Pis’ma v ZhETF, vol. 109, iss. 12, pp. 833 - 834

June 25

Dimension effects in insulating NbTiN disordered films and asymptotic

freedom of Cooper pairs

M. V. Burdastyh^a,b, S. V. Postolova^a,c, I. A. Derbezov^a, A. V. Gaisler^a, M. C. Diamantini^d, C. A. Trugenberger^e,

V. M. Vinokur^f , A. Yu. Mironov^a,b1)

^aA. V. Rzhanov Institute of Semiconductor Physics, 630090 Novosibirsk, Russia

^bNovosibirsk State University, 630090 Novosibirsk, Russia

^cInstitute for Physics of Microstructures of the Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia

^dNoise in Physical Systems Laboratory, Istituto Nazionale di Fisica Nucleare and Dipartimento di Fisica e Geologia,

University of Perugia, I-06100 Perugia, Italy

^eSwissScientific Technologies SA, CH-1204 Geneva, Switzerland

^fMaterials Science Division, Argonne National Laboratory, Argonne, 60637 Illinois, USA

Submitted 24 April 2019

Resubmitted

2 May 2019

Accepted 4

May 2019

DOI: 10.1134/S0370274X19120063

Building on the measurements of NbTiN film

which is on the insulating side of the disorder-driven

superconductor-insulator transition

[1-4], we show

that the threshold voltage of measured step-like I(V )

curves is a linear function of the distance between the

measuring electrodes. The measurements are taken on

polycrystalline film with 10 nm thickness obtained by

the atomic layer deposition [5] at temperature 350^◦C.

At temperatures T > 0.8 K, R_□(T ) collapse on top

of each other and are well fit the activation behavior.

At T < 0.8 K, R_□(T ) for larger samples demonstrate

hyperactivation growth characteristic to superinsulator,

the curves for different sizes converging, smaller systems

revealing the tendency of bending down. Decreasing the

film size leads to suppression of the insulating proper-

ties down to quasi-metallic saturation of the resistance.

Analysis of the R_□(T ) dependence for the largest film

in terms of the charge Berezinsky-Kosterlitz-Thouless

Рис. 1. (Color online) Dependence of the threshold voltage

(CBKT) transition shows deviation from the CBKT

upon the distance between electrodes. The dashed line is

criticality at lowest temperatures, manifesting the finite

the fit V_th = 0.78L. Inset: Evolution of the I-V curves

electrostatic screening length λ (the perfect criticality

of the insulating NbTiN film as function of the distance

would have been observed for λ > 10⁵ [1]).

between electrodes taken at 20 mK

To explore possible consequences of the string model

of a superinsulator, we measure I-V curves demonstrat-

ing the threshold behavior on the device comprising the

strongly-interacting 2D electron system in silicon MOS-

parts of the same film but with different distances be-

FETs and were successfully described by the depinning

tween measuring electrodes, see Fig. 1. Interestingly, the

of the 2D Wigner crystal [6, 7]. However, in our case

similar threshold I(V ) dependencies were observed in

of the Mott Cooper pair insulator in strongly disor-

dered films, formation of the Cooper pair Wigner crys-

¹⁾e-mail: mironov@isp.nsc.ru

tal that requires weak pinning, seems unlikely, and is

Письма в ЖЭТФ том 109 вып. 11 - 12

2019

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M. V. Burdastyh, S. V. Postolova, I. A. Derbezov et al.

not supported by the data that do not follow the for-

the maximum of the potential k_BT_CBKT ln(r/ℓ) - σ_eff L,

mulas of [6, 7]. The current jumps become less sharp

where σeff ≃ 2eE - σ, E being the applied electric

upon shortening the distance between electrodes, while

field. This gives the estimate for σ_eff

≈ T_CBKT/r^∗,

threshold voltage dependence upon L exhibits linear be-

r^∗ ≈ 3ℓ. One expects σ_eff ≃ σ. Taking ℓ ≈ 10 nm and

havior, implying that the threshold electric field E_th =

T_CBKT ≃ 0.4 K, one obtains σ_eff ≃ 1.3 · 10⁵ K/cm. Con-

= V_th/L is independent on the distance between elec-

sidering the crudeness of the model, one can view this

trodes. This was only observed in gases and liquids [8]

as an excellent agreement of the obtained experimen-

or in nano-scale solids with L < 1 µm [9], i.e., in sam-

tal data with the theoretical expectations. Thus, the far

ples which are over three orders of magnitude shorter

reaching implication of our findings is that one can use

than those examined in our work. At the same time,

the desktop measurements in order to determine exper-

the observed linear dependence is in accordance with

imentally the linear tension of the Polyakov’s electric

the prediction of [10], see Fig. 1.

strings.

We now can compare the experimental findings with

The work on transport measurements at Novosi-

the theoretical predictions of [10]. The asymptotic free-

birsk is supported by Russian Science Foundation

dom scale, i.e., the minimum string size below which

#18-72-10056. Analysis of the experimental data

the linear confinement of charges in a superinsulator

(V. M. Vinokur) are supported by the U.S. Department

vanishes estimated as [10]

of Energy, Office of Science, Basic Energy Sciences,

Materials Sciences and Engineering Division.

ℏv

λ_string ≃

(1)

Full text of the paper is published in JETP Letters

k_BT_CBKT

journal. DOI: 10.1134/S0021364019120014

where v = c/√ε is the light velocity in the material,

c is the light velocity in vacuum, and ε is the bare di-

electric constant of NbTiN near the SIT. Taking the

1. A. Yu. Mironov, D. M. Silevitch, T. Proslier,

experimental values of T_CBKT ≃ 400 mK and ε ≃ 800,

S. V. Postolova, M. V. Burdastyh, A. K. Gutakovskii,

T. F. Rosenbaum, V. V. Vinokur, and T. I. Baturina,

we find λ_string ≃ 0.2 mm which agrees fairly well with

Sci. Rep. 8, 4082 (2018).

the observed size of 0.2 mm at which the temperature

2. M. Ovadia, D. Kalok, I. Tamir, S. Mitra, B. Sacépé, and

dependence of ln R_□(T ) vs. 1/T turns to saturation.

D. Shahar, Sci. Rep. 5, 13503 (2015).

With using a non-relativistic version of the

3. Y.-H. Lin, J. Nelson, and A. M. Goldman, Physica C

Polyakov’s compact QED model [11, 12] we can show

514, 130 (2015).

that the superinsulating state is the state in which

4. M. V. Burdastyh, S. V. Postolova, T. I. Baturina,

Cooper pairs are localized and the electrostatic poten-

T. Proslier, V. M. Vinokur, and A. Yu. Mironov, JETP

tial of the charges is screened by the Bose condensate

Lett. 106(11), 749 (2017).

of vortices. To estimate the threshold field, we use

5. T. Proslier, J. A. Klug, N. C. Becker, J. W. Elam, and

interpolation formula for the attractive interaction

M. J. Pellin, ECS Transactions 41, 237 (2011).

between charges of the opposite sign separated by the

6. P. Brussarski, S. Li, S. V. Kravchenko, A. A. Shashkin,

distance r:

and M. P. Sarachik, Nat. Comm. 9, 3803 (2018).

U (r) ≃ σ(T )r + k_BT_dcln(r/ℓ),

(2)

7. A. A. Shashkin and S. Kravchenko, Appl. Sci. 9, 1169

(2019).

where ℓ is the ultraviolet (UV) cutoff of the theory iden-

8. P. Kubek, Acta Energetica 3, 124 (2017).

tified with the film thickness, σ is the linear tension of

9. C. Neusel and G. A. Schneider, Journal of the Mechanics

the confining electric string, T_dc is the deconfinement

and Physics of Solids 63, 201 (2014).

temperature coinciding with T_CBKT. The threshold volt-

10. M. C. Diamantini, C. A. Trugenberger, and V. M. Vi-

age marks the voltage that capable to overcome the con-

nokur, Comm. Phys. 1, 77 (2018); DOI: 10.1038/s42005-

finement, i.e., break the electric string. This gives the es-

018-0073-9.

timate σ ≃ 2eV_th/L. Using the experimental values from

11. A. M. Polyakov, Phys. Lett. 59, 82 (1975).

Fig. 1, one estimates σ ≃ 10⁵ K/cm. At the same time,

12. A. M. Polyakov, Gauge Fields and Strings, Harwood

in order to deconfine, the charges have to overcome

Academic Publisher, Chur (Switzerland) (1987).

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2019