I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia
National Research University Higher School of Economics, 101000 Moscow, Russia
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia
Department of Physics, Moscow Pedagogical State University, 119435 Moscow, Russia
National Research University Higher School of Economics, 101000 Moscow, Russia
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology (KIT), 76021 Karlsruhe, Germany
^{1}I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia
^{2}National Research University Higher School of Economics, 101000 Moscow, Russia
^{3}Department of Physics, Moscow Pedagogical State University, 119435 Moscow, Russia
^{4}Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology (KIT), 76021 Karlsruhe, Germany
Guest Editor: A. S. Sidorenko
Beilstein J. Nanotechnol. 2020, 11, 1402–1408. https://doi.org/10.3762/bjnano.11.124
Received 31 May 2020,
Accepted 27 Aug 2020,
Published 14 Sep 2020
A non-peer-reviewed version of this article has been posted as a preprint https://doi.org/10.3762/bxiv.2020.65.v1
We investigate superconductor–insulator quantum phase transitions in ultrathin capacitively coupled superconducting nanowires with proliferating quantum phase slips. We derive a set of coupled Berezinskii–Kosterlitz–Thouless-like renormalization group equations demonstrating that interaction between quantum phase slips in one of the wires gets modified due to the effect of plasma modes propagating in another wire. As a result, the superconductor–insulator phase transition in each of the wires is controlled not only by its own parameters but also by those of the neighboring wire as well as by mutual capacitance. We argue that superconducting nanowires with properly chosen parameters may turn insulating once they are brought sufficiently close to each other.
Keywords: quantum phase slips; quantum phase transitions; RG equations
Quantum fluctuations dominate the physics of superconducting nanowires at sufficiently low temperatures making their behavior markedly different from that of bulk superconductors [1-4]. Many interesting properties of such nanowires are attributed to the effect of quantum phase slips (QPSs) which correspond to fluctuation-induced local temporal suppression of the superconducting order parameter inside the wire accompanied by the phase slippage process and quantum fluctuations of the voltage in the form of pulses. By applying a bias current the symmetry between positive and negative voltage pulses is broken and, as a result, a superconducting nanowire acquires a non-vanishing electrical resistance down to the lowest temperatures [5,6]. This effect was directly observed in a number of experiments [7-10].
Likewise, quantum phase slips in superconducting nanowires yield shot noise of the voltage [11] which originates from the process of quantum tunneling of magnetic flux quanta across the wire. One can also proceed beyond the voltage–voltage correlator and evaluate all cumulants of the voltage operator, thus deriving full counting statistics of quantum phase slips [12]. This theory enables one to obtain a complete description of superconducting fluctuations in such nanowires. Interesting QPS-related effects also occur in superconducting nanorings which can be employed, for example, for possible realization of superconducting qubits [13]. Such effects were investigated theoretically [14] and observed experimentally [15,16].
Each quantum phase slip generates sound-like plasma modes [17] which propagate along the wire and interact with other QPSs. The exchange of such Mooij–Schön plasmons produces a logarithmic interaction in space–time between different QPSs where the magnitude is controlled by the wire diameter (cross section) [5]. For sufficiently thick wires this interaction is strong and the QPSs are bound in close pairs. Accordingly, the (linear) resistance of such wires tends to zero at T → 0, thus demonstrating a superconducting-like behavior in this limit. On the other hand, inter-QPS interaction in ultrathin wires is weak, quantum phase slips are unbound and the superconducting phase fluctuates strongly along the wire. In this case the wire looses long-scale superconducting properties, its total resistance remains non-zero and even tends to increase with decreasing temperature thus indicating an insulating behavior at T → 0. At zero temperature the transition between these two types of behavior comes as a quantum phase transition (QPT) driven by the wire diameter [5]. Below we will also refer to this QPT as a superconductor–insulator transition (SIT).
In this work we will show that this SIT can be substantially modified in a system of capacitively coupled superconducting nanowires even without any direct electrical contact between them. In our previous work [18] we already elucidated some non-local QPS-related effects in such nanowires which yield non-equilibrium voltage fluctuations in the system which exhibit a non-trivial dependence on frequency and bias current. Here we will demonstrate that quantum fluctuations in one of the two wires effectively ”add up” to those of another one, thereby shifting the QPT in each of the wires in a way to increase the parameter range for the insulating phase. Qualitatively the same effect is expected to occur in a single superconducting nanowire that has the form of a meander frequently used in experiments.
We first consider the system of two long superconducting nanowires parallel to each other, as schematically shown in Figure 1a.
The wires are described by geometric capacitances C_{1} and C_{2} (per unit wire length) and kinetic inductances and (times length) effectively representing the two transmission lines. Capacitive coupling between these two nanowires is accounted for by the mutual capacitance C_{m}. The corresponding contribution to the system Hamiltonian that keeps track of both electric and magnetic energies in these coupled transmission lines reads
where x is the coordinate along the wires, and C_{ij} denote the matrix elements of the inductance and capacitance matrices
and Φ_{0} = π/e is the superconducting flux quantum. Note that for the sake of simplicity here and below we set Planck constant ℏ, speed of light, c, and Boltzmann constant, k_{B}, equal to unity.
The Hamiltonian (Equation 1) is expressed in terms of the dual operators and [14] which obey the canonical commutation relation
and are related to the charge density and the local phase operators, and respectively, by means of the following equations
Physically, represents the magnetic flux operator, while the operator is proportional to that of the total charge that has passed through the point x of the ith wire up to the time moment t, i.e.,
Provided that the wires are thick enough, the low energy Hamiltonian in Equation 1 is sufficient. However, for thinner wires, one should also account for the effect of quantum phase slips. The corresponding contribution to the total Hamiltonian for our system can be expressed in the form [14]
where
denotes the QPS amplitudes per unit wire length [6], g_{j}_{ξ} = R_{q}/R_{j}_{ξ} is the dimensionless conductance of the jth wire segment of length equal to the superconducting coherence length ξ (here and below R_{q} = 2π/e^{2} ≃ 25.8 kΩ is the quantum resistance unit and R_{j}_{ξ} is the normal state resistance of the corresponding wire segment), Δ is the superconducting order parameter and a ≈ 1 is a numerical prefactor. We also note that the Hamiltonian (Equation 5) describes tunneling of the magnetic flux quantum, Φ_{0}, across the wire and can be viewed as a linear combination of creation and annihilation operators for the flux quantum Φ_{0}.
It is obvious from Equation 4 that QPS events cause redistribution of charges inside the wire and generate pairs of voltage pulses moving simultaneously in the opposite direction (cf., Figure 1a)
Clearly, in the presence of capacitive coupling quantum phase slips in one of the wires also generate voltage pulses in another one.
To summarize the above considerations, the total Hamiltonian for our system is defined as a sum of the two terms in Equation 1 and Equation 5,
representing an effective sine-Gordon model that will be treated below.
In order to quantitatively describe QPT in coupled superconducting wires we will employ the renormalization group (RG) analysis. This approach is well developed and was successfully applied to a variety of problems in condensed matter theory, such as, the problem of weak Coulomb blockade in tunnel [19-22] and non-tunnel [23-25] barriers between normal metals or that of a dissipative phase transition in resistively shunted Josephson junctions [19,26-28]. In the case of superconducting nanowires QPT was described [5] with the aid of RG equations equivalent to those initially developed for two-dimensional superconducting films [29] which exhibit classical Berezinskii–Kosterlitz–Thouless (BKT) phase transition driven by temperature. In contrast, quantum SIT in quasi-one-dimensional superconducting wires [5] with geometric capacitance C and kinetic inductance is controlled by the parameter [5]
which is proportional to the square root of the wire cross section, s.
It follows immediately from the analysis of [5] that, provided the two superconducting wires depicted in Figure 1a are decoupled from each other (i.e., for C_{m} → 0), one should expect two independent QPTs to occur in these two wires respectively at λ_{1} = 2 and at λ_{2} = 2 where, according to Equation 9, we define The task at hand is to investigate the effect of capacitive coupling between the wires on these two QPTs.
For this purpose let us express the grand partition function of our system in terms of the path integral
where
is the effective action corresponding to the Hamiltonian (Equation 8) and
denotes the effective fugacity for the gas of quantum phase slips in the ith wire. Note that, having in mind that the QPS core size in x- and τ-directions is respectively x_{0} ∼ ξ and τ_{0} ∼ Δ^{−1}, in Equation 11 for the sake of convenience we rescaled the spatial coordinate in units of x_{0}, i.e. x → xξ and the time coordinate in units of τ_{0}, i.e. τ → τ/Δ.
In the spirit of Wilson’s RG approach we routinely divide the χ-variables into fast and slow components, , where
Setting δΛ/Λ ≪ 1, expanding in the fast field components and integrating them out we proceed perturbatively in y_{1,2} and observe that in order to account for the leading order corrections it is necessary to evaluate the matrix Green function at coincident points which reads
where and is the velocity matrix for plasmon modes propagating along the wires. The matrix has the form
where is the velocity of the Mooij–Schön modes in the ith wire in the absence of capacitive coupling between the wires, i.e. for C_{m} → 0.
Following the standard procedure [29] and proceeding to bigger and bigger scales Λ, we eventually arrive at the following RG equations for the QPS fugacities y_{1} and y_{2}:
where λ_{11} and λ_{22} are diagonal elements of the matrix (Equation 15). Note that here we restrict our RG analysis to the lowest order in y_{1,2} which is sufficient for our purposes. As long as one keeps only the linear y_{1,2} terms in the RG equations, all other parameters of our problem, e.g., λ_{ii}, remain un-renormalized.
As it can be observed from Equation 16, our system exhibits two BKT-like QPTs at λ_{11} = 2 and λ_{22} = 2. In the limit C_{m} → 0 the wires are independent from each other, λ_{11(22)} → λ_{1(2)} and these QPTs obviously reduce to that predicted in [5]. However, for non-zero capacitive coupling between the wires, the two QPTs occur at the values of λ_{1,2} exceeding 2. For the first wire the corresponding phase transition point is fixed by the condition
The same condition for the second wire is obtained from (Equation 17) by interchanging the indices 1 ↔ 2.
The above results allow us to conclude that in the presence of capacitive coupling SIT in both wires occurs at larger values of λ_{1,2} than in the absence of such coupling. In other words, quantum fluctuations in one of these wires effectively decrease the superconducting properties of the other one.
It follows from Equation 17 that the magnitude of such mutual influence depends on the ratio of the plasmon velocities in the two wires v_{1}/v_{2} and on the strength of the capacitive coupling controlled by C_{m}. Provided the wire cross sections s_{1} and s_{2} differ strongly the plasmon velocities also differ considerably. Assume, for instance, that the first wire is much thinner than the second one. In this limit we have v_{1} ≪ v_{2} and, hence, the QPT condition (Equation 17) in the first wire remains almost unaffected for any capacitive coupling strength. If, on the contrary, the first wire is much thicker than the second one, then one has v_{1} ≫ v_{2} and the condition (Equation 17) reduces to demonstrating that the critical value λ_{1} can exceed 2 considerably for sufficiently large C_{m} values.
It is obvious that the strength of capacitive coupling depends on the distance between the wires. At large distances this coupling is negligible C_{m} → 0. However, as the wires get closer to each other the value C_{m} increases and, hence, their mutual influence increases as well. Let us choose the wire parameters in such a way that for C_{m} = 0 both these wires remain in the superconducting phase being relatively close to SIT. In this case the parameters λ_{1} and λ_{2} should be just slightly larger than 2. Moving the wires closer to each other we ”turn on” the capacitive coupling between them, thus, decreasing both values λ_{1} and λ_{2} to less than 2. As a result, two superconducting wires become insulating as soon as they are brought sufficiently close to each other. This remarkable physical phenomenon is illustrated by the phase diagram in Figure 2b.
In order to complete this part of our analysis, we point out that transport properties can be investigated in exactly the same manner as was done in [5] in the case of a single nanowire. Generalization of the technique [5] to the case of two capacitively coupled superconducting nanowires is straightforward. For a linear resistance of the ith wire R_{i}(T) and for λ_{ii} > 2 (or for any λ_{ii} at sufficiently high temperatures) we obtain
The effects discussed here can be observed in a variety of structures involving superconducting nanowires. For instance, superconducting nanowires in the form of a meander (see Figure 1b) are frequently employed in experiments [30]. In this case different segments of the wire are parallel to each other being close enough to develop electromagnetic coupling. Having in mind the above analysis, one expects that the wire of such a geometry would be ”less superconducting” than the same wire that has the form of a straight line.
For illustration, let us mimic the behavior of the wire depicted in Figure 1b by considering three identical capacitively coupled superconducting nanowires parallel to each other. For simplicity we will assume the nearest neighbor interaction, that is, the second (central) nanowire is coupled to both the first and the third nanowires via the mutual capacitance, C_{m}, whereas the latter two are decoupled from each other. We again assume that the wires are thin enough and quantum phase slips may proliferate in each of these wires.
The quantum properties of this system are described by the same effective action (Equation 11) where the inductance and capacitance matrices now take the form
and the summation runs over the indices i,j = 1,2,3. Proceeding along the same lines as in the previous section we again arrive at Equation 14, where the diagonal elements of the matrix now read
and the QPS interaction parameter λ is defined in Equation 9. We again arrive at the RG equations of the form (Equation 16) (now with i = 1,2,3). Being combined with Equation 20 and Equation 21 these RG equations demonstrate that in the presence of capacitive coupling SETs occur at λ_{ii} = 2 implying λ > 2 for each of the three wires. This observation is fully consistent with our previous results derived for two coupled nanowires.
Furthermore, the RG equation (Equation 16) with i = 2 combined with Equation 20 also describes the effect of interacting quantum phase slips and QPTs in the wire having the form of a meander (Figure 1b). In this case, within the approximation of the nearest neighbor, capacitive interaction between the wire segments QPT occurs at
that is, the critical value of the parameter λ exceeds 2 as soon as the mutual capacitance C_{m} differs from zero. As it is clear from Equation 20 and Equation 21, the approximation of the nearest neighbor interaction appears to be well justified in the limit C_{m} ≪ C. For stronger interactions with C_{m} ≈ C this approximation most likely becomes insufficient for a quantitative analysis. However, on a qualitative level our key observations should hold also in this case: A nanowire in the form of a straight line with λ slightly exceeding the critical value 2 should demonstrate superconducting-like behavior with R(T) ∝ T^{2λ−3} [5] whereas a wire with exactly the same parameters may turn insulating provided it has the form of a meander with capacitive coupling between its segments.
We have analyzed the effect of quantum fluctuations in capacitively coupled superconducting nanowires. We have demonstrated that plasma modes propagating in one such nanowire play the role of an effective quantum environment for another one, modifying the logarithmic interaction between quantum phase slips in this wire. As a result, the superconductor–insulator quantum phase transition gets shifted in a way to increase the parameter range for the insulating phase. Hence, superconducting nanowires may turn insulating provided they are brought close enough to each other. It would be interesting to observe this effect in forthcoming experiments with superconducting nanowires.
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