Determination of the diffusion constant in disordered mesoscopic Au rings from the temperature dependence of the current

Highlights

• The diffusion constant is obtained from the temperature dependence of the current.

• A modified data-fitting function gives a better diffusion constant.

• The method can be used to determine the diffusion constant in a closed-loop system.

Abstract

A simple calculation to determine the diffusion constant D for electrons in disorder mesoscopic Au rings from the temperature dependence of the current obtained by Chandrasekhar et al. (1991) [1] gives larger D than the diffusion constant $D\prime$ obtained from Einstein's relation via resistance measurements. In addition, the curve-fitting function employed by them does not agree well with the temperature-dependent data below 10 mK. Using a more precise calculation based on the quasi-particle picture, we show that the discrepancy between the fitting function and the experimental data below 10 mK is reduced and that a newly obtained D is close to $D\prime$. The method proposed here is very simple to determine the diffusion constant in a small closed loop via the curve-fitting of the temperature-dependent data, which needs no resistance measurements for the use of Einstein's relation.

Introduction

To obtain precise data of very small currents in a disordered mesoscopic wire that is set between macroscopic electrodes, we should eliminate the influence of these macroscopic electrodes because we cannot easily extract very small current signals in the mesoscopic wire from the entire system with large resistance of the electrodes. To detect the small current signals after eliminating the electrodes, we should curl up the wire to a ring. In this case, the size of the current in the ring can easily be determined from a magnetic response by the application of a magnetic field to the ring.

This type of experiment was performed for a micron-sized normal-metal (Au) loop at low temperature, and the size of the current in the loop was obtained from the measured magnetization with a dc-SQUID magnetometer [1]. In this measurement, the temperature (T) dependence of the current amplitude $I_{h/e}$ for a period of ${\varphi }_0=h/e$ was obtained for a ring with a diameter of 2.4 µm and a loop with a rectangle of 1.4 µm×2.6 µm in Fig. 3(a) of Ref. [1], which is replotted in Fig. 1 in this paper. To obtain physical parameters (e.g., the diffusion constant) from the data, a theoretical curve $I_{h/e}=(\mathit{Ce}/{\tau }_D)e^{-L/l_T}$ was employed in Ref. [1], where C is a constant, e is the electric charge, L is the circumference of the loop, ${\tau }_D=L^2/D$ is the time required for an electron to diffuse around the loop (D is the diffusion constant), and l_T is the thermal diffusion length defined by $l_T={(\mathit{\hslash }D/k_{\mathrm{B}}T)}^{1/2}$ [2] (ℏ is the Planck constant h divided by $2\pi$ and $k_{\mathrm{B}}$ is the Boltzmann constant).

Since the experimental data showed little temperature dependence below $T_0=10$ mK ($T_0^{1/2}=0.1$ K${}^{1/2}$), a data-fit was performed at a temperature-dependent slope beyond 10 mK [1] by use of a function

$$I_{h/e}=C\frac{e}{{\tau }_D}{e}^{-L/l_T}=C\frac{e}{{\tau }_D}{e}^{-L{(k_{\mathrm{B}}T/\mathit{\hslash }D)}^{1/2}},$$

which is a straight slope when a logarithmic plot of $I_{h/e}$ is made with a $T^{1/2}$ scale, as indicated in Fig. 1 by the solid straight line. Clearly, this fit does not agree well with the experimental data below 10 mK (or 0.1 K ${}^{1/2}$). Thus, it is considered inappropriate to extract physical parameters from it.

Nonetheless, if one uses the above straight slope to make the data-fit, one obtains $L/l_T=14.4T^{1/2}$ via the method of least squares, which gives the relation

$$D=\frac{k_{\mathrm{B}}{L}^2}{14.4^2\mathit{\hslash }}.$$

Eq. (2) provides the size of the diffusion constant D by the insertion of L=7.54 µm of a 2.4 µm ring or L=8 µm of a $1.4\mu \mathrm{m}\times 2.6\mu \mathrm{m}$ rectangular loop, $k_{\mathrm{B}}=1.38\times {10}^{-23}$ J/K, and $\mathit{\hslash }=1.05\times {10}^{-34}$ J s. We then obtain $D=3.60\times {10}^{-2}$ m²/s for the ring or $D=4.06\times {10}^{-2}$ m²/s for the rectangular loop.

To check whether those diffusion constants (D) are relevant or not, we compare them to the diffusion constant $D\prime$ obtained from Einstein's relation [3] with a measured resistance of $R_{\square }=0.2$ Ω for an Au film [1]:

$$D\prime =\frac{1}{e^{2}N(0)R_{\square }{L}_z}=3.25\times {10}^{-2}{\mathrm{m}}^2/\mathrm{s},$$

where

$$N(0)=\frac{4\pi }{h^3}{(2m_0)}^{3/2}{E}_{\mathrm{F}}^{1/2}$$

is the density of states at the Fermi level (The Fermi energy in Au is $E_{\mathrm{F}}=5.51\mathrm{eV}$ [4]), and L_z (=60 nm) is the thickness of the Au film [1]. Since the Au-film thickness (L_z=60 nm) is much greater than the electron wavelength (or the Fermi wavelength) ${\lambda }_{\mathrm{F}}=0.52$ nm, the three dimensional density of states in Eq. (4) is employed, where ${\lambda }_{\mathrm{F}}(=h/m_0{v}_{\mathrm{F}})$ is calculated from the Planck constant $h=6.63\times {10}^{-34}$ J s [5], the electron mass $m_0=9.11\times {10}^{-31}\mathrm{kg}$, and the Fermi velocity $v_{\mathrm{F}}=1.39\times {10}^6\mathrm{m}/\mathrm{s}$ in Au [6].

The obtained D=3.60–$4.06\times {10}^{-2}$ m²/s via Eq. (1) is 10.8–24.9% greater than $D\prime =3.25$ m²/s in Eq. (3). In addition, as mentioned above, a gap exists between Eq. (1) and the experimental data below 10 mK (0.1 K ${}^{1/2}$).