spectroscopy in superconductors
In this chapter the basic concepts of conventional quasiparticle tunneling spectroscopic studies of superconductors are reviewed. Additionally, how these analyses are extended to fitting the quasi- particle tunneling spectra of cuprate superconductors are discussed. It will become clear that the assumption of pure superconductivity in the ground state of cuprate superconductors fails to con- sistently account for a wide variety of quasiparticle tunneling spectra in the cuprates. These anal- yses together with overwhelming empirical evidences of competing orders in the cuprates therefore suggest that modifications to the conventional theory for quasiparticle tunneling spectroscopy in superconductors are necessary.
2.1 Physics of normal metal(N)-insulator(I)-superconductor(S)
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was directly measured, and it was supposed that the measured quantum tunneling current was determined solely by the quasiparticle density of states of the superconductor. Bardeen confirmed this interpretation through his transfer Hamiltonian formalism.
The Bardeen formalism assumes that two systems, representing two materials, are separated by a tunneling barrier and that the quantum transmission through the barrier is small enough to treat the transmission as a perturbation. The formalism predicts the quantum mechanical transmission current through an insulating barrier from one system to another, and, for simplicity, we describe the formalism for the simple one-dimensional scenario shown in Fig. 2.1. The transmission is assumed to be small enough that the left- and right-hand systems may be treated as almost independent in Bardeen’s approach. Specifically, the density of states on each side is considered approximately equivalent to the case where the systems are completely separate, and a small perturbation transfer Hamiltonian, HT, is used in time-dependent perturbation theory to describe the transition proba- bilities for the left and right states to quantum mechanically tunnel through the barrier.
The Hamiltonian for the time-dependent treatment is given by
H =Hl+Hr+HT =H0+HT, H0Ψl=ElΨl,
H0Ψr=ErΨr,
(2.1)
where Ψl is the wavefunction of the left system, Ψris the wavefunction of the right system, andHT is the transfer Hamiltonian. The transfer probability for the left system from the initial state Ψl to the final state Ψr is calculated using the time-dependent Schr¨ondinger equation:
HΨ(t) =i~dΨ(t)
dt . (2.2)
Using theAnsatz
Ψ(t) =c(t)Ψle−iElt/~+d(t)Ψre−iErt/~ (2.3)
~
whereNr is the density of states on the right system andMrl is the tunneling matrix..
To fully describe the quantum tunneling current from left to right electrodes when a voltage bias (eV) is applied between the two systems separated by a tunneling barrier, as illustrated in Fig. 2.2, the Fermi function must be accounted for in Eq. 2.5. The transmitted current from left to right must be modified to account for the availability of density of states in the right system into which the electrons from the left system may tunnel, and the result is
jt=2π
~ |Mrl|2Nr(1−fr), (2.6)
wherefris the Fermi function in the right system,fr= 1
eE/kB T+1, andE=−F is the energy, , measured relative to the Fermi level,F. The total current from left to right with a voltage bias of eV is
jrl∝ Z ∞
−∞
|M |2Nl(E)Nr(E+eV)fl(E)[1−fr(E+eV)]dE. (2.7)
The transition of the right system from the initial state, Ψr, to the final state, Ψl, follows similarly, such that the net current is
j=jrl−jlr∝ Z ∞
−∞
|M |2Nl(E)Nr(E+eV)[fl(E)−fr(E+eV)]dE. (2.8)
For energy, E, near the Fermi level and at low temperatures, the value of the matrix element squared | M |2 is assumed to be a constant that depends on the separation of the left and right
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Figure 2.1: Simple schematic of quantum tunneling in one-dimension: The figure shows a simple one-dimensional rectangular potential for modeling quantum mechanical tunneling in one-dimension between two systems (left and right), using Bardeen’s transfer Hamiltonian approach [88]. The approach is a time-dependent perturbation theoretical treatment to predict the quantum tunneling current that transfers between the left and right systems. The wavefunction of the left system, Ψl, and the wavefunction of the right system, Ψr, are labeled. For the left system, the formalism describes the transition rate from the initial state Ψlto the final state, Ψr, through interaction with the transfer Hamiltonian, HT. The transition rate of the right-hand system is treated similarly.
systems. More concretely, the left and right systems may be considered to represent the tip (t) and sample (s) in an STM junction, separated by distance,z, and the tunneling matrix for a three- dimensional treatment is then given by
Mts=−~2 2m
Z
dS·(Ψ∗t∇Ψs−Ψs∇Ψ∗t), (2.9)
according to Ref. [88]. Here, dS is a surface integral over a surface within the tunneling barrier separating the tip and sample. Following the simple analysis of Refs. [91, 92], it is found that
|M |2∝exp(−2
√2mφ
~
z), (2.10)
where φ is the convoluted work function, or effective tunneling barrier, between the tip and the sample. The work function of simple metals, such as gold, are typically on the order of a feweV.
Assuming, in Eq. 2.8, that the tip is a simple metal with constant density of states and that
Figure 2.2: Bandstructure picture of quantum tunneling between two systems: (a) Schematic band- structure picture of quantum tunneling between two metals at T = 0. The Fermi level is denoted EF. Metals are expected to have a constant density of states that are filled toEF at T = 0. (b) Schematic bandstructure picture of quantum tunneling between a metal and a superconductor at T
= 0. The quasiparticle density of states for the superconductor is gapped by ∆, such that quantum tunneling cannot occur for|eV|<∆.
|M|2 obeys Eq. 2.10, we may take|M|2andNt(E) outside of the integral in Eq. 2.8. The tunneling current at low temperature is then approximately given by
I∝ Z ∞
−∞
Ns(E+eV)[Θ(E+eV)−Θ(E)]dE= Z eV
0
Ns(E+eV)dE, (2.11)
where Θ(x) is the heaviside function. The derivative of Eq. 2.11 for a superconductor is directly proportional to the quasiparticle density of states of the superconductor,Ns(eV):
dI
dV ∝Ns(eV). (2.12)
To compute the density of states at finite temperatures, the Fermi functions must be retained in Eq. 2.8. The measured density of states is then approximately given by
dI/dV ' Z ∞
−∞
| df(E+eV)
dT | dI
dV(E)dE. (2.13)
We measure the quasiparticle tunneling density of states in high-temperature cuprate supercon- ductors for temperatures TTC to T>TC. Based on Eqs. 2.12 and 2.13, we are then measuring the quasiparticle density of states of the superconductors.
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