Relaxation of the resistive superconducting state in boron-doped diamond films

A. Kardakova, A. Shishkin, A. Semenov, G. N. Goltsman, S. Ryabchun, T. M. Klapwijk, J. Bousquet, D. Eon, B. Sacépé, Th. Klein, and E. Bustarret

Phys. Rev. B 93, 064506 – Published 8 February 2016

Abstract

We report a study of the relaxation time of the restoration of the resistive superconducting state in single crystalline boron-doped diamond using amplitude-modulated absorption of (sub-)THz radiation (AMAR). The films grown on an insulating diamond substrate have a low carrier density of about $2.5 \times 10^{21} {cm}^{- 3}$ and a critical temperature of about $2 K$ . By changing the modulation frequency we find a high-frequency rolloff which we associate with the characteristic time of energy relaxation between the electron and the phonon systems or the relaxation time for nonequilibrium superconductivity. Our main result is that the electron-phonon scattering time varies clearly as $T^{- 2}$ , over the accessible temperature range of $1.7$ to $2.2$ K. In addition, we find, upon approaching the critical temperature $T_{c}$ , evidence for an increasing relaxation time on both sides of $T_{c}$ .

Received 1 October 2015
Revised 12 January 2016

DOI:https://doi.org/10.1103/PhysRevB.93.064506

Physics Subject Headings (PhySH)

Electron-phonon coupling Superconductivity

Thin films

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. Kardakova^* and A. Shishkin

Physics Department, Moscow State Pedagogical University, Russia

A. Semenov

Physics Department, Moscow State Pedagogical University, Russia and Moscow Institute of Physics and Technology, Russia

G. N. Goltsman^† and S. Ryabchun

Physics Department, Moscow State Pedagogical University, Russia and National Research University Higher School of Economics, Russia

T. M. Klapwijk^‡

Physics Department, Moscow State Pedagogical University, Russia; Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands; and Donostia International Physics Center, Donostia-San Sebastian, Spain

J. Bousquet, D. Eon, B. Sacépé, Th. Klein, and E. Bustarret

Institut Néel, CNRS, Grenoble, France and Université Grenoble Alpes, Grenoble, France

^*kardakova@rplab.ru
^†goltsman@rplab.ru
^‡t.m.klapwijk@tudelft.nl

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Issue

Vol. 93, Iss. 6 — 1 February 2016

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Images

Figure 1
Current-voltage curves at different temperatures near the critical temperature $T_{c}^{r}$ in zero magnetic field. Under the RF power an operation point ( $I = 20 μ A$ and $V = 1 mV$ ) shifts along the equivalent load resistance line that produces the voltage signal $δ V$ . The inset shows the equivalent circuit for determining an output signal of the superconducting film upon the absorption of the amplitude-modulated radiation.
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Figure 2
Coupling between the thermodynamic subsystems in the case of (a) a thin metal or superconducting (SC) film on an insulating substrate; (b) a boron-doped diamond film on a diamond substrate, illustrating, in comparison to (a), the absence of a Kapitza resistance. The electron reservoir under illumination can be described by the Fermi-Dirac distribution function $f (E)$ with an effective electron temperature $T_{e}$ exceeding the phonon temperature $T_{p h}$ . In practice, the metal film is in a resistive superconducting state which is due to the vortices, phase and/or amplitude fluctuations. It is assumed that the electron temperature $T_{e}$ , increased by DC power and by RF power, controls the changes of the resistivity of the superconducting state.
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Figure 3
The frequency dependence of the sample response at different bath temperatures $T_{b}$ . The experimental data are measured at a temperature in the middle of the superconducting transition (where $\partial R / \partial T = \max$ ) at the same bias current. The temperature of the resistive transition shifts when a magnetic field normal to the film plane is applied. The data of each curve were normalized to 0 dB for convenience. The solid lines are a least-square fit with Eq. (2). The fit standard error of the rolloff frequency does not exceed 10%. The inset shows the energy relaxation time vs the normalized temperature ( $T / T_{c}^{r}$ ), where the critical temperature $T_{c}^{r}$ is the temperature of the middle point of the resistive transition. The experimental results correspond to both types of measurements: case A (red triangles) and case B (black squares).
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Figure 4
The temperature dependence of the energy-relaxation time at low temperatures far from $T_{c}$ ( $0.75 T_{c}^{r} < T < 0.95 T_{c}^{r}$ ). The full lines represent the theoretical estimate for $τ_{e - p h} (T)$ according to Eq. (6). For comparison the dashed line represents a $T^{- 3}$ dependence.
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Figure 5
The inverse square of the relaxation time ( $τ^{- 2}$ ) and the film resistance as a function of the temperature. The dashed line corresponds to the longitudinal relaxation time, which diverges as ${(T_{c}^{L} / (T_{c}^{L} - T))}^{1 / 2}$ , where $T_{c}^{L}$ is determined as the temperature at which the superconducting gap is completely suppressed and $τ^{- 2} = 0$ . The values of $T_{c}^{L}$ , indicated in the legend, are close to the temperature $T_{c}^{r}$ determined from the resistive transition.
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Figure 6
The relaxation time vs $T - T_{c}^{p}$ , where the temperature $T_{c}^{p}$ correspond to a temperature at the peak separating regime II and regime III. On both sides of the temperature $T_{c}^{p}$ the observed relaxation time increases, suggesting a divergent behavior upon approaching $T_{c}^{p}$ . For sample N1, since the last three points were measured in the limit where the film is normal, we left them out of the discussion.
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