Dynamical properties of superconducting nanowires

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2006, Vol. 102, No. 2, pp. 327–333. © Pleiades Publishing, Inc., 2006Original Russian Text © S.V. Nikolaev, K.N. Yugay, 2006, published in Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, 2006, Vol. 129, No. 2, pp. 371–377.

327

1. INTRODUCTION

One-dimensional superconductors (nanowires) fea-ture unusual physical phenomena not observed in bulksuperconductors. In particular, a nonzero resistanceappears in the absence of an external magnetic field ata current density much greater than the critical value.This phenomenon was observed for the first time in theexperiments reported in [1, 2]. Subsequent experimen-tal investigations [3–8] led to the conclusion that animportant role in this phenomenon is played by thermalfluctuations, which induce the process of phase slip-page. At temperatures close to the critical value,the main role is played by the thermally activated phaseslip [4–8], while at significantly lower temperatures,the main factor is the quantum tunneling of the super-conductor order parameter via the free energy barrier,or the quantum phase slip [3, 5–9].

The important role of thermal fluctuations in one-dimensional superconducting systems was originallypointed out in a theoretical study by Little [10]. Thisidea has received further development in the classicalpapers [11, 12], where it was theoretically demon-strated that large, low-probable thermal fluctuations inone-dimensional superconducting systems give rise toa phase slip due to the quantum tunneling of the orderparameter via the Ginzburg–Landau free energy bar-rier. Duan and Renn [13, 14] determined the tempera-ture dependence of the resistance of a thin supercon-ducting wire related to the thermally activated phaseslip in the presence of electric and magnetic fields. Itwas shown that the fields make a small contribution to

suppression of the superconducting order parameterunder the phase slip conditions. Subsequent investiga-tions [15–17] confirmed that the quantum phase slipdetermines the temperature dependence of the resis-tance of thin superconducting wires at low tempera-tures.

The situation significantly changes if the current ina nanowire is close to the critical value. In this case, thesystem exhibits time-dependent behavior even for aconstant current. Recently, it was demonstrated [18]that the phase slip process in a superconducting nano-wire under such conditions is determined, in contrast tothe case of small currents, by the dynamic phenomena.The dynamic mechanism of suppression of the super-conducting order parameter is related to the internalnonlinear properties of the system under consideration,which are not manifested when the current density issignificantly below the critical level (this case was stud-ied previously [1–17]).

In this study, as well as in [18], we consider a super-conducting nanowire under conditions in which thedynamical mechanism of the phase slip is operative.However, in contrast to [18], we will study the electro-magnetic waves generated by a superconductingnanowire featuring the phase slip. The analysis is basedon numerical solutions of a one-dimensional time-dependent Ginzburg–Landau equation with an orderparameter. The obtained solutions showed that, for cer-tain values of the order parameter

u

characteristic of a“pure” superconductor (

u

> 1), there are two criticalcurrent density values,

j

c

1

and

j

c

2

. As the order parame-

Dynamical Properties of Superconducting Nanowires

S. V. Nikolaev and K. N. Yugay

Omsk State University, Omsk, 644077 Russiae-mail: [email protected]

Received July 11, 2005

Abstract

—The dynamical properties of thin superconducting wires (nanowires) are studied using numericalsimulations based on a one-dimensional time-dependent Ginzburg–Landau equation, which is modified byintroducing an order parameter

u

characterizing the “purity” of the superconductor material. It is establishedthat relatively long nanowires (with lengths much greater than the coherence length) made of a “pure” super-conductor (

u

> 1) are characterized by two critical current density values:

j

c

1

and

j

c

2

. For

j

<

j

c

1

, the total currentis entirely superconducting, whereas for

j

>

j

c

2

, the current is purely normal. In the intermediate region of cur-rent densities,

j

c

1

<

j

<

j

c

2

, the total current contains both superconducting and normal components (mixed state)and the nanowire exhibits the generation of high-frequency electromagnetic waves. The current–voltage char-acteristics are constructed and the radiation spectrum is obtained. The properties of short superconductingnanowires (with lengths on the order of the coherence length) coincide with those of the Josephson junction. Inthe case of an “impure” superconductor (

u

< 1), the nanowire is characterized by a single critical currentdensity.

PACS numbers: 74.20.De, 74.40.+k, 74.25.Sv

DOI:

10.1134/S1063776106020130

ELECTRONIC PROPERTIESOF SOLIDS

328

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 102

No. 2

2006

NIKOLAEV, YUGAY

ter

u

decreases, the

j

c

2

value approaches

j

c

1

. In the inter-mediate region of current densities,

j

c

1

<

j

<

j

c

2

, the cur-rent–voltage characteristic exhibits an anomaly and thesuperconducting nanowire occurs in a mixed state. Inthis region, the nanowire was found to generate high-frequency electromagnetic waves. We propose a mech-anism responsible for this radiation, which is based onthe theory of Josephson junctions and on the notion ofdynamical phase slip centers.

2. CURRENT–VOLTAGE CHARACTERISTICSOF A SUPERCONDUCTING NANOWIRE

Evolution of the order parameter in a superconduct-ing nanowire carrying a constant current is describedusing a one-dimensional time-dependent Ginzburg–Landau equation that can be written as [19]

(1)

where the variables are normalized as follows:

(2)

Here,

ξ

is the coherence length,

∆

0

is the equilibriumenergy gap,

t

0

is the order parameter relaxation time,

µ

is the chemical potential,

is the corresponding normalization factor,

j

is the totalcurrent density,

m

is the electron mass, and

N

is the den-sity of conduction electrons.

The order parameter

u

introduced into Eq. (1) isdefined as follows:

(3)

where

τ

l

=

l

/

ν

F

,

l

is the electron mean free path and

ν

F

isthe electron velocity on the Fermi surface. Formula (3)can be also written as

(4)

∂∂t-----ψ iµψ+

∂2

∂x2--------ψ ψ ψ 2ψ,–+=

j Im ψ*∂

∂x------ψ⎝ ⎠

⎛ ⎞ u∂

∂x------µ,–=

xxξ--, ψ ∆

∆0-----, t

tt0---,

t0π

8k Tc T–( )--------------------------, µ µ

µ0-----,=

jjj0----, j0

eN2mξ----------.=

µ08k Tc T–( )

π--------------------------=

u8k Tc T–( )τl

π------------------------------,=

u 0.46l

ξ0----- 1 T

Tc

-----–⎝ ⎠⎛ ⎞ ,=

where

(5)

This expression reveals the physical meaning of theorder parameter as characterizing the nanowire mate-rial purity, where the cases of

u

< 1 (

l

ξ

0

) and

u

> 1(

l

ξ0) correspond to an “impure” and “pure” semi-conductor, respectively.

The voltage drop on the nanowire is defined as

(6)

where Φ is the electric potential and L is the nanowirelength normalized to ξ. Taking into account that µ =2eΦ and normalizing this value to µ0, we obtain

(7)

The normalization of this value to

yields an expression for the dimensionless voltagedrop:

(8)

Now let us determine the initial and boundary con-ditions for system of equations (1). The boundary con-ditions can be written as

(9)

where ϕ0 is the phase of the wavefunction of supercon-ducting electrons at the nanowire boundary. Accordingto the second Josephson relationship, the electrochem-ical potential at the superconductor boundary mustobey the condition

(10)

We assume that the nanowire occurs at the initialmoment in the superconducting state and carries asuperconducting current equal to the total current.Therefore, the initial conditions can be written as

(11)

For the numerical integration, it is convenient topass to the Cartesian variables ψ = R + iI, for which the

ξ0 0.18νF

kbTc

----------.=

V Φ 0( ) Φ L( ),–=

V4k Tc T–( )

πe-------------------------- µ 0( ) µ L( )–( ).=

V '4k Tc T–( )

πe--------------------------,=

V µ 0( ) µ L( ).–=

ψ 0( ) 1, ψ L( ) iϕ0( ),exp= =

µ 0( ) 0, µ L( )∂ϕ0

∂t---------.–= =

ψ x( ) 1, ϕ x( ) jx, µ x( ) 0.= = =

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 102 No. 2 2006

DYNAMICAL PROPERTIES OF SUPERCONDUCTING NANOWIRES 329

boundary and initial conditions (9)–(11) take the fol-lowing form:

(12)

(13)

The Cartesian representation for ψ is preferable tothe usual representation in the polar coordinates (ψ =fexp(iχ)) in that the point R = I = |ψ| = 0 for Eqs. (12)is not singular and the integration over the phase trajec-tory passing through this point can be performed in aregular manner.

The numerical integration of the obtained system ofequations was carried out using two methods for thesolution of parabolic equations: (i) an explicit finite-dif-ference scheme and (ii) the method of straight lines.The explicit scheme is conditionally stable, providedthat

,

where a is a coefficient (in the case under consideration,a =1). This condition is imposed on the grid character-istics with respect to τ (time step) and h (coordinatestep). According to the method of straight lines, theCauchy problem is solved using a fourth-order Runge–Kutta scheme. In this scheme, the stability wasachieved by selecting the proper time integration stepby means of a trial procedure, which eventually yieldedthe same condition as above. Since the results obtainedusing the two schemes coincided to within a high pre-cision, the explicit scheme was preferred as providingthe optimum combination of accuracy and convergencerate. Thus, the results presented below were obtainedusing the explicit scheme, and some of these data wereverified using the method of straight lines.

Figure 1 shows a series of the current–voltage char-acteristics (solid curves) of a superconducting nano-wire, which were determined using numerical calcu-lations based on Eqs. (12) with the boundary condi-tions (13) in the limiting case of a short wire (with alength on the order of ξ). The data presented in Fig. 1indicate that the system under consideration corre-

∂∂t-----R

∂2

∂x2--------R 1 R2– I2–( )R µI ,+ +=

∂∂t----- I

∂2

∂x2-------- I 1 R2– I2–( )I µR,–+=

µ 1u--- R

∂∂x------ I I

∂∂x------R–⎝ ⎠

⎛ ⎞ xd

0

x

∫ jx–⎩ ⎭⎨ ⎬⎧ ⎫

,=

R 0 t,( ) 1, I 0 t,( ) 0,= =

R L t,( ) iI L t,( )+ iϕ0( ),exp=

R x 0,( ) iI x 0,( )+ ijx( ).exp=

a2τh2-------- 1

2---≤

sponds to the Josephson junction. For comparison,Fig. 1 also presents the current–voltage characteristics(dashed lines) of a normal nanowire obeying the Ohmlaw. As can be seen, the curves for the superconductingnanowire asymptotically tend to straight lines parallelto the dashed lines determined by the Ohm law. Thisshift is well known from the theory of Josephson junc-tions and referred to as the “excess current” effect,which is related to the so-called neighborhood effect atthe nanowire boundary [20].

It was especially interesting to determine the cur-rent–voltage characteristics of superconducting nanow-ires with the lengths greater than ξ. The results of ournumerical calculations showed that such relatively longnanowires representing a “pure” superconductor (u > 1)are characterized by two characteristic current densityvalues, which are denoted here as jc1 and jc2. The mean-ing of these thresholds is as follows. For j < jc1, the totalcurrent is entirely superconducting, whereas for j > jc2,the current is purely normal. In the intermediate regionof current densities, jc1 < j < jc2, the total current con-tains both superconducting and normal components(mixed state). In other words, jc1 and jc2 have the mean-ing of critical currents. The electric potential is zero atj = jc1; as the current density is increased above thisthreshold, the potential also grows to reach Vc2 at j = jc2 .This value can also be considered as critical. Assumingthat the electric field strength is constant along the wire,we can write

(14)j ΩuVL---,=

10 20 30 40 50

V

0.5

0

jc1 = 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0J

u = 0.10.20.4

2.5

Fig. 1. A series of the current–voltage characteristics of(solid curves) a superconducting nanowire with L = 1 and(dashed lines) a normal nanowire obeying the Ohm law forvarious values of the order (“purity”) parameter u (indicatedat the curves).

330


NIKOLAEV, YUGAY

where L is the length of the nanowire, V is the voltage, and

These relations show that, under otherwise identicalconditions, Vc2 ∝ L.

Figure 2 shows the current–voltage characteristicsof long superconducting nanowires and indicates(vertical line) the critical voltage Vc2. According to for-mula (14), this value is two times smaller for L = 25than for L = 50. The intersections of this line with thecurrent–voltage characteristics determine the secondcritical current density (jc2), thus dividing the curvesinto two parts corresponding to V < Vc2 (resistiveregion) and V > Vc2 (normal region). It is important tonote that, according to the results of our numerical cal-culations, the nanowires generate electromagneticwaves only in the resistive region. It should be notedthat the current–voltage characteristics of nanowiresexhibit a characteristic bending in the resistive region.

In addition, our calculations showed that the electro-magnetic waves are not generated when u < 1. As canbe seen from Fig. 2, the values of critical currents at V =Vc2 in this range of u values coincide (jc1 = jc2) and theresistive region vanishes. This behavior of criticalparameters can be also clearly traced in Fig. 3, whichshows plots of the critical current densities versus theorder parameter u. The width of the resistive region var-ies in proportion to the u value and vanishes when theorder parameter approaches unity.

3. RADIATION SPECTRUM

In order to study the spectrum of generated radia-tion, we performed a direct Fourier transform of the

Ω eπξ4k Tc T–( )--------------------------.=

voltage as a function of the time and calculated the fre-quency spectra, which are depicted in Figs. 4 and 5. Ascan be seen from these spectra, there is a certain funda-mental frequency ω0 and several multiple frequenciesω = nω0. It should be noted that the amplitude and fre-quency in the spectra in Figs. 4 and 5 are normalized as

(15)

The radiation spectrum represents a sequence ofharmonics, in which amplitudes decrease with increas-ing harmonic number. A comparison of the spectra in

AAV '-----, V '

4k Tc T–( )πe

--------------------------;=

ω ωω '-----, ω '

8k Tc T–( )π

--------------------------.=

(a)

10 20 30 40 50 600V

0.5

1.0

1.5

2.0

2.5J

jc1 = 0.38Vc2

u = 2.5

510

0.4

jc2u = 5

jc2u = 2.5

0

0.5

1.0

1.5

2.0

2.5J

jc1 = 0.38

jc2u = 5

jc2u = 2.5

(b)

u = 2.5

Vc2

10 5

0.4

20 40 60 80 100V

Fig. 2. Current–voltage characteristics (solid curves) of superconducting nanowires with L = 25 (a) and 50 (b) and (dashed lines) anormal nanowire obeying the Ohm law for various values of the order (“purity”) parameter u (indicated at the curves).

2

1

4 6 8 10u

0

2

3

4

5

0.875

0.38

jc1, jc2

jc1

jc2

Fig. 3. Plots of the critical current densities jc1 and jc2 ver-sus order parameter u.



Figs. 4a and 4b shows that, as the u value increases, thesignal frequency and amplitude decrease in inverse pro-portion to the change in u. The spectra in Figs. 4c and4d correspond to different values of the nanowirelength (L = 25 and 50, respectively). As can be seen, thechange in the wire length does not influence the charac-ter of the spectrum.

It is interesting to note that the radiation spectrumsignificantly depends on the current density. The spec-tra presented in Figs. 4a and 4b correspond to j = 0.6(i.e., to the current densities below of within the regionof bending on the current–voltage characteristic con-sidered in the preceding section. The fundamental fre-quencies in these spectra are not very clearly mani-fested and the spectra exhibit a considerable noise. Incontrast, the spectra corresponding to the current densi-ties above the region of bending on the current–voltagecharacteristics show clearly defined frequencies(Figs. 4c and 4d). In addition, an increase in the currentdensity leads to a growth in the signal frequency with adecrease in the number of harmonics (cf. Figs. 4band 4c).

Let us consider the spectrum of radiation generatedby a nanowire with L = 1 (Fig. 5). This system can beexpected to exhibit Josephson generation, since thenanowire with a length on the order of ξ is equivalent tothe Josephson junction. This behavior was actually

observed and, in contrast to the case of long nanowires,it took pace in the entire range of current densitiesabove the critical level (j > jc). Moreover, the generationin this system is observed at any value of the orderparameter u. Figure 5 presents the spectra for two val-ues of u, which show that (in accordance with the lawsof Josephson generation), the amplitudes of harmonicsin these spectra decrease so as to obey the law of geo-metric progression [21].

In addition, the data in Fig. 5 show that, as the uvalue increases, the signal frequency and amplitudeexhibit a decrease. This behavior is consistent with thewell-known formulas for the Josephson contact:

(16)

where R is the contact resistance in the normal state, Iand Ic are the total and critical currents, respectively,

and is the average voltage on the junction. The resis-tance R is inversely proportional to the conductivitywhich, in turn, is proportional to the mean free path l.As can be seen from the definition of the order param-eter u (see formula (4)), the resistance R is inversely pro-portional to u. Then, taking into account relations (16),

ω02eV

----------, V R I2 Ic

2– ,= =

V

0.20

0.15

0.10

0.05

0

A L = 25, u = 5, j = 0.6

(a)

0.1 0.2 0.3 0.4 0.5 0.6ω

0.20

0.15

0.10

0.05

0

A L = 25, u = 2.5, j = 0.8

(c)

0.1 0.2 0.3 0.4 0.5 0.6ω

0.20

0.15

0.10

0.05

0

A L = 25, u = 5, j = 0.6

(b)

0.1 0.2 0.3 0.4 0.5 0.6ω

0.20

0.15

0.10

0.05

0

A L = 50, u = 2.5, j = 0.8

(d)

0.1 0.2 0.3 0.4 0.5 0.6ω

Fig. 4. The spectra of radiation generated by a superconducting nanowires with different lengths (L = 25 and 50) at various valuesof the order parameter u and the current density j.

332


NIKOLAEV, YUGAY

we conclude that the signal frequency and amplitudemust inversely proportional to the u value.

An analysis of the spectra of radiation generated bythe superconducting wires of various lengths showsthat the dependence of the spectral characteristics (fre-quency and amplitude) on the nanowire length, currentdensity, and u value for the short wire (L = 1) are qual-itatively similar to those for the long wires (L = 25and 50).

Considering the results and taking into account thedynamic behavior the phase slip centers in the resistiveregion [18], we may suggest that the mechanism ofradiation in long nanowires is related to the dynamicappearance of phase slip centers, which possess theproperties of Josephson junctions. These very junctionsgenerate the radiation observed in long superconduct-ing nanowires.

There is a remarkable analogy between the super-conducting nanowires and the superconductors oftypes I and II with respect to the factors responsible forthe assignment of superconductors to these types.Indeed, the boundary between these types is deter-mined by the parameter κ = λ/ξ of the Ginzburg–Lan-

dau theory: κ 1 (more precisely, κ > 1/ ) corre-sponds to the superconductor of type II, while κ 1

(more precisely, κ < 1/ ) corresponds to the super-conductor of type I. In our case, the parameter κ isreplaced by the order parameter u, which distinguishesthe “pure” and “impure” superconductors. As is known,the superconductors of type II are characterized by twocritical values of the magnetic field strength (Hc1, Hc2),

2

2

whereas the type I superconductors have a single criti-cal field (Hcm). In the interval of magnetic fieldsbetween the two critical values, the superconductoroccurs in a mixed state. Returning to the problem understudy and considering formulas (16), we can also deter-mine the interval of field strengths (in this case, for theelectric field), in which the “pure” superconductoroccurs in a mixed state, which is accompanied by theappearance of phase slip centers and manifested in thegeneration of electromagnetic waves. By the sametoken, the “impure” superconducting nanowires (liketype I superconductors) exhibit a single critical value ofthe electric field strength.

4. CONCLUSIONS

An analysis of the simulated radiation spectra andthe current–voltage characteristics of superconductingnanowires with various length leads us to the followingconclusions:

(i) In a “pure” superconducting nanowire (u > 1), theresistive state arises in the region of constant currentsbounded by the critical values jc1 and jc2. A long wire(L 1) occurring in this resistive state generates elec-tromagnetic waves. Outside the interval between jc1 andjc2, the radiation is absent.

(ii) The radiation spectrum of a long superconduct-ing nanowire is noisy.

(iii) The current–voltage characteristics of shortsuperconducting nanowires (L = 1) coincide with thoseof the Josephson junctions.

(iv) The current–voltage characteristics of all super-conducting nanowires (irrespective of length) exhibitthe “excess current” effect, whereby the curves areshifted by a certain constant value relative the ohmiccharacteristics of nanowires in the normal state ratherthan merge with them.

(v) In the case of an “impure” superconductor (u <1), the resistive state in a superconducting nanowirecarrying a constant current is absent and, hence, theradiation is not generated in the entire range of currentsbelow a single critical value.

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0.1

0.5

0.2 0.3 0.4 0.5

(b)

ω0

1.0

1.5

2.0

2.5

3.0A

0.5

0.5

1.0 1.5 2.0

(a)

ω0

1.0

1.5

2.0

2.5

3.0A

Fig. 5. The spectra of radiation generated by a supercon-ducting nanowire with L = 1 (Josephson junction) at j = 1.5and u = 0.4 (a) and 2.5 (b).



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Translated by P. Pozdeev

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Dynamical properties of superconducting nanowires