Shot noise in ultrathin superconducting wires
NNoname manuscript No. (will be inserted by the editor)
Shot noise in ultrathin superconducting wires
Andrew G. Semenov · Andrei D. Zaikin
Received: date / Accepted: date
Abstract
Quantum phase slips (QPS) may produce non-equilibrium voltage fluctuations in current-biased supercon-ducting nanowires. Making use of the Keldysh techniqueand employing the phase-charge duality arguments we in-vestigate such fluctuations within the four-point measure-ment scheme and demonstrate that shot noise of the voltagedetected in such nanowires may essentially depend on theparticular measurement setup. In long wires the shot noisepower decreases with increasing frequency Ω and vanishesbeyond a threshold value of Ω at T → Keywords
Quantum phase slips · Shot noise · Ultrathinsuperconductors
PACS · · Electric current can flow through a superconducting mate-rial without any resistance. This is perhaps the most fun-damental property of any bulk superconductor which prop-erties are usually well described by means of the standardmean field theory approach. The situation changes, however,provided (some) superconductor dimensions become suffi-ciently small. In this case thermal and/or quantum fluctua-
A.G. Semenov1) I.E.Tamm Department of Theoretical Physics, P.N.Lebedev PhysicsInstitute, 119991 Moscow, Russia2) National Research University Higher School of Economics, 101000Moscow, RussiaE-mail: [email protected]. Zaikin1) Institute of Nanotechnology, Karlsruhe Institute of Technology(KIT), 76021 Karlsruhe, Germany2) I.E.Tamm Department of Theoretical Physics, P.N.Lebedev PhysicsInstitute, 119991 Moscow, RussiaE-mail: [email protected] tions may set in and the system properties may qualitativelydiffer from those of bulk superconducting structures.In what follows we will specifically address fluctuationeffects in ultrathin superconducting wires. In the low tem-perature limit thermal fluctuations in such wires are of littleimportance and their behaviour is dominated by the quantumphase slippage process [1] which causes local temporal sup-pression of the superconducting order parameter ∆ = | ∆ | e i ϕ inside the wire. Each such quantum phase slip (QPS) eventimplies the net phase jump by δ ϕ = ± π accompanied by avoltage pulse δ V = ˙ ϕ / e as well as tunneling of one mag-netic flux quantum Φ ≡ π / e = (cid:82) | δ V ( t ) | dt across the wirenormally to its axis. Different QPS events can be viewed aslogarithmically interacting quantum particles [2] forming a2d gas in space-time characterized by an effective fugacityproportional to the QPS tunneling amplitude per unit wirelength [3] γ QPS ∼ ( g ξ ∆ / ξ ) exp ( − ag ξ ) , a ∼ . (1)Here g ξ = πσ N s / ( e ξ ) (cid:29) ξ , ∆ is the mean field order parametervalue, σ N and s are respectively the wire Drude conductanceand the wire cross section.In the zero temperature limit a quantum phase transitionoccurs in long superconducting wires [2] controlled by theparameter λ ∝ √ s to be specified below. In thinnest wireswith λ < T = λ > R decreaseswith T and takes the form [2] R = d (cid:104) ˆ V (cid:105) dI ∝ (cid:40) γ QPS T λ − , T (cid:29) Φ I , γ QPS I λ − , T (cid:28) Φ I . (2) a r X i v : . [ c ond - m a t . s up r- c on ] M a y Andrew G. Semenov, Andrei D. Zaikin
Superconductor V I x x -L/2 L/2 Fig. 1
The system under consideration.
Hence, the wire non-linear resistance remains non-zero downto lowest temperatures, just as it was observed in a numberof experiments [4,5,6,7].The result (2) combined with the fluctuation-dissipationtheorem (FDT) implies that equilibrium voltage fluctuationsdevelop in superconducting nanowires in the presence ofquantum phase slips. One can also proceed beyond FDTand demonstrate [8,9] that quantum phase slips may gener-ate non-equilibrium voltage fluctuations in ultrathin super-conducting wires. Such fluctuations are associated with theprocess of quantum tunneling of magnetic flux quanta Φ and turn out to obey Poisson statistics. The QPS-inducedshot noise in such wires is characterized by a highly non-trivial dependence of its power spectrum on temperature,frequency and external current.Note that in Refs. [8,9] we addressed a specific noisemeasurement scheme with a voltage detector placed at oneend of a superconducting nanowire while its opposite endwas considered grounded. The main goal of our present workis to demonstrate that quantum shot noise of the voltagein such nanowires may essentially depend on the particularmeasurement setup. Below we will evaluate QPS-inducedvoltage noise within the four-point measurement scheme in-volving two voltage detectors and compare our results withderived earlier [8,9]. Let us consider the system depicted in Fig. 1. It consists ofa superconducting nanowire attached to a current source I and two voltage probes located in the points x and x . Thewire contains a thinner segment of length L region wherequantum phase slips can occur with the amplitude (1).In order to proceed with our analysis of voltage fluctua-tions we will make use of the duality arguments [8,9]. Theeffective dual low-energy Hamiltonian of our system has theformˆ H wire = ˆ H TL + ˆ H QPS . (3)The termˆ H TL = ∞ (cid:90) − ∞ dx (cid:18) L kin (cid:0) ˆ Φ ( x ) + L kin I (cid:1) + e ( ∇ ˆ χ ( x )) C w Φ (cid:19) (4)defines the wire Hamiltonian in the absence of quantum phaseslips. It describes an effective transmission line in terms of two canonically conjugate operators ˆ Φ ( x ) and ˆ χ ( x ) obeyingthe commutation relation [ ˆ Φ ( x ) , ˆ χ ( x (cid:48) )] = − i Φ δ ( x − x (cid:48) ) . (5)Here and below C w denotes geometric capacitance per unitwire length and L kin = / ( πσ N ∆ s ) is the wire kinetic in-ductance (times length). In Eqs. (4), (5) and below ˆ Φ ( x ) isthe magnetic flux operator while the quantum field ˆ χ ( x ) isproportional to the total charge ˆ q ( x ) that has passed throughthe point x up to the some time moment t , i.e. ˆ q ( x ) = − ˆ χ ( x ) / Φ .Hence, the local charge density ˆ ρ ( x ) and the phase differ-ence between the two wire points x and x can be definedasˆ ρ ( x ) = ∇ ˆ χ ( x ) / Φ , (6)ˆ ϕ ( x ) − ˆ ϕ ( x ) = e L kin I ( x − x ) + e x (cid:90) x dx ˆ Φ ( x ) . (7)Employing the expression for the local charge density (or,alternatively, the Josephson relation) it is easy to recover theexpression for the operator corresponding to the voltage dif-ference between the points x and x . It readsˆ V = Φ C w ( ∇ ˆ χ ( x ) − ∇ ˆ χ ( x )) . (8)The term ˆ H QPS in Eq. (3) accounts for QPS effects andhas the form [10]ˆ H QPS = − γ QPS (cid:90) L / − L / dx cos ( ˆ χ ( x )) . (9)It is easy to note that this term is exactly dual to that de-scribing the Josephson coupling energy in spatially extendedJosephson tunnel junctions. In order to investigate QPS-induced voltage fluctuations inour system we will employ the Keldysh path integral tech-nique. We routinely define the variables of interest on theforward and backward time parts of the Keldysh contour, χ F , B ( x , t ) , and introduce the “classical” and “quantum” vari-ables, respectively χ + ( x , t ) = ( χ F ( x , t ) + χ B ( x , t )) / χ − ( x , t ) = χ F ( x , t ) − χ B ( x , t ) . (11)Any general correlator of voltages can be represented in theform [9] (cid:104) V ( t ) V ( t ) ... V ( t n ) (cid:105) = (cid:10) V + ( t ) V + ( t ) ... V + ( t n ) e iS QPS (cid:11) , (12) hot noise in ultrathin superconducting wires 3 where V + ( t ) = Φ C w ( ∇ χ + ( x , t ) − ∇ χ + ( x , t )) , (13) S QPS = − γ QPS (cid:90) dt L / (cid:90) − L / dx sin ( χ + ( x , t )) sin ( χ − ( x , t ) / ) (14)and (cid:104) ... (cid:105) = (cid:90) D χ ( x , t )( ... ) e iS [ χ ] (15)indicates averaging with the effective action S [ χ ] corre-sponding to the Hamiltonian ˆ H TL . It is important to empha-size that Eq. (12) defines the symmetrized voltage correla-tors. E.g., for the voltage-voltage correlator one has (cid:104) V ( t ) V ( t ) (cid:105) = (cid:104) ˆ V ( t ) ˆ V ( t ) + ˆ V ( t ) ˆ V ( t ) (cid:105) , (16)In order to evaluate formally exact expressions for thevoltage correlators (12), (16) one employ the perturbationtheory expanding these expressions in powers of the QPSamplitude γ QPS . It is easy to verify that the first order termsin this expansion vanish and one should proceed up to thesecond order in γ QPS . Due to the quadratic structure of theaction S the result is expressed in terms of the average (cid:104) χ + ( x , t ) (cid:105) = Φ It and the Green functions G K ( x − x (cid:48) , t − t (cid:48) ) = − i (cid:104) χ + ( x , t ) χ + ( x (cid:48) , t (cid:48) ) (cid:105) + i (cid:104) χ + ( x , t ) (cid:105) (cid:104) χ + ( x (cid:48) , t (cid:48) ) (cid:105) , (17) G R ( x − x (cid:48) , t − t (cid:48) ) = − i (cid:104) χ + ( x , t ) χ − ( x (cid:48) , t (cid:48) ) (cid:105) . (18)The Keldysh function G K can also be expressed in the form G K ( x , ω ) =
12 coth (cid:16) ω T (cid:17) (cid:0) G R ( x , ω ) − G R ( x , − ω ) (cid:1) . (19)A simple calculation allows to explicitly evaluate the re-tarded Green function, which in our case reads G R ( x , ω ) = − π i λω + i e i ω | x − x (cid:48)| v . (20)Here v = / √ L kin C w is the plasmon velocity [11] and theparameter λ is defined as λ = R Q / ( Z w ) , where R Q = π / ( e ) is the "superconducting" quantum resistance unit and Z w = (cid:112) L kin / C w being the wire impedance. The above expressions allow to directly evaluate voltage cor-relators perturbatively in γ QPS . In the case of the four-pointmeasurement scheme of Fig. 1 the calculation is similar toone already carried out for the two-point measurements [8,9]. Therefore we can directly proceed to our final results.Evaluating the first moment of the voltage operator (cid:104) ˆ V (cid:105) weagain reproduce the results [2,8] which yield Eq. (2). For thevoltage noise power spectrum S Ω we obtain S Ω = (cid:90) dte i Ω t (cid:104) V ( t ) V ( ) (cid:105) = S ( ) Ω + S QPS Ω , (21)where the term S ( ) Ω describes equilibrium voltage noise inthe absence of QPS (which is of no interest for us here) and S QPS Ω = γ QPS e coth (cid:0) Ω T (cid:1) π C (cid:90) dk π M k ( Ω ) (cid:0) P k ( π I / e ) − ¯ P k ( − π I / e ) + P k ( − π I / e ) − ¯ P k ( π I / e ) (cid:1) + γ QPS e coth (cid:0) Ω T (cid:1) π C (cid:90) dk π S k ( Ω ) S − k ( Ω ) (cid:0) ¯ P k ( − Ω − π I / e ) − P k ( Ω + π I / e ) + ¯ P k ( − Ω + π I / e ) − P k ( − Ω − π I / e ) (cid:1) + γ QPS e (cid:16) coth (cid:16) Ω + π I / e T (cid:17) − coth (cid:0) Ω T (cid:1)(cid:17) π C (cid:90) dk π S k ( Ω ) × S − k ( − Ω ) (cid:0) P k ( Ω + π I / e ) + ¯ P k ( Ω + π I / e ) − P k ( − Ω − π I / e ) − ¯ P k ( − Ω − π I / e ) (cid:1) + (cid:0) Ω → − Ω (cid:1) (22)is the voltage noise power spectrum generated by quantumphase slips. Eq. (22) contains the function P k ( ω ) = (cid:90) dxe ikx ∞ (cid:90) dte i ω t e iG K ( x , t ) − iG K ( , )+ i G R ( x , t ) (23)and geometric form-factors M k ( Ω ) and S k ( Ω ) which ex-plicitly depend on x and x . E.g., for x = L / x = − L / M k ( Ω ) = ( πλ ) e i Ω Lv sin (cid:0) kL (cid:1) vk (cid:32) (cid:0) kL (cid:1) vk + sin (cid:16) ( Ω + vk ) L v (cid:17) Ω + vk + sin (cid:16) ( Ω − vk ) L v (cid:17) Ω − vk (cid:33) , (24) S k ( Ω ) = πλ e i Ω L v (cid:32) sin (cid:16) ( Ω + vk ) L v (cid:17) Ω + vk + sin (cid:16) ( Ω − vk ) L v (cid:17) Ω − vk (cid:33) . (25)We observe that these form-factors oscillate as functions of Ω . Such oscillating behavior stems from the interferenceeffect at the boundaries of a thinner wire segment making Andrew G. Semenov, Andrei D. Zaikin the result for the shot noise in general substantially differ-ent as compared to that measured by means of a setup withone voltage detector [8,9]. In the long wire limit and for Ω (cid:29) v / L one has M k ( Ω ) ≈ π L ( πλ ) v e i Ω Lv δ ( k ) , (26) S k ( Ω ) S − k ( Ω ) ≈ ( πλ ) v e i Ω Lv (cid:18) π L δ (cid:18) k + Ω v (cid:19) + π L δ (cid:18) k − Ω v (cid:19)(cid:19) , (27) S k ( Ω ) S − k ( − Ω ) ≈ ( πλ ) v (cid:18) π L δ (cid:18) k + Ω v (cid:19) + π L δ (cid:18) k − Ω v (cid:19)(cid:19) . (28)Neglecting the contributions (26) and (27) containing fastoscillating factors e i Ω Lv and combining the remaining term(28) with Eq. (22), we obtain S QPS Ω = L π γ QPS e sinh (cid:16) Φ I T (cid:17) ς (cid:16) Φ I (cid:17) sinh (cid:0) Ω T (cid:1) × (cid:18) ς (cid:18) Φ I − Ω (cid:19) − ς (cid:18) Φ I + Ω (cid:19)(cid:19) . (29)where ς ( ω ) = τ λ ( π T ) λ − Γ (cid:16) λ − i ω π T (cid:17) Γ (cid:16) λ + i ω π T (cid:17) Γ ( λ ) , (30) τ ∼ / ∆ is the QPS core size in time and Γ ( x ) is the Eu-ler Gamma-function. Note that the result (29) turns out tobe two times smaller as compared that derived within an-other measurement scheme [8,9] in the corresponding limit.In other words, shot noise measured by each of our two de-tectors is 4 times smaller than that detected with the aid ofthe setup [8,9]. The result (29) is illustrated in Fig. 2.At T → S QPS Ω ∝ (cid:40) I λ − ( I − Ω / Φ ) λ − , Ω < Φ I / , , Ω > Φ I / . (31)In order to interpret this threshold behaviour let us bear inmind that at T = N plas-mons ( N = , ... ) with total energy E = Φ I and total zeromomentum. The left and the right moving plasmons (eachgroup carrying total energy E /
2) eventually reach respec-tively the left and the right voltage probes which then detectvoltage fluctuations with frequency Ω . As in the long wirelimit and for non-zero Ω these two groups of plasmons be-come totally uncorrelated, it is obvious that at T = Fig. 2
The dependence of QPS noise power S Ω (29) at λ = . Ω and temperature T . noise can only be detected at Ω < E / T → Φ I / References
1. K.Yu. Arutyunov, D.S. Golubev, and A.D. Zaikin, Superonductiv-ity in one dimension, Phys. Rep. , 1 (2008).2. A.D. Zaikin et al. , Quantum phase slips and transport in ultrathinsuperconducting wires, Phys. Rev. Lett. , 1552 (1997).3. D.S. Golubev and A.D. Zaikin, Quantum tunneling of the orderparameter in superconducting nanowires, Phys. Rev. B , 014504(2001).4. A. Bezryadin, C.N. Lau, and M. Tinkham, Quantum suppres-sion of superconductivity in ultrathin nanowires, Nature , 971(2000).5. C.N. Lau et al. , Quantum phase slips in superconductingnanowires, Phys. Rev. Lett. , 217003 (2001).6. M. Zgirski et al. , Quantum fluctuations in ultranarrow super-cooducting aluminum nanowires, Phys. Rev. B , 054508 (2008).7. X.D.A. Baumans et al. , Thermal and quantum depletion of super-condutivity in narrow junctions created by controlled electromi-gration, Nat. Commun. , 10560 (2016).8. A.G. Semenov and A.D. Zaikin, Quantum phase slip noise, Phys.Rev. B , 014512 (2016).hot noise in ultrathin superconducting wires 59. A.G. Semenov and A.D. Zaikin, Quantum phase slips and volt-age fluctuations in superconducting nanowires, Fortschr. Phys. ,XXX (2016).10. A.G. Semenov and A.D. Zaikin, Persistent currents in quantumphase slip rings, Phys. Rev. B , 054505 (2013).11. J.E. Mooij and G. Schön, Propagating plasma mode in thin super-conducting filaments, Phys. Rev. Lett.55