Abstract
A superconducting quantum interference device (SQUID) is the most sensitive magnetic flux sensor currently known. The SQUID can be seen as a flux to voltage converter, and it can generally be used to sense any quantity that can be transduced into a magnetic flux, such as electrical current, voltage, position, etc. The extreme sensitivity of the SQUID is utilized in many different fields of applications, including biomagnetism, materials science, metrology, astronomy and geophysics. The heart of a squid magnetometer is a tunnel junction between two superconductors called a Josephson junction. Understanding the work of these devices rests fundamentally on the BCS theory of superconductivity. In this chapter, we introduce the notion of local potential and confinement in superconductivity. We show how BCS ground state is formed from interaction of wave packets confined to these local potential wells. The starting point of the BCS theory of superconductivity is a phonon-mediated second-order term that describes scattering of electron pair at Fermi surface with momentum k i , − k i and energy 2 ℏ ω i to k j , − k j with energy 2 ℏ ω j . The transition amplitude is M = − d 2 ω d ω i − ω j 2 − ω d 2 , where d is the phonon scattering rate and ω d is the Debye frequency. However, in the presence of offset ω i − ω j , there is also a present transition between states k i , − k i and k j , − k i of sizable amplitude much larger than M . How are we justified in neglecting this term and only retaining M ? In this chapter, we show all this is justified if we consider phonon-mediated transition between wave packets of finite width instead of electron waves. These wave packets are in their local potentials and interact with other wave packets in the same well to form a local BCS state we also call BCS molecule. Finally, we apply the formalism of superconductivity in finite size wave packets to high Tc in cuprates. The copper electrons in narrow d-band live as packets to minimize the repulsion energy. The phonon-mediated coupling between wave packets (of width Debye energy) is proportional to the number of k-states in a packet, which becomes large in narrow d-band (10 times s-band); hence, d-wave Tc is larger (10 times s-wave). At increased doping, packet size increases beyond the Debye energy, and phonon-mediated coupling develops a repulsive part, destroying superconductivity at large doping levels.
Keywords
- local potentials
- superconductivity
- phonons
- Josephson junctions
- squids
1. Introduction
There is a very interesting phenomenon that takes place in solid-state physics when certain metals are cooled below critical temperature of order of few Kelvin. The resistance of these metals completely disappears and they become superconducting. How does this happen? One may guess that maybe at low temperatures there are no phonons. That is not true, as we have low frequency phonons present. Why do we then lose all resistivity? Electrons bind together to form a molecule by phonon-mediated interaction. The essence of this interaction is that electron can pull on the lattice which pulls on another electron. This phonon-mediated bond is not very strong for only few meV, but at low temperatures, this is good enough; we cannot break it with collisions with phonons which only carry
This phenomenon whereby many materials exhibit complete loss of electrical resistance when cooled below a characteristic critical temperature [1, 2] is called superconductivity. It was discovered in mercury by Dutch physicist Onnes in 1911. For decades, a fundamental understanding of this phenomenon eluded the many scientists who were working in the field. Then, in the 1950s and 1960s, a remarkably complete and satisfactory theoretical picture of the classic superconductors emerged in terms of the Bardeen-Cooper-Schrieffer (BCS) theory [3]. Before we talk about the BCS theory, let us introduce the notion of local potentials.
Shown in Figure 1 is a bar of metal. How are electrons in this metal bar? Solid-state physics texts start by putting these electrons in a periodic potential [4, 5, 6]. But that is not the complete story.
Shown in
Figure 2
is a periodic array of metal ions. Periodic arrangement divides the region into cells (region bounded by dashed lines in
Figure 2
) such that the potential in the
where
Coming back to a more realistic estimate of the kinetic energy, electron wave function is confined to length
With this energy bandwidth, electrons are all well confined by the confining potential. In fact we do not need a potential of depth 50 eV; to confine the electrons, we can just do it with a depth of ∼10 eV which means a length of around
In this chapter, we spell out the main ideas of the BCS theory. The BCS theory tells us how to use phonon-mediated interaction to bind electrons together, so that we have big molecule and we call the BCS ground state or the BCS molecule. At low temperatures, phonons do not have energy to break the bonds in the molecule; hence, electrons in the molecule do not scatter phonons. So, let us see how BCS binds these electrons into something big.
2. Cooper pairs and binding
Let us take two electrons, both at the Fermi surface, one with momentum
We proceed into the interaction frame of the natural Hamiltonian (system energies) by transformation
This gives for
The first integral averages to zero, while the second integral
Evaluating it explicitly, we get for our system that second-order integral is
which couples levels 1 and 3 and drives transition between them at rate
Observe
How does all this help. Suppose
has energy
3. Electron-phonon collisions
Recall we are interested in studying how a BCS molecule scatters phonons. For this we first understand how a normal electron scatters of a thermal phonon. We also derive electron–phonon interaction (
Consider phonons in a crystalline solid. We first develop the concept of a phonon packet. To fix ideas, we start with the case of one-dimensional lattice potential. Consider a periodic potential with period
where
The potential is shown in Figure 7 .
Now, consider how potential changes when we perturb the lattice sites from their equilibrium position, due to lattice vibrations:
For a phonon mode with wavenumber
we have
where
Using Fourier series, we can write
We can determine
where
We do not worry much about
where
Then, we get
At temperature of
with
We considered one phonon mode. Now, consider a phonon wave packet (which can also be thought of as a mode, localized in space) which takes the form
where
This deformation potential due to phonon wave packet is shown in
Figure 8
. The maximum value of the potential is around
3.1 Time dynamics and collisions
Of course phonons have a time dynamics given by their dispersion relation:
With the phonon dispersion relation
The deformation potential travels with velocity of sound and collides with an incoming electron. To understand this collision, consider a phonon packet as in Eq. (14) centred at the origin. The packet is like a potential hill. A electron comes along say at velocity
In the above, we assumed phonon packet is stationary; however, it moves with velocity
In the phonon frame the electron travels towards it with velocity
and by conservation of energy, the phonon has lost energy, lowering its temperature.
In the second case, electron and phonon are traveling in the same direction. This is shown in
Figure 11
. In the frame of phonon, electron travels towards the phonon with velocity
Thus, we have shown that electron and phonon can exchange energy due to collisions. Now, everything is true as in statistical mechanics, and we can go on to derive
All our analysis has been in one dimension. In two or three dimensions, the phonon packets are phonon tides (as in ocean tides). Let us fix ideas with two dimensions, and three dimensions follow directly. Consider a two-dimensional periodic potential with period
Now, consider how potential changes when we perturb the lattice sites from their equilibrium position, due to lattice vibrations:
Let us consider phonons propagating along
We have due to
where
Note
Using Fourier series, we can write
We can determine
which gives us a deformation potential as before:
which is same along
Since deformation potential is a tide, electron–phonon collisions do not have to be head on; they can happen at oblique angles, as shown in
Figure 13
in a top view (looking down). The velocity of electron parallel to tide remains unchanged, while velocity perpendicular to tide gets reflected. If the perpendicular velocity is large enough, the electron can jump over the tide and continue as shown by a dotted line in
Figure 13
. Imagining the tide in three dimensions is straightforward. In three dimensions, the deformation potential takes the form a wind gust moving in say
We described how a normal electron scatters phonons. Now, let us go back to our discussion on electron-phonon interaction and recall a phonon
where
Using
Thus, electron-phonon coupling Hamiltonian is of form
where
Using a cosine potential with
We just derived an expression for the electron-phonon interaction (
We said there are only two states,
it has energy
The states do not have to be exactly on a Fermi surface as shown in
Figure 14A
; rather, they can be in an annulus around the Fermi sphere as shown in
Figure 14B
. When
But before we proceed, a note of caution is in order when we use the formula
We proceed into the interaction frame of the natural Hamiltonian (system energies) by transformation:
This gives for
We evaluate the effective evolution of
Let us calculate
Similarly
and finally
Then, from Eq. (27), we get
where
Observe in the above the term
We have been talking about electron waves in this section. Earlier, we spent considerable time showing how electrons are wave packets confined to local potentials. We now look for phonon-mediated interaction between wave packets. A wave packet is built from many k-states (k-points). These states have slightly different energies (frequencies) which make the packet moves. We call these different frequencies
has binding energy
The wave packet in a potential well shuttles back and forth, which averages the offsets
We saw how two electrons bind to form a Cooper pair. However, for a big molecule, we need to bind many electrons. How this works will be discussed now. The basic idea is with many electrons; we need space for electron wave packets to scatter to. For example, when there was only one packet pair at the Fermi surface, it could scatter into all possible other packet pairs, and we saw how we could then form a superposition of these states. Now, suppose we have 2
4. BCS ground state
Let
As we will see soon,
As shown in last section, the Cooper pair
The binding energy of this state is
Next, we study how low-frequency thermal phonons try to break the BCS molecule. The electron wave packet collides with the phonon and gets deflected, which means the Cooper pair gets broken. Then, the superconducting state constitutes
which gives
When
5. Molecular orbitals of BCS states
We now come to interaction of neighboring BCS molecules. In our picture of
If
where
We talked about two BCS states separated by a thin insulator in a Josephson junction. In an actual superconductor, we have an array (lattice) of such localized BCS states as shown in
Figure 20
. Different phases
If
where
What we have now is a new lattice of potential wells as shown in
Figure 20
with spacing of
is the ground state of new lattice (Eq. 40). A state like
has a momentum and constitutes the supercurrent. It has energy
In the presence of electric field
Now, consider the local BCS states in
Figure 20
put in a loop. If we turn on a magnetic field (say in time
where
giving
This is the magnetic quantum flux. When one deals with the superconducting loop or a hole in a bulk superconductor, it turns out that the magnetic flux threading such a hole/loop is quantized [9, 10] as just shown.
Figure 21
depicts the schematic of a superconducting quantum interference device (SQUID) where two superconductors
6. Meissner effect
When a superconductor placed in an magnetic field is cooled below its critical
German physicists Walther Meissner and Robert Ochsenfeld discovered this phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin and lead samples. The samples, in the presence of an applied magnetic field, were cooled below their superconducting transition temperature, whereupon the samples canceled nearly all interior magnetic fields. A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In type-I superconductor if the magnetic field is above certain threshold
We talked about how wave packets shuttle back and forth in local potentials and get bound by phonons to form a BCS molecule. In the presence of a magnetic field, they do not shuttle. Instead, they do cyclotron motion with frequency
7. Giaever tunnelling
When we bring two metals in proximity, separated by a thin-insulating barrier, apply a tiny voltage nd then the current will flow in the circuit. There is a thin-insulating barrier, but electrons will tunnel through the barrier. Now, what will happen if these metals are replaced by a superconductor? These are a set of experiments carried out by Norwegian-American physicist Ivar Giaever who shared the Nobel Prize in Physics in 1973 with Leo Esaki and Brian Josephson “for their discoveries regarding tunnelling phenomena in solids”. What he found was that if one of the metals is a superconductor, the electron cannot just come in, as there is an energy barrier of
Recall in our discussion of superconducting state that we had 2
8. High Tc
in cuprates
High-temperature superconductors (abbreviated high-
Whereas “ordinary” or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K (243.2°C) and must be cooled using liquid helium in order to achieve superconductivity, HTS have been observed with transition temperatures as high as 138 K (135°C) and can be cooled to superconductivity using liquid nitrogen. Compounds of copper and oxygen (so-called cuprates) are known to have HTS properties, and the term high-temperature superconductor was used interchangeably with cuprate superconductor. Examples are compounds such as lanthanum strontium copper oxide (LASCO) and neodymium cerium copper oxide (NSCO).
Let us take lanthanum copper oxide
When we hole/electron dope, we remove/add electron to
References
- 1.
Tinkham M. Introduction to Superconductivity. 2nd ed. New York, USA: McGraw Hill; 1996 - 2.
De Gennes PG. Superconductivity of Metals and Alloys. New York, USA: W.A. Benjamin, Inc; 1966 - 3.
Bardeen J, Cooper L, Schriffer JR. Theory of superconductivity. Physical Review. 1957; 108 (5):1175 - 4.
Kittel C. Introduction to Solid State Physics. 8th ed. John Wiley and Sons; 2005 - 5.
Ashcroft NW, Mermin D. On a new method in the theory of superconductivity. Solid State Physics. Harcourt College Publishers; 1976 - 6.
Simon S. Oxford Solid State Basics. New Delhi, India: Oxford University Press; 2013 - 7.
Kittel C, Kroemer H. Thermal Physics. Orlando, USA: Freeman and Co; 2002 - 8.
Bogoliubov NN. Nuovo Cimento. 1958; 7 :794 - 9.
Deaver B, Fairbank W. Experimental evidence for quantized flux in superconducting cylinders. Physical Review Letters. 1961; 7 (2):43-46 - 10.
Doll R, Näbauer M. Experimental proof of magnetic flux quantization in a superconducting ring. Physical Review Letters. 1961; 7 (2):51-52 - 11.
Khomskii D. Transition Metal Compounds. Cambridge University Press; 2014