Luiz A.M. (ed.) Superconductor

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Phase Dynamics of Superconducting Junctions

under Microwave Excitation in Phase Diffusive Regime

321

modulation in the state energy that

cos

EE eV t

+ for an unperturbed wavefunction

. Then the change in wavefunction is simply on the dynamical phase such that,

()

exp cos

in t

tEeVtdt

Jxe e

ϕω

−

⎡

⎤

′

Ψ= +

⎢

⎥

⎣

⎦

∫

∑



. (12)

Fig. 9. The energy levels generated according to Eq. (12) in the presence of a microwave

field. Adopted from (Tucker & Feldman 1985).



Fig. 10. A schematic of the enhanced macroscopic quantum tunneling in a single Josephson

junction due to photon excitation. In each potential valley, the quantum states may form a

harmonic oscillator ladder with a spacing of

 . The absorption of photon energy may

lead to an inter-valley resonant tunneling Γ

, and a tunneling followed by an inner-valley

excitation,

. The enhancement in macroscopic quantum tunneling results in a reduction of

junction critical current.

Again

xeV

=  as defined before. This expression is useful in the tunneling-

Hamiltonian formalism applicable to the high-impedance devices such as single electron

Superconductor

322

transistors and quantum dots. One may include additional tunneling events through the

side-band states with energies of

 , namely the photon-assisted tunneling(PAT).

The PAT is a simple way to probe the quantum levels in the junctions. For example, if the

Josephson coupling energy E

is relatively large, a single potential valley of the washboard

potential can be viewed as a parabolic one. In this case the system energy spectrum has a

structure of simple harmonic ladder as shown in Fig. 10. The microwave excitation enhances

the tunneling of the phase to adjacent valleys, also called macroscopic quantum tunneling

when the photon energy matches the inter-level spacing. This can be observed in the

reduction of junction critical current.

6. Experiments on ultra-small junctions

The Josephson junction under the microwave excitation has been studied for decades and

much works have contributed to the topic, however mostly on low impedance junctions.

(Tinkham 1996) Here the main focus is the junctions with small dissipation, namely, with

environmental impedance Re

ZR≥ . Although a large junction tunneling resistance as well

as small junction capacitance can be obtained by using advanced sub-micron lithography,

the realization of the high impedance condition remains a challenge to single junctions

because of large parasitic capacitance between electrodes. Tasks have been made by using

electrodes of high impedance to reduce the effective shunted resistance and

capacitance.(Kuzmin & Haviland 1991) Another approach to this problem can be made by

using systems in a moderate phase diffusive regime by thermal fluctuation, namely,

EkT . Koval et al performed experiments on sub-micron Nb/AlO

/Nb junctions and

found a smooth and incoherent enhancement of Josephson phase diffusion by microwaves.

(Koval, Fistul et al. 2004) This enhancement is manifested by a pronounced current peak at

the voltage

VP∝ . Recently experiments on untrasmall Nb/Al/Nb long SNS junctions

have found that the critical current increases when the ac frequency is larger than the

inverse diffusion time in the normal metal, whereas the retrapping current is strongly

modified when the excitation frequency is above the electron-phonon rate in the normal

metal. (Chiodi, Aprili et al. 2009)

Double junctions, also called Bloch transistors and junction arrays are much easier for

experimentalists to realize the high impedance (low dissipation) condition. The pioneer

work by Eiles and Martinis provided the Shapiro step height versus the microwave

amplitude in ultra-small double junctions.(Eiles & Martinis 1994) Several works found that

the step height satisfies a square law,

(

)

(

)

sn Cn

IV IJ x= instead of the RCSJ result,

()

sn Cn

IV I J x= .(Eiles & Martinis 1994; Liou, Kuo et al. 2007) In the one-dimensional(1D)

junction arrays, the supercurrent as a function of microwave amplitude can be found to

obey

()

(

)

IIJx= at high frequencies,

> , although no Shapiro steps were seen. At

low frequencies, the current obeys the classical detector result as in expression (8) even for

quasi-particle tunneling. (Liou, Kuo et al. 2008) Therefore a direct and primary detection

scheme was proposed by using the 1D junction arrays.

In single junctions with a high environmental impedance, people has reported observation

of structures in

IV curves at 22Ie

, featuring the Bloch oscillations due to pronounced

charge blockade. (Kuzmin & Haviland 1991) The 1D arrays also demonstrate similar

Phase Dynamics of Superconducting Junctions

under Microwave Excitation in Phase Diffusive Regime

323

interesting behavior signifying time-correlated single charge tunneling when driven by

external microwaves. This behavior yields a junction current of

Ine

as what was

found in the single junctions.(Delsing, Likharev et al. 1989; Andersson, Delsing et al. 2000)

Recently, the Bloch oscillations are directly observed in the “quantronium” device and a

current-to-frequency conversion was realized. (Nguyen, Boulant et al. 2007)

-0.2 0.0 0.2

-100

-50

100

150

200

250

300

05101520

I (pA)

(mV)

(μV)

ν (GHz)

w/o rf

-36dBm

-33dBm

-30dBm

-25dBm

-20.8dBm

Fig. 11. The

IV curves of a double junction under microwave irradiation clearly show

Shapiro steps. Inset illustrate the step voltages obey the theoretical prediction. Adopted

from (Liou, Kuo et al. 2007)

PAT is an ideal method to probe the quantum levels or band gaps in a quantum system. For

the charge dominant system(

, R>R

) as an example, Flees et al. (Flees, Han et al. 1997)

studied the reduction of critical current of a Bloch transistor under a microwave excitation.

The lowest photon frequency corresponding to the band gap in the transistor was found to

reduce as the gate voltage tuned to the energy degeneracy point for two charge states. In

another work, Nakamura et al. biased the transistor at the Josephson-quasiparticle (JQP)

point. The irradiating microwaves produced a photon-assisted JQP current at certain gate

voltages, providing an estimation of the energy-level splitting between two macroscopic

quantum states of charge coherently superposed by Josephson coupling.(Nakamura, Chen

et al. 1997) For the phase dominant systems, enhanced macroscopic quantum tunneling

were observed in system of single Josephson junction (Martinis, Devoret et al. 1985; Clarke,

Cleland et al. 1988) and superconducting quantum interference devices(SQUIDs)(Friedman,

Patel et al. 2000; van der Wal, Ter Haar et al. 2000). Recently, devices based on Josephson

junctions, such as SQUIDs, charge boxes, and single junctions have been demonstrated as an

ideal artificial two-level system for quantum computation applications by using the

microwave spectrometry.(Makhlin, Schon et al. 2001)

Superconductor

324

(pA)

100

8.00.0 2.0 4.0 6.0

2eV

gmax

n=0

n=1

n=2

n=3

Fig. 12. The Shapiro height as a function of the microwave amplitude

observed in a

double junction system obeys the square law, a feature of incoherent photon absorption in

this system. Adopted from(Liou, Kuo et al. 2007).

7. Conclusion

We have discussed the dc response of a Josephson junction under the microwave excitation

in the phase diffusion regime theoretically as well as summarized recent experimental

findings. In relative low impedance cases, the classical description (in phase) is plausible to

explain the observed Shapiro steps and incoherent photon absorption. The quantum

mechanical approaches may provide a more precise description for the experimental results

of higher impedance cases such as Bloch oscillations and photon-assisted tunneling. In

extremely high impedance cases, single charge tunneling prevails and a classical description

in charge, such as charging effect can be an ideal approach.

8. Acknowledgement

The authors thank National Chung Hsing University and the Taiwan National Science

Council Grant NSC-96-2112-M-005-003-MY3 for the support of this research.

9. References

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Phys 149: 374.

Chiodi, F., M. Aprili, et al. (2009). "Evidence for two time scales in long SNS junctions."

Physical Review Letters 103(17): 177002.

Clarke, J., A. Cleland, et al. (1988). "Quantum mechanics of a macroscopic variable: the

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Phase Dynamics of Superconducting Junctions

under Microwave Excitation in Phase Diffusive Regime

325

Corlevi, S., W. Guichard, et al. (2006). "Phase-Charge Duality of a Josephson Junction in a

Fluctuating Electromagnetic Environment." Physical Review Letters 97(9): 096802.

Delsing, P., K. K. Likharev, et al. (1989). "Time-correlated single-electron tunneling in one-

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1861.

Devoret, M., D. Esteve, et al. (1990). "Effect of the electromagnetic environment on the

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1824-1827.

Eiles, T. M. and J. M. Martinis (1994). "Combined Josephson and charging behavior of the

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Bloch Transistors." Physical Review Letters 78(25): 4817.

Friedman, J. R., V. Patel, et al. (2000). "Quantum superposition of distinct macroscopic

states." Nature 406(6791): 43-46.

Ingold, G.-L. and Y. V. Nazarov (1991).

Single Charge Tunneling. H. Grabert and M. H.

Devoret. New York, Plenum. 294.

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microwaves." Physical Review Letters 93(8): 87004.

Kuo, W., C. S. Wu, et al. (2006). "Parity effect in a superconducting island in a tunable

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dimensional arrays of superconductor/ insulator/superconductor junctions." New

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Makhlin, Y., G. Schon, et al. (2001). "Quantum-state engineering with Josephson-junction

devices." Reviews of Modern Physics 73(2): 357.

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State of a Current-Biased Josephson Junction." Physical Review Letters 55(15): 1543.

Nakamura, Y., C. Chen, et al. (1997). "Spectroscopy of energy-level splitting between two

macroscopic quantum states of charge coherently superposed by Josephson

coupling." Physical Review Letters 79(12): 2328-2331.

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Circuit." Physical Review Letters 99(18): 187005.

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Determination of the Local Crystal-Chemical

Features of Complex Chalcogenides by Copper,

Antimony, and Arsenic NQR

R.R. Gainov

1,2

, A.V. Dooglav

, I.N. Pen’kov

, A.Yu. Orlova

I.A. Evlampiev

, N.N. Mozgova

, and R.R. Khasanov

Department of Physics, Kazan Federal University, Kazan, Kremlevskaya str. 18, 420008,

Department of Geology, Kazan Federal University, Kazan, Kremlevskaya str. 4/5, 420111,

Institute of geology of ore deposits, petrography, mineralogy and geochemistry of the

Russian Academy of Science, Staromonetny per. 35, Moscow 109017,

Russian Federation

1. Introduction

In recent years, a number of reviews have appeared in scientific literature on the application

of different physical methods in the structural-chemical and physical studies of solid state

materials, for example, high-temperature superconductors (HTSC). An important place

among them is occupied by nuclear-resonance methods, particular, nuclear magnetic

resonance (NMR), nuclear quadrupole resonance (NQR), nuclear gamma resonance (NGR

or Mössbauer spectroscopy) and other. Particularly, this article shows the feasibility of using

the NQR method, in some instance, in the study of those features of material structure and

chemistry, which are difficult or impossible to attack by conventional methods. The key

advantage of NQR is a possibility to probe directly the electric (quadrupole) interactions

between nuclei and their local environment. In general, the quadrupole interaction describes

the coupling of the nuclear quadrupole moment to the electric field gradient (EFG). The EFG

is determined by bond directions and electronic charge distribution, therefore the

quadrupole interaction is a very sensitive tool for structural-chemical studies of condensed

matter. Notably, it is possible to analyze NQR data (both the spectroscopic and the

relaxation) in materials containing different structural copper (

63,65

Cu), arsenic (

As), and

antimony (

121,123

Sb) complexes. This circumstance is especially important in the light of

existence of two types of HTSC: Cu-oxide materials (cuprates) (Rigamonti et al., 1998) and

As- and Sb-bearing pnictides (Wilson, 2009).

Our article is devoted to brief review of some important spectroscopic experimental results

obtained during the studies of binary chalcogenide CuS (also known as covellite),

representing the unusual low-temperature superconductor, and ternary material Ag

SbS

(referred to as a stephanite). Examples are taken mainly from the studies of the authors′

research group.

The article is organized as follows. Section I contains necessary NQR background for

63,65

Cu,

As,

121,123

Sb nuclei. In Section II we report our results concerning the covellite CuS and

Superconductor

328

some related materials with subsequent discussions. Section III is devoted to studies of

stephanite Ag

SbS

in connection with other related compounds. The paper concludes with

a summary in Section IV.

2. NQR background

The NQR background will be reviewed only for the sake of clarity (Grechishkin, 1973). The

nuclear spins I interact with their electronic environment through quadrupole (i.e. electric)

and magnetic hyperfine couplings. In general, the NQR spectrum is described by the

following Hamiltonian:

222

{3 ( 1) ( )}

4(2 1) 2

eQV

HIIIII

+−

=−+++

−

(1)

Hamiltonian H

refers to as the coupling of nuclear quadrupole moment eQ to the local

crystal electric field gradient (EFG) with V

the largest component of the crystal EFG

tensor,

=|V

- V

|/V

the asymmetry parameter showing the deviation of the EFG

symmetry from the axial one, i.e. the value of

lies in the range [0,1]. The EFG components

satisfy Laplace equation: V

+ V

= 0.

The pure NQR spectrum is observed in the case of absence of the external (H

=0) and

internal (H

int

= 0) static magnetic fields. The number of NQR lines is defined (i) by the

amount of crystallographically nonequivalent positions of quadrupole nucleus (I > ½) in the

crystal structure, (ii) by the local geometry (i.e. symmetry) of quadrupole nucleus

surrounding, (iii) by the magnitude of nuclear spin I and (iv) by the number of naturally

available isotopes of quadrupole nucleus.

In case of copper, only two naturally available isotopes exist:

Cu (69.2% natural

abundance,

/2π=1.128 kHz/Oe, Q=-0.22 barn) and

Cu (30.8% natural abundance,

/2π=1.209 kHz/Oe, Q=-0.204 barn) both having spin I=3/2 and thus two doubly

degenerate ±1/2 and ±3/2 energy levels. Thus, for each isotope a transition between these

levels yields a single NQR signal at a frequency:

63,65

63,65 2

eQV

⋅

=⋅+. (2)

Arsenic has a single total-lot isotope

As (I = 3/2, 100% natural abundance,

/2π=0.729 kHz/Oe, Q=0.3 barn), therefore the single arsenic crystal-chemical position gives

only one NQR signal at a frequency:

75 2

eQV

⋅

=⋅+. (3)

Antimony has two naturally occurring isotopes

121

Sb (I = 5/2, 57.3% natural abundance,

/2π=10,188 MHz/T, Q=-1.3 barn) and

123

Sb (I = 7/2, 45.7% natural abundance,

/2π=5,517 MHz/T, Q=-0.18 barn). The

121

Sb exhibits two NQR signals at frequencies (

< 0.2):

121

121 2

(3/2 1/2)

359

(1 )

20 54

eQV

±↔±

⋅

⋅+ ⋅ , (4a)

Determination of the Local Crystal-Chemical Features

of Complex Chalcogenides by Copper, Antimony, and Arsenic NQR

329

121

121 2

(5/2 3/2)

611

(1 )

20 54

eQV

±↔±

⋅

⋅− ⋅ . (4b)

The

123

Sb with I=7/2 has three NQR signals at frequencies (

< 0.2):

123

123 2

(3/2 1/2)

109

(1 )

14 30

eQV

±↔±

⋅

⋅+ ⋅ , (4c)

121

123 2

(5/2 3/2)

217

(1 )

14 30

eQV

±↔±

⋅

=⋅−⋅, (4d)

121

123 2

(7/2 5/2)

(1 )

14 10

eQV

±↔±

⋅

=⋅−⋅. (4e)

Thus, the single antimony crystal-chemical position in the structure of material under study

yields five NQR signals.

As one can see, NQR frequency ν

depends on the certain arrangement of the surrounding

ions through the parameters V

and

. The value of ν

is defined by two parameters in

Eq.(4): V

and

; so it is impossible to obtain both parameters experimentally from the

NQR spectra consisting of only one line for I=3/2 (copper and arsenic). Usually it is done

from the angular dependence of the NMR spectrum taken for a single crystal or from

numerical simulations of NMR spectrum for the unaligned powder. In some cases, for

determination of

the 2D nutation NQR studies also could help.

In a semi-empirical approach (Kaufmann & Vianden, 1979), it is assumed that components

of EFG tensor at Cu nuclei site can be written as the sum of two terms – lattice and valence

contributions:

(1 ) (1 )

ZZ latt val val

VVRV

∞

=− ⋅ +− ⋅ (5)

The parameters γ

∞

, R

val

are the Sternheimer antishielding factors. The first contribution

arises from all ion charges outside the ion under consideration and can be calculated using

the model of point-charges (MPC):

(3 cos 1)

latt

⋅

⋅−

=∑ , (6)

where q

and r

are the charge and the position of the i-th ion, respectively, θ

– angle between

main axis of symmetry and direction into the neighboring ion.

The computation of valence contribution V

val

(and, as consequence, total V

) constitutes a

complex problem, since it requires a detailed knowledge of the structure and electronic

density distribution. On the other hand, the comparison of ν

and V

, deduced from

theoretical calculations on the basis of different structural models and approaches (see

Eq.(5)), with experimental values (Eq.(4)) permits to determine the individual features of

local electronic arrangement and peculiarities of chemical bonds. Actually, the lattice

contribution V

val

is defined by crystal symmetry, whereas the valence contribution V

val

determined by the local symmetry of nearest environment, i.e. according to NQR

frequencies and line-widths it is possible to investigate the following EFG characteristics: the

Superconductor

330

electronic density distribution inside of different coordination complexes and their

symmetry, the extent of ionic (covalent) bonds and angles between them. Every factor,

which is connected with reorganization of atomic or molecular group, can change

significantly the EFG value on the nucleus inside of a such group and, as consequence, NQR

is very sensitive to different crystal-chemical transformations in crystal lattice. Therefore, the

studies of temperature dependences of NQR spectroscopic parameters give the possibility to

find phase transitions and investigate their nature.

3. Towards to electronic structure of binary copper chalcogenides

There is a great number of compounds in the system of binary copper chalcogenides Cu-S,

Cu-Se, and Cu-Te. In particular, materials eleven natural minerals and two unstable

synthetic phases among only binary copper-sulfide have been discovered up to date (Goble,

1985; Gablina, 1997). Many of them are widely used in optoelectronics (Abrikosov et al.,

1975; Isac et al., 2007) and some of them represent the industrial mineral specimens in

copper-ore deposits (Izoitko, 1997). The more significant involving of these compounds into

the practical application is stopped by the fact that currently available standard

instrumental techniques for studying crystal-chemical problems such as X-ray structural

diagnosis, optical spectroscopy, and electronic microscopy have inherent limitations

(Gablina, 1997). These manifest themselves in being unable to characterize the chalcogenides

samples fully, especially if they are polycrystalline. These methods are mainly ineffective

and little informative in case of multiphase natural samples (minerals) (Gablina, 1997).

This Section describes briefly the spectroscopic studies of layered low-temperature

superconductor CuS (covellite, originally published in (Gainov et al., 2009)) and related,

potentially perspective, but insufficiently known compounds: non-stoichiometric sulfide

1.6

S (naturally occurring as geerite) and selenium analog of CuS – α-CuSe (klockmannite).

The description of experimental methodic and synthesis conditions can be found in (Gainov

et al., 2009).

3.1 Crystal-chemistry of CuS, Cu

1.6

S, and α-CuSe

Covellite CuS represents the metal with excellent hole conductivity, which becomes

anisotropically superconducting below T

= 1.6 K (Meissner, 1929; Buckel & Hilsch, 1950;

Saito et al., 1997; Di Benedetto et al., 2006). The elementary cell of CuS at room-T has a

hexagonal symmetry (space group P6

/mmc) (Berry, 1954; Evans & Konnert, 1976; Ohmasa

et al., 1977). The structure of CuS can be described as a “sandwich”, which consists of three

alternating layers A

-B-A

(Fig. 1). The layer B represents the net of [Cu(1)-S(1)

]–triangles,

whereas the layers A

and A

are made up of [Cu(2)-S(1)S(2)

]–tetrahedra. The units A

-B-A

are connected together along the c-axis by S(2)-S(2) bonds. It was revealed that the second-

order phase transition (PT) is occurred at about T

= 55 K (Fjellvåg et al., 1988). This

transition can be presented as a little shift of the layer B with respect to the layers A

and A

perpendicular to the c-axis with the change of the Cu(2)-S(1)-Cu(2) bonding angle from 180º

at room-T to 170º below T

The crystal structure of klockmannite α-CuSe is not identified in detail up to date. As a first

approach, it is assumed that the structure of α-CuSe is somewhat similar to those in CuS

(Berry, 1954; Peiris et al., 1998; Milman, 2002): there are the same space group and layered

motive A

-B-A

with the CuSe

, CuSe

, and Se-Se complexes. At the same time, it is

proposed the occurrence of CuS-based superlattice in klockmannite (Berry, 1954).