scientists, including Onnes, felt that a cold wire's

resistance would dissipate. This suggested that there would be a steady decrease in

electrical resistance, allowing better conduction of electricity. At some very low

temperature point, scientists felt that there would be a leveling off as the resistance

reached some ill-defined minimum value allowing the current to flow with little or no

resistance. Onnes passed the current through a very pure mercury (Hg) wire and

measured its resistance as he steadily lowered the temperature. Much to his surprise

there was no leveling off of resistance, let alone the stopping of electrons as suggested

by Kelvin. At a temperature of 4.2 K, called the superconducting transition

temperature Tc, the resistance suddenly vanished [1]. Current was flowing through the

mercury wire and nothing was stopping it, the resistance was zero. According to

Onnes, "Mercury has passed into a new state, which on account of its extraordinary

electrical properties may be called the superconductive state". The experiment left no

doubt about the disappearance of the resistance of a mercury wire. Kamerlingh Onnes

called this newly discovered state, Superconductivity.

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1.2 Fundamentals of superconductors

The theoretical understanding of superconductivity is extremely complicated and

involved. Superconductors have the ability to conduct electricity without the loss of

energy. When current flows in an ordinary conductor, for example copper wire, some

energy is lost. In a light bulb or electric heater, the electrical resistance creates light

and heat. In metals such as copper (Cu) and aluminium (Al), electricity is conducted as

outer energy level electrons migrate as individuals from one atom to another. These

atoms form a vibrating lattice within the metal conductor; the warmer the metal the

more it vibrates. As the electrons begin moving through the maze, they collide with

tiny impurities or imperfections in the lattice. When the electrons bump into these

obstacles they fly off in all directions and lose energy in the form of heat. Inside a

superconductor the behaviour of electrons are vastly different. The impurities and

lattice are still there, but the movement of the superconducting electrons through the

obstacle course is quite different. As the superconducting electrons travel through the

conductor they pass freely through the complex lattice. Because they bump into

nothing and create no friction, they can transmit electricity with no appreciable loss in

the current and no loss of energy. In this section fundamental terms and phenomena of

superconductvity will be discussed.

1.2.1 Meissner effect

In 1933, Walther Meissner and R. Ochsenfeld discovered that superconductors

are more than a perfect conductor of electricity and they also have an interesting

magnetic property of excluding a magnetic field. When a superconductor is cooled

below its transition temperature in a magnetic field, it excludes the magnetic flux. This

phenomenon, known as Meissner effect, was discovered by Meissner and Ochsenfeld

[2]. The Meissner effect will occur only if the magnetic field is relatively small. In a

weak applied field, a superconductor "expels" nearly all magnetic flux. It does this by

setting up electric currents near its surface. The magnetic field of these surface currents

cancels the applied magnetic field within the bulk of the superconductor. As the field

expulsion, or cancellation, does not change with time, the currents producing this

effect (called persistent currents) do not decay with time. Therefore the conductivity

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can be thought of as infinite: a superconductor. The sequence of events is shown in

Figure 1.1. If the magnetic field becomes too great, it penetrates the interior of the

metal and the metal loses its superconductivity.

Figure 1.1 Meissner effect in a superconducting sphere cooled in a constant applied

magnetic field. Below the transition temperature the magnetic flux are ejected from the

sphere.

1.2.2 Characteristics of superconductors

Critical magnetic field (Bc)

A superconductor has the property of perfect diamagnetism, also called the

Meissner effect, means that the magnetic susceptibility has the value c = -1. So the

superconducting state cannot exist in the presence of a magnetic field greater than a

critical value, even at absolute zero. This critical magnetic field is strongly correlated

with the critical temperature for the superconductor. The critical magnetic field at any

temperature below the critical temperature is given by the relationship,

Bc (T) = Bc (0) [1 − (T⁄Tc )2] (1.1)

Above this critical magnetic field the flux penetrate into the material and the material

goes to normal state.

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Critical current density (Jc)

Within two years of the discovery of superconductivity in mercury, Onnes

recorded that there was a ‘‘threshold value’’ of the current density in mercury, above

which the zero resistance state disappeared. This critical value was temperature

dependent which increases as the temperature was reduced below the critical

temperature, according to the expression,

Jc (T) = Jc (0) (Tc − T)⁄Tc (1.2)

A common way to estimate Jc is to measure a hysteresis loop in high field at a constant

temperature and use of the Bean-model formula,

Jc = 1.59 × 106

µ0 ∆M

d

(1.3)

Where ∆M = M+ − M_ is the difference in magnetization between the top and bottom

of the hysteresis at a particular magnetic field, µ0 = 4π´10-7 N/A2

is the permeability of

free space and d is the diameter of the sample grains in meter.

London penetration depth (λL)

For a superconductor in an applied magnetic field, the screening currents which

circulate to cancel the magnetic flux inside it must flow within a finite surface layer.

Consequently, the flux density does not vanish abruptly to zero at the boundary of the

superconductor. It penetrates up to a region in which the screening currents flow and

the width of this region is known as the London penetration depth of the

superconductor. This is illustrated in Figure 1.2.

Figure 1.2 A schematic representation of the penetration depth (λL) of a

superconductor in an applied magnetic field.

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One of the theoretical approaches to the description of the superconducting

state is the London equation. It relates the curl of the current density J to the magnetic

field:

∇̄̄ × J = − 1

(1.4) µ0 hL

The nature of the decay depends upon the superconducting electron density n:

λ1 = J s0 mc2

ne2 (1.5)

Where, ε0 = 8.854´10-12 F/m is the vacuum permittivity

m = mass of an electron

c = velocity of light

n = electron density

e = charge of an electron

In the superconducting state, the only field allowed is exponentially damped as

we go in from the external surface and it is given by,

B(x) = B(0) exp(-x/λL) (1.6)

Where B(0) is the field at the plane boundary. This implies that the magnetic field

penetrates up to a length scale λL from the surface into the interior of the

superconductor, giving the penetration depth, a fundamental length scale, from the

Londons equations. The temperature dependence of λL can be expressed by the

empirical equation:

λ (T) = λ(0) [1 - (T/Tc)4]

-1/2 (1.7)

The typical value of the penetration depth for most of the elemental superconductors

ranges between 10-2000 nm.

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Coherence length (ξ0)

The coherence length is a measure of distance within which the superconducting

electron density cannot change drastically in a spatially-varying magnetic field. That is

the superconducting electron density cannot change quickly and there is a minimum

length over which a given change can be made, lest it destroy the superconducting

state. For example, a transition from the superconducting state to a normal state will

have a transition layer of finite thickness which is related to the coherence length. This

coherence length is related to the Fermi velocity (vF) for the material and the energy

gap (Eg) associated with the condensation to the superconducting state.

ÇO = 2ħrF

nEg

(1.8)

1.2.3 BCS theory of superconductivity

The ability of electrons to pass through superconducting material has puzzled

scientists for many years. The warmer a substance is the more it vibrates. Conversely,

the colder a substance is the less it vibrates. Early researchers suggested that fewer

atomic vibrations would permit electrons to pass more easily. However this predicts a

slow decrease of resistivity with temperature. It soon became apparent that these

simple ideas could not explain superconductivity. It is much more complicated than

that.

The understanding of superconductivity was advanced in 1957 by three

American Physicists-John Bardeen, Leon Cooper and John Schrieffer, through their

Theories of Superconductivity, known as the BCS Theory. The BCS theory explains

superconductivity at temperatures close to absolute zero. The basis of this theory is that

even a very weak attractive interaction between electrons, mediated by phonons,

creates a bound pair of electrons, called the Cooper pair, occupying states with equal

and opposite momentum and spin (i.e. k↑, -k↓). The formation of the bound states

creates instability in the ground state of the Fermi sea of electrons and a gap (∆(Τ))

opens up at the Fermi level. The minimum energy Eg required to break a Cooper pair

to create two quasi-particle excitations is Eg = 2∆(T).

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The formation of the Cooper pairs mediated by the phonons is illustrated in

Figure 1.3. An electron with momentum k travelling through the lattice will polarize it,

thereby creating a local positive charge. A second electron with momentum –k

travelling through this lattice will be attracted to the local positive charge, thereby,

getting attracted to the first electron. This leads to the formation of the Cooper pairs.

Figure 1.3 Schematic illustration of the formation of a cooper pair between two

electrons travelling with momentum k and –k, mediated by the lattice.

The key consequences of the BCS theory include a connection of the gap

parameter Δ to the transition temperature Tc,

2∆ = 3.52 kB Tc (1.9)

and the Debye temperature (ΘD),

Tc = 1.14 ΘD e 1⁄N(O)V (1.10)

where N(0) is the density of states at the Fermi level and V is the attractive electron-

phonon interaction potential. Tc is in part determined by the Debye temperature so that

an observable shift in ΘD should accompany an alteration of Tc. Such a change in ΘD

can be accomplished by replacing one element in the material with a different isotope

of the same element. Indeed, measurements demonstrating such a shift in Tc, termed

the isotope effect, provided convincing support for the BCS model of

superconductivity.

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For weak coupling superconductors, the reduced gap ∆(T)/∆(0) is a universal

function of the reduced temperature T/Tc, near the critical temperature Tc,

∆ (T) = 1.76 (1 −

T )

(1.11) ∆ (O) Tc

So that the energy gap approaches zero continuously as T → Tc as shown in Figure 1.4.

Figure 1.4 Variation of the reduced gap ∆(T)/∆(0) with the reduced temperature T/Tc

according to the BCS theory.

1.2.4 Type I and Type II superconductors

Figure 1.5a shows the magnetization versus applied magnetic field for a bulk

superconductor which exhibits complete Meissner effect (perfect diamagnetism). A

superconductor with this behaviour is called Type I superconductor. Above the critical

field Hc the specimen is a normal conductor and the magnetization is too small. Very

pure samples of lead (Pb), mercury (Hg) and tin (Sn) are examples of Type I

superconductors. Type II superconductors have superconducting electrical properties

up to a field denoted by Hc2 (Figure 1.5b). Between the lower critical field Hc1 and the

upper critical field Hc2 the flux density B ≠ 0 and the Meissner effect is incomplete in

this region. The flux starts to penetrate the specimen at a field Hc1 lower than the

thermodynamic critical field Hc. In the region between Hc1 and Hc2 the superconductor

is threaded by flux lines and is said to be in the vortex state. A schematic of the mixed

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state is shown in Figure 1.6. The normal regions in the mixed state are in the form of

cylinders with their axis along the direction of the magnetic field. Current vortices

circulate around these normal cores to generate the flux within. The direction of this

current is opposite to the main surface shielding current which makes the flux in the

superconducting region zero. With the increase in the magnetic field beyond Hc1, the

distance between the normal cores decreases. At a field equal to the upper critical field

Hc2, there is complete overlap of the normal cores and the superconductor goes over

completely to the normal state.

Figure 1.5 Magnetization versus applied magnetic field for (a) Type I and (b) Type II

superconductors.

Figure 1.6 Schematic representation of the mixed state of a Type II superconductor.

The white cylindrical regions denote the normal cores where the flux penetrates. The

normal cores are separated by superconducting regions.

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An important difference between Type I and Type II superconductors is the

mean free path of the conduction electrons in the normal state. If the coherence length

is longer than the penetration depth, the superconductor will be Type I, with k < 1; k =

λL/ξ. But when the mean free path is short, the coherence length is short and the

penetration depth is great with k > 1, and the superconductor will be Type II. Type I

superconductors are conventional superconductors and they are well described by

the BCS theory.

1.3 High Tc Superconductors

Until 1986, Physicists had believed that BCS theory forbade superconductivity at

temperatures above about 30 K. In that year, Bednorz and Müller discovered

superconductivity in a lanthanum-based cuprate (La2-xBaxCuO4) perovskite material,

which had a transition temperature of 30 K [3]. It was soon found that replacing the

lanthanum with yttrium (i.e., making YBCO) raised the critical temperature to 92 K

[4], which was important because liquid nitrogen could then be used as a refrigerant

(the boiling point of nitrogen is 77 K at atmospheric pressure). This remarkable

discovery has renewed the interest in superconductivity research. Soon after that many

related materials which came to be known as cuprates were discovered to show

superconductivity at high-Tc values. The highest critical Tc of 135 K was achieved in

1993 in HgBa2Ca2Cu3O8 [5] compound (Tc = 164 K at high pressure). In the year 2001,

three interesting discoveries were made: (i) MgCNi3, a completely surprising analogy

to the oxide perovskites. It is based on the combination of a light electropositive metal

(Mg) with another light element (C), with the addition of a transition element (Ni),

having Tc at 8 K [6]. (ii) MgB2 was found supercondcuting at 39 K [7], this is the

highest Tc in a simple binary material. (iii) Another family of oxide supercondcutors

AuBa2Can-1CunO2n+3 (n = 3, 4) with Tc at 99 K was discovered by Kopnin et al [8] in

the same year. These high Tc superconductors or Type II superconductors are called

unconventional superconductors and they do not fit with the conventional BCS theory

of superconductivity.

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The cuprate superconductors are technologically important since the Tc varies up

to 135 K. They are generally considered to be quasi-two-dimensional materials with

their superconducting properties determined by electrons moving within weakly

coupled copper-oxide (CuO2) layers. Neighboring layers containing ions such as

lanthanum (La), barium (Ba), strontium (Sr) or other atoms act to stabilize the structure

and doping electrons or holes onto the copper-oxide layers. The basic structure of the

cuprate superconductors is a CuO2 plane separated by intervening planes composed of

metal donor ions and oxygens. The simplest of these is the La2-xSrxCuO4, and related

materials. The structure of this material (Figure 1.7) consists of single atomic planes of

CuO2 separated by two atomic planes of La-Sr oxide. One of the properties of the

crystal structure of oxide superconductors is an alternating multi-layer of CuO2 planes

with superconductivity taking place between these layers. The more layers of CuO2 the

higher the Tc. This structure causes a large anisotropy in normal conducting and

superconducting properties, since electrical currents are carried by holes induced in the

oxygen sites of the CuO2 sheets. The electrical conduction is highly anisotropic, with a

much higher conductivity parallel to the CuO2 plane than in the perpendicular

direction. Generally, critical temperatures depend on the chemical compositions,

cations substitutions and oxygen content.

Figure 1.7 Crystal structure of La2-xSrxCuO4.

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The phase diagram of the high Tc cuprate superconductors consists of several

distinct regions (Figure 1.8). The stoichiometric parent compounds are

antiferromagnetic Mott insulators dominated by strong electronic interactions. In this

region of the phase diagram each site of the two-dimensional CuO2 plane is occupied

by single charge carrier and the strong Coulomb repulsion efficiently blocks the

movement of charges. Despite the immobility of the charges, the kinetic energy gain

due to virtual nearest hopping processes favors an antiferromagnetic ordering of the

spins. Upon doping of either electrons or holes the charge carriers gain mobility and

the material becomes conducting and at higher doping even superconducting. On the

hole doped side of the phase diagram the transition region form the antiferromagnetic

Mott insulating to the superconducting state is called the pseudogap regime where

strong antiferromagnetic fluctuations are still present although long range magnetic

order is no longer maintained. The strongly momentum dependent spin fluctuations are

believed to be important for the appearance of d-wave superconductivity in the

cuprates and are even prominent candidates for mediating the superconducting pairing

interaction. Even though enormous experimental as well as theoretical efforts have

been made on the pseudogap regime, it is not yet fully understood.

Figure 1.8 Schematic phase diagram of high Tc cuprates.

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1.4 Applications of superconductors

Soon after Kamerlingh Onnes discovered superconductivity, scientists began

dreaming up practical applications for this strange phenomenon. Powerful

superconducting magnets could be made much smaller than a resistive magnet,

because the windings could carry large currents with no energy loss. Generators wound

with superconductors could generate the same amount of electricity with smaller

equipment and less energy. Once the electricity was generated it could be distributed

through superconducting wires. Energy could be stored in superconducting coils for

long periods of time without significant loss.

The subsequent discovery of high temperature superconductors brings us a giant

step closer to the dream of early scientists. Applications currently being explored are

mostly extensions of current technology used with the low temperature

superconductors. Current applications of high temperature superconductors include;

magnetic shielding devices, medical imaging systems, superconducting quantum

interference devices (SQUIDs), infrared sensors, analog signal processing devices and

microwave devices. As our understanding of the properties of superconducting

material increases, applications such as; power transmission, superconducting magnets

in generators, energy storage devices, particle accelerators, levitated vehicle

transportation, rotating machinery and magnetic separators will become more practical.

The ability of superconductors to conduct electricity with zero resistance can be

exploited in the use of electrical transmission lines. Currently, a substantial fraction of

electricity is lost as heat through resistance associated with traditional conductors such

as copper or aluminum. A large scale shift to superconductivity technology depends on

whether wires can be prepared from the brittle ceramics that retain their

superconductivity at 77 K while supporting large current densities.

The field of electronics holds great promise for practical applications of

superconductors. The miniaturization and increased speed of computer chips are

limited by the generation of heat and the charging time of capacitors due to the

resistance of the interconnecting metal films. The use of new superconductive films

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may result in more densely packed chips which could transmit information more

rapidly by several orders of magnitude. Superconducting electronics have achieved

impressive accomplishments in the field of digital electronics. Logic delays of 13

picoseconds and switching times of 9 picoseconds have been experimentally

demonstrated. Through the use of basic Josephson junctions scientists are able to make

very sensitive microwave detectors, magnetometers, SQUIDs and very stable voltage

sources.

The use of superconductors for transportation has already been established using

liquid helium as a refrigerant. Prototype levitated trains have been constructed in Japan

by using superconducting magnets.

Superconducting magnets are already crucial components of several

technologies. Magnetic resonance imaging (MRI) is playing an ever increasing role in

diagnostic medicine. The intense magnetic fields that are needed for these instruments

are a perfect application of superconductors. Similarly, particle accelerators used in

high-energy physics studies are very dependent on high-field superconducting

magnets. The recent controversy surrounding the continued funding for the

superconducting super collider (SSC) illustrates the political ramifications of the

applications of new technologies.

1.5 Limitations of superconductors

Despite many scientists believing that superconductors are the way of the future,

there are still a number of limitations to their design.

Ø The first of these is the restricted range for operating temperature. Since the

world record for the highest critical temperature stands at 135 K, there is still a

long way to go before superconductors are available to the average user at room

temperature. It is impractical for handheld, consumer devices to have liquid

nitrogen running through them.

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Ø Even if we decide to try and cool some devices continually with liquid

nitrogen, it is very impractical to cool thousands of kilometres of underground

electrical wiring connected to the power grid. More work must be done before

they become a practical room temperature device.

Ø Also, like most ceramics, Type II superconductors are extremely brittle and

therefore impractical unless methods are developed to reduce the brittle nature

of these superconductors.

Ø Type I superconductors, whilst not brittle, are not able to be cooled with liquid

nitrogen (77 K) and their critical temperatures are nowhere near as feasible as

their Type II counterparts.

Ø The other noticeable limitation to superconductors is the fact that they are quite

sensitive to a changing magnetic field, meaning that AC current will not work

effectively with superconductors. As a result, devices such as transformers,

which only work with AC current, will be more difficult to implement into a

DC oriented world when superconductors become a reality.

1.6 Superconductivity at room temperature

Room temperature superconductivity is the holy grail of solid state physics. It is

becoming increasingly obvious to scientists all over the world that superconductors are

the future in terms of transmission and applications with electricity. To make

superconductors a feasible option for the electrical devices, scientists must put their

effort into a number of key problem areas. The first of these is getting superconductors

to work at room temperature. This may entail creating devices that contain a cooling

agent or it could mean that scientists need to find new compounds that work at even

higher critical temperatures than those currently available. So while superconductors

are a very viable future solution in so many applications, much work must be done

before it becomes feasible. This induces me to choose the research problem in

superconductivity. In order to achieve room temperature superconductor much efforts

have been put by the researcher all around the world. By joining in this race, I made an

effort to synthesise new superconducting materials with novel physical and chemical

properties. Also, the superconductivity of unconventional superconductors cannot be

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described by the conventional BCS theory and each requiring a different physical

mechanism. The chosen research problem may enlighten the superconducting research

by supporting the explanation of the mechanism of already existing superconductors.