Eg = 2∆√
1−T opens at φ=π. The backscattering of electrons leads to an avoided crossing of the Andreev bound states. Independently of the transmission, the Andreev bound states reach the gap edgeEA= ∆ atφ→0 andφ→2π. The current carried by the Andreev bound states is calculated by
I(φ) =−2e
~ X
s=±
∂EAs
∂φ tanh(βEAs/2). (2.35) withβ = 1/kBT. The last term im Eq. (2.35) contains the occupation of the Andreev bound states.
2.4 Noise
The measurement of fluctuations in macroscopic observables provides information about the microscopic dynamics not accessible by the measurement of averaged quan-tities. This is the case for the averaged current and the noise in nanocontacts. In these contacts, different sources contribute to fluctuations. At non-zero temperature, thermal noise causes the occupation number to fluctuate. Shot noise is a further fundamental source of fluctuations and appears due to the discreteness of the charge and a finite probability of transmission at the interface. This kind of noise occurs in systems driven out-of equilibrium. In a stationary system, we define the non-symmetrized noise as [12]
S(ω) = 2Z ∞
−∞
d(t−t0)eiω(t−t0)hIˆ(t)ˆI(t0)i, (2.36) with the current operator ˆI(t). Since operators in general do not commute, we have thatS(ω)6=S(−ω). In the following, we discuss the thermal and shot noise and study in Sec. 2.4.1 as an example the noise in a two terminal contact. The description of the thermal noise and the shot noise follows the Refs. [12] and [17].
Thermal noise
In a conductor the available states are filled with electrons and a state at energy ε is either occupied (nε = 1) or empty (nε = 0). The statistical average of the occupation at temperature T is determined by the Fermi function f(ε) = hnεi = 1/[1 + exp(ε/kBT)] such that the probability of an occupied state at energyεis given byf(ε). Fluctuations around this value can be calculated byh∆nεi=h(nε−hnεi)2i= hn2εi − hnεi2. Taking into account that the states are either occupied or empty, the fluctuations can be simplified to h∆nεi = f(ε)(1−f(ε)). Hence, the thermal fluctuations are given by the probability that a state is occupiedf(ε) multiplied with the probability that the state is empty 1−f(ε). At zero temperature the fluctuations of the occupation vanish and the fluctuations reduce to the Boltzmann distribution at large temperature (kBT ε).
Shot noise
The shot noise (partition noise) is a fundamental source of noise and appears due to the discreteness of the charge in systems driven out of equilibrium (eV 6= 0) and a finite transmission between, for instance, a two terminal contact. To explain the origin of shot noise, we consider a two terminal contact with one channel and a scattering region in between. We want to calculate the probabilityPN that out ofN∆t attempts in the time interval ∆t, N electrons are transferred through the scattering region. An electron incoming at the scattering region can be either transmitted with probability T or reflected with probability 1−T. Under the assumption that the tunneling events are independent, the probability that N electrons tunnel through the contact and Nat−N are reflected, is given byTN(1−T)Nat−N. Since there are
Nat
N
!
different combinations of theN transmission andNat−N reflections, the total probability thatN electrons are transfered through the contact is [12]
PN = Nat
N
!
TN(1−T)Nat−N (0≤N ≤Nat) , (2.37) which is known as the binomial distribution. From this distribution, we can calculate the average of the transmitted electronshNiand the varianceh∆N2iwhich are given by hNi = NatT and h∆N2i = hNi(1−T), respectively. The transmitted electrons give rise to a current which is I =N e/twith the chargeeof the electrons. The noise at zero frequency in Eq. (2.36) is then calculated by
S(0) = 2ehIi(1−T) . (2.38) If the number of electrons tunneling through the contact goes to infinity and the probability for transmission goes to zero (T 1), the binomial distribution reduces to the Poission distribution. At small transmission, the time interval of two elec-trons tunneling through the scattering region becomes large, and hence the Poission distribution describes the transfer of uncorrelated electrons. In this limit, the zero-frequency noise is given by S(0) = 2ehIi which is known as the Schottky formula.
Deviation from the Schottky value of noise are manifested in the Fano factor F = S(0)
2ehIi, (2.39)
defined as the ratio between the zero-frequency noise and the Schottky value. If F >1, the noise is called superpoissonian, ifF <1 subpoissonian.
2.4.1 Noise in a two terminal contact
In this section, we consider as an example a two terminal contact and discuss the zero-frequency and the frequency-dependent noise. The contact consist of a number of energy-independent channels Ti. A derivation of the noise by using the scattering formalism is given in Refs. [12] and [17]. The discussion in this section is useful to understand the results in chapter 6 in which we study the current noise in normal and superconducting contacts by the Green’s function technique.
2.4 Noise
Zero-frequency noise
The zero-frequency noise in a two terminal contact is given by [17]
S(0) = 4e2 the thermal noiseS(0) = (8e2/h)kBTPiTi. Via the fluctuation-dissipation theorem, the thermal equilibrium noise is directly related to the conductanceG= (2e2/h)PTi
of the system byS(0) = 4GkBT. Hence, the measurement of the thermal noise will give the same information as the measurement of the conductance. In the opposite limit of zero temperature (T = 0), the noise is given by the shot noise
S(0) = (4e2/h)X
i
Ti(1−Ti)eV . (2.41) In the limit of small transmission, Ti 1, we obtain the Schottky formula [18], S(0) = 2ehIi. The Fano factor is determined by
The frequency-dependent noise and the shot noise are similar expression except that the energies in the Fermi function are shifted by±ω. A calculation of the frequency-dependent noise for a two terminal contact gives the result [17]
S(ω) = 4e2 At zero temperature the noise can be written as (eV >0)
S(ω) = 2G F given by Eq. (2.39). The frequency-dependent noise for one channel and different