Why We Study Granular Cr films

In this thesis, we study four the Al/AlOx/Cr planar tunnel junctions comprising of granular Cr electrodes. The non-magnetic insulating barrier AlOx was intentionally used, instead of the magnetic CrOx, to rule out the possible influence of the magnetic nature of barrier on the tunneling signals. Besides, the attempt to answer the puzzle in history, we also have a great interest in the granular conductors.

Experimentally, it is intriguing that electrical-transport measurements on granular systems with strong intergrain coupling often revealed ln T dependence of resistivity [21, 22], rather than of conductivity [23, 24] as the theory predicted (see Sec.2.4 and Fig. 1.3 (a)).

Therefore, it still awaits convincing experimental test. Moreover, to the best of our knowledge, there is still no experimental observation in three-dimensional (3D) granular films concerning the electronic DOS. Thus, we are motived to study Al/AlOx/granular Cr tunnel junctions. The dimensionality of the granular arrays (the Cr electrodes) was made to have d = 3, and both conductivities and tunneling DOS of the Cr electrodes (with intergrain tunneling conductance spanning from g ≅ 1 to g ≫ 1) are measured to verified the recent theory of granular metals [25-27]. We would like to mention that previously in our experiment, we found a Efros-Shklovskii type temperature dependence in the nanocontacts formed with granular Cr films, where the σ ∝ exp(−√T₀/T ) behavior was observed in the broad temperature interval of 1 − 100 K [6] (see Fig. 1.3 (b)), while the conductivity at low temperatures is theoretically established to possess this type-like dependence in the opposite limit of weak intergrain coupling (g ≪ 1), for T₀ is a characteristic temperature [23,28-31]. In that case, we experimentally realized the regime g ≪ 1 [32].

This thesis is organized as follows: In Chap. 2, besides the granular system, the the EEI effects and the property of tunnel junctions are also briefly introduced. We discuss our experimental considerations, methods and setups in Chap. 3. Chapter 4 contains our

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experimental results and discussions. We interpret our measured conductivity and tunneling DOS in terms of the theory of granular metals, and rule out the WL and EEI effects developed for weakly disordered homogeneous conductors to be the origins of our observations. Finally, our conclusion is presented in Chap. 5.

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(a)

Fig. 1.3 (a) The temperature dependence electrical transport behaviors of the granular systems in previous studies: (Up) A logarithmic temperature dependence of the resistivity of the granular NbN cermet film (in a sample with (300 K) ~ 100 Ω/ ) [21]. (Down) A logarithmic temperature dependence of the conductivity of the granular Pt/C nanowires (ρ(300 K) ~ 4 mΩ cm) in a wide range of up to 200 K [24].

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(b)

Fig. 1.3 (b) Previous,y, the regime g ≪ 1 in the granular has been verified to have the Efros-Shklovskii type temperature dependence, where the σ ∝ exp(−√T₀/𝑇 ) behavior, in our experiment. In this experiment, the nanocontacts formed with granular Cr films, and the difference between the two curve is the contact resistance. And it was observed in the broad temperature interval of 1 − 100 K, log R is a function of T^-1/2 [5].

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Chapter 2

Background and Theory

There are three main parts will be introduced in this chapter. In this thesis, we measure the differential conductance to investigate the DOSs of the granular Cr electrode, so we start with Sec. 2.1 the general tunneling behavior in metal-insulator-metal junctions and Sec. 2.2 the BDR Model describes the background tunneling conductance in the junctions. Then from Sec. 2.3 to 2.4, theoretical predictions related to our system will be introduced. Since the granular conductor is also a disorder system, the electron-electron interaction (EEI) effects, which have well explained weak disorder systems, will be mentioned in Sec.2.3. And finally, the theoretical prediction for the granular electronic system is in Sec. 2.4.

2.1 Tunneling in Metal-Insulator-Metal Tunnel Junctions

In this thesis, we measure the tunneling current through the metal-insulator –metal (M-I-M) tunnel junctions to investigate the DOS of granular conductors. In order to know the relation between the DOS of electrodes and the tunneling current, we start with the basis of the tunneling mechanism and extend to obtain the tunneling current in such a junction structure. Then, the relation of the DOS of electrodes and the differential conductance will be derived.

In classical view, we know that there is no way for a particle passing through a barrier

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with a potential higher than its total energy. However, in quantum mechanism, whenever a particle treated as a wave runs into a barrier potential, there will be both probabilities to penetrate it and to be reflected. In our cases, we can consider the energy diagram in a M-I-M tunnel junction similar to Fig. 2.1 (a): x < 0 (the left of the barrier), 0 < x < t (within the barrier), t < x (the right of the barrier). The wave incident from the left is expressed to be e^𝑖𝑘𝑥 for k =√2𝑚𝐸_𝑥/ℏ (𝐸_𝑥: total energy of the tunneling particle in the x direction, m: the effective mass) and it decays in an exponential form, e^−𝜅𝑥 for κ = √2𝑚(𝜙(𝑥) − 𝐸_𝑥) (𝜙(𝑥): the barrier potential), inside the barrier. And, the wave penetrated to the right is Te^𝑖𝑞𝑥. The transmission coefficient, D, is a ratio of the probability flux transmitted through the barrier to the probability flux incident upon the barrier. For an extremely small transmission [33], D can be reduced to be

D(E_𝑥) = ge^−2𝐾, from Metal 1 to Metal 2 can be expressed as [33]

𝐽_1,2(V) = − 2e

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Then, the net current density would be

𝐽 = 𝐽_2,1(V) − 𝐽_1,2(V) involve the electrons of energy within the transport window opened by eV, so

𝐽 = −2e𝜌_𝑡

ℎ ∫ ∫ 𝑑𝐸_𝑥𝑑𝐸_𝑟𝐷(𝐸_𝑥, 𝑉)

𝐸_𝑟 𝐸_𝑥

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(a)

Fig. 2.1 (a) The schematic energy diagram of a metal-insulator-metal (M1-I-M2) tunnel junction with a finite bias V. ϕ(x) describes the barrier potential. The net tunneling current flow through the junction should be the summation of current from left to right and that from right to left, i.e. I = I₁₂+ I_21.

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2.2 The BDR Model

-- the Background Tunneling Conductance in the Metal-Insulator-Metal Tunnel Junction Figure 2.2 (a) shows the schematic energy diagram in a metal-insulator-metal tunnel junction. As such a structure is made, the Fermi energies of two metals become equal after the system reaches the equilibrium, and the barrier is asymmetric due to the different work function in the metals. That is, the barrier height seen from two electrodes is different, i.e.

ϕ₁ for Metal 1 and ϕ₂ for Metal 2 as showed.

In order to see how the voltage bias affects the tunneling behavior in this tunnel junction, Brinkaman, Dynes and Rowell [34] (BDR model) describe the asymmetric barrier height in a simple form (corresponding to Fig. 2.2 (b)):

ϕ(x, V) = 𝜙₁+𝑥 G(V) depands on the bias approximately in a parabolic form under the BDR model, and we notice that the minimum value takes place at the zero bias for Δϕ = 0 or when the barrier is symmetric.

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(a)

(b)

Figs. 2.2 (a-b) The schematic energy diagram of a metal-insulator-metal (M1-I-M2) tunnel junction with (a) zero-bias (b) a finite bias V. Under BDR model, the potential of the barrier 𝜙(𝑥) can be described to be ϕ(x, V) = 𝜙1+^𝑥_𝑡(𝜙₂− 𝑒𝑉 − 𝜙₁) (so that ) ϕ(x, 0) = 𝜙₁+^𝑥_𝑡(𝜙₂− 𝜙₁) for (a) the zero-bias.

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2.3 The Electron-Electron Interaction in Disorder System

The electron-electron interaction (EEI) and weak localization effects in disordered systems have been studied for decades [9-13, 35]. Although both the two effects contribute significantly to the electrical conduction properties of a given system, it is possible to understand the individual contribution. Indeed, the EEI effect can be isolated by studying the differential conductances, G(V, T) = dI(V, T)/dV , of a metal-insulator-metal tunnel junction at low temperatures. Here the (left) reference electrode is made of a clean metal while the (right) electrode is often made of a weakly disordered metal to be investigated.

According to Altshuler and coworkers [9-13], the EEI effect in the presence of weak disorder will lead to suppression in the electron density of states (DOS) near the Fermi level.

Moreover, such a DOS singularity is predicted to be sensitive to both bias voltage and temperature. At a low temperature (to ignore the thermal smearing effect), and at the weak-disorder limit, i.e.,k ≫ 1, where k is the Fermi wavenumber and is the elastic mean free path of electrons, the EEI-induced correction to the DOS, δNd(E), depends strongly on the effective dimensionality d of the sample and has the following forms:

for d = 2, the correction is given by [9, 11, 12]

- 19 - relative to the Fermi energy. Besides, the theory predicts that a crossover from two to three dimensions should occur at a characteristic bias energy eVc = Ec ≈ (2π)²(ℏD/t²). This critical energy corresponds to an EEI characteristic length of Lc =√ℏD/Ec ≈ t/2π. Notice that the characteristic bias voltage Vc scales with D/t², and hence it reduces with increasing disorder.

For a metal-insulator-metal tunnel junction comprised of a clean metal and a disordered metal, at low temperatures, the variation in the differential conductance G(V) directly reflects the energy dependence of DOS of the disordered metal [33]:

G(V) = PN_𝑐(0)N_𝑑(eV), where P is the tunneling rate which depends on the barrier properties (barrier height and thickness), N_𝑐 is the DOS of the clean metal which depends weakly on energy and can be approximated as the value at Fermi level N_𝑐(0), and N𝑑 is the

Experimentally, from measurements of the G(V, T) of metal-insulator-metal tunnel

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junctions, the DOS singularities have been investigated by several groups. Both weakly [36-41]

and strongly [15, 16, 42] disordered metal electrodes have been employed. In recent years, the EEI theory has further been critically tested by high-resolution photoemission spectroscopy measurements [43]. In general, it is found that the EEI theory is fairly successful in explaining the experimentally observed DOS singularities in weakly disordered conductors.

In addition, the EEI prediction of a crossover from the two-dimensional G(V) ∝ lnV law [11, 12] to the three-dimensional G(V) ∝ √V law [13] as the bias voltage increases has been confirmed by several experiments [36-39, 41] where electrodes of metal films in the tunnel junctions were used.

2.4 Theoretical Prediction for Granular Electronic Systems

Granular conductors, which are composite materials of metallic granules and dielectric

components, have recently attracted much renewed theoretical attention as tunable systems for addressing mesoscopic physics problems [25, 26]. In contrast to disordered

“homogeneous systems”, the electronic transport properties of granular conductors are largely governed by the strength of the intergrain tunneling [25]. Theoretically, a granular conductor is characterized by a number of physical quantities: the mean energy level spacing in a single granule, δ, the dimensionless tunneling conductance between neighboring granules, g (i.e., the average tunneling conductance between neighboring grains expressed in units of 2e²/h), and the single-grain Coulomb charging energy, Ec. For strong intergrain coupling (g ≫ 1) and in the not-too-low temperature interval g ≪ k_𝐵T ≪ Ec (where k_𝐵 is the Boltzmann constant), charging effects are important yet the quantum-interference weak-localization (WL) effects are suppressed. This unique regime provides a tempting

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opportunity to probe the electronic conduction properties due to the many-body Coulomb interaction effects in the presence of granularity. The electrical conductivity σ is predicted to obey the law [26, 44]

σ(T) = σ(0)[1 −

_{2𝜋𝑔𝑑}¹

ln (

_𝑘^𝑔𝐸𝑐

𝐵𝑇

)]

where d is the dimensionality of the granular array, and σ(0) = 2(e²/h)ga^2−𝑑 is the classical conductivity without the Coulomb interaction (i.e., the system conductivity at temperature k_𝐵T ≫ Ec), and a is the radius of the (spherical) grain. It is important to note that, unlike that due to the WL and electron-electron interaction (EEI) effects in weakly disordered homogeneous systems [9], this σ ∝ ln 𝑇 law is predicted to hold for all dimensions, since the dimensionality d only enters the prefactor of the logarithmic correction term. On contrary, the functional form of the tunneling electronic density of states (DOS) is predicted to depend critically on sample dimensionality:

for d = 3 [26, 44],

ν

₃

(ϵ) = ν

₀

[1 −

_4𝜋𝑔^A

ln (

_max(𝑘^𝑔𝐸𝑐

𝐵𝑇,𝜖)

)]

where ϵ is the tunneling electron energy measured from the Fermi energy level (E_F), 𝜈₀ is the DOS in the absence of Coulomb interaction, and A is a numerical prefactor;

for d = 2

ν

₂

(ϵ) = ν

₀

exp[− 1

16𝜋

²

𝑔 ln

²

( 𝑔𝐸𝑐

max(𝑘

_𝐵

𝑇, 𝜖) )]

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The underlying physics which leads to the conductivity and DOS corrections is due to the presence of local voltage fluctuations between neighboring granules.

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Chapter 3

Experimental Method

3.1 Sample Fabrication

To study the conductivity and the tunneling DOS of granular Cr films, we fabricated four Al/AlOx/Cr planar tunnel junctions by using the standard thermal evaporation method:

1. First, a set of parallel, relatively clean 0.8 or 1 mm wide and 25 nm thick Al films were deposited on glass substrates held at room temperature.

2. Then, surfaces of the as-deposited Al films were subsequently oxidized by utilizing plasma discharge to produce a ≈ 1.5– 2 nm − thick AlOx layer.

3. Finally, a long Cr electrode (1 mm wide, and 15– 30 nm thick) was then deposited across the parallel AlOx coated Al strips to complete the tunnel junction geometries. At the same time, the Cr electrode was attached with leads appropriate for four-probe electrical measurements.

Figures 3.1 (a-c) show the schematic diagrams of our sample during the steps described

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above. The resistivities of our Al reference electrodes were typically 13 (16) µΩ cm at 4 (300) K, corresponding to the product k ≈ 54 at 4 K, where k is the Fermi wavenumber and is the electron mean free path. The conductivity of the Cr electrode in each set of junctions was adjusted by varying the mean Cr film thickness and the deposition rate between 0.01 and 1.5 nm/s. To achieve a very low conductivity in the junction D, the Cr film was deposited onto a cold substrate held at liquid-nitrogen temperature, by employing a very low deposition rate of ∼ 0.01 nm/s.

The reason for selecting Cr as our electrode is because Cr films deposited in a vacuum often form granular, rather than uniform and continuous, layers [6-8]. For example, a 10 − nm − thick Cr film deposited by thermal evaporation on a mica substrate showed a distribution of disk-shaped granules with a diameter of ~ a few tens of nanometer and a height of ≈ 2– 6 nm, as was evidenced from atomic force microscopy (AFM) studies (see Fig.

3.1 (d) [6]). Varying the deposition rate modified the average grain size [6]. Even thermally evaporated in a vacuum having a background pressure as low as ∼ 1 × 10⁻⁶ mbar, the surfaces of Cr granules became oxidized and formed thin dielectric layers of CrOx [8]. In this work, we carried out our thermal evaporation deposition at a pressure of ∼ 5 × 10⁻⁶ mbar so that our films were guaranteed to form metallic Cr granules separated by thin CrOx dielectric layers. Table 1 lists the values for the relevant parameters of the four Cr electrodes comprising the tunnel junctions A–D studied in this work.

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(a)

(b)

(c)

Figs. 3.1 (a-c) Schematic diagrams of the top views (left) and side views (right) of our sample during the fabrication progresses: (a) The evaporated Al film; (b) Utilizing O2 plasma to oxidize the surface of the deposited Al film; (c) Depositing Cr film to cross the AlOx/Al structure. The side view shows the junction area where the red dotted lines enclose on the left top view figure.

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(d)

Fig. 3.1 (d) The AFM image of a thin Cr layer, prepared by thermal evaporation deposition on a mica substrate, shows the granular property. The surface profile below is along the solid line indicated in the AFM image [6].

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Table 1 Values for relevant parameters of the Cr electrodes in Al/AlOx/Cr tunnel junctions.

is the junction area and is the junction resistance at 300 K. t is the thickness, ρ ( ) is the resistivity (sheet resistance) at 2.5 K, and N𝑐𝑟,𝑑(0) is the DOS at the Fermi energy. ^k was calculated by using the Drude model. The diffusion constant D was evaluated through the Einstein relation ρ⁻¹ = N_{𝑐𝑟,𝑑}(0)𝑒²D. The values of k ℓ and D were evaluated for 2.5 K. Ec, σ0, g and a are defined in Sec. 2.4. Notice that the values of a are only listed for reference, because our Cr granules are disk-shaped rather than spherical.

Sample 𝐍_{𝒄𝒓,𝒅}(𝟎)

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3.2 Experimental Methods

In our experiments, we measured the resistivity of Cr films with the four-probe method, while the tunneling differential conductances, G(V, T) = dI(V, T)/dV, across the junctions were measured by utilizing the standard lock-in technique, where I is the tunneling current between the Al and Cr electrodes, and V is the voltage dropped across the insulating barrier (see Fig. 3.2 (a) for a schematic diagram). Figure 3.2 (b) shows the equivalent circuit of measuring the differential conductance to have the DOS of Cr (see Sec. 2.1).

To ensure the quality of each tunnel junction, we measured the superconducting gap of the clean Al electrode at 0.25 K before performing detailed measurements of G(V, T) curves. Our Al electrodes became superconducting at ≈ 1.8 – 2 K, and Fig. 3.2 (c) shows one of the results in Sample D. Due to the great change of the superconducting gap in the differential conductance, the experiment, different from measurement for the G(V) = dI(V)/dV of Cr electrodes, is performed by applying V to measure I. The equivalent circuit is in Fig. 3.2 (d).

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(a)

Fig. 3.2 (a) Diagrams depicting (up) four-probe resistance measurements of Cr electrodes, and (down) differential conductance measurements of Al/AlOx/Cr tunnel junctions. Black strips stand for Al films, and green (gray) strips for Cr electrodes.

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(b)

Fig. 3.2 (b) The equivalent circuit for the differential conductance measurements for Cr electrodes.

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(c)

Fig. 3.2 (c) The differential conductance measurement of the most disordered Cr electrode, sample D, at temperature T = 0.25, 0.5, 1, 2.2 K where the Al electrode has already been superconducting. The barrier quantity can be tested and verity by the signature of the superconducting gap, since G(V) = dI(V)/dV is a function of N_𝐶𝑟(V) and N_𝐴𝑙(V).

The measured result reveals the superconducting gap at T = 0.25 K is ≲ 0.5 meV, while the calculated energy gap for Al being superconducting at 1.14 K is 0.34 meV. The data (■) shows the superconductivity is destroyed by the applied magnetic field.

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(d)

Fig. 3.2 (d) The equivalent circuit of the G(V) = dI(V)/dV measurements for the superconductor energy gap. Due to the great and sudden change at the edge limit of the gap, we send V to measure I in contrary to Fig. 3.2 (b).

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3.3 Experimental Setups

In our experiments, we measured and the DOS of Cr electrodes under low temperature with ³He cryostats (see Fig. 3.3 (a)) which will be introduced together with the superconducting magnet (Fig. 3.3 (b)) in this section.

Before starting to cool down our system, we first launch our sample on the sample holder just below ³He pot, and then seal the parts below IVC (inner vacuum chamber) flange with an IVC shell. After pumping out the air inside IVC space (to be vacuum), we put some ⁴He gas utilized to be exchange gas. There are three stages to cool the system:

1. From room temperature to 4.2K

The cryostats first puts into the LN2 for pre-cooling until T ~ 80 K. Then move the cryostats into LHe4 to be T ~ 4.2 K. During this stage, there are only a few exchange gases inside the IVC shell, therefore the temperature will not go down too quickly.

2. From 4.2 K to 1.8 K

At this stage, we lower the temperature by lowering the pressure above the LHe4 less than 1 atm. In Fig. 3.3 (a), 1.5 K condenser is above ³He pot. We pump the LHe4 in and out of a small tube attached or connected to the 1.5 K condenser with different pumping rates, so that the temperature can be decreased to be ~ 1.5 K as the pressure is lower.

Through the thermal contact (from 1.5 K condenser to the sample holder), the sample can be cooled down.

3. From 1.8 K to 0.3 K

This stage uses the same method in stage 2 but with LHe3 to cool the system down to 0.3 K. Activated carbon, cable of absorbing the gases, is used as the sorption pump, and

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this ability depends on temperature (the higher temperature it is, the larger kinetic energy the gas has, and the poorer ability of the sorption pump to absorb gas). The sorption pump is connected to ³He pot (which is just above the sample). At beginning, we control the sorb to be ~45 K where ³He should be gas since its condensation point at 1 atm is ~2.8 K. As the system cools to be ~1.8 K at stage 2, ³He should be condensed to be liquid. If we lower the temperature of the sorb at this time, then it will absorb gas more efficiently, so that the gas pressure in ³He pot decreases. It leads to the already condensed LHe3

evaporate! Then the ³He pot temperature cools downs again until reaches the lowest temperature ~ 0.3 K.

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(a)

Fig. 3.3 (a) The photo (left) and schematic figure (right) of He³ cryostats.

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Chapter 4

Result and Discussion

In this chapter, we first demonstrate our measured results in Sec. 4.1 temperature dependence of conductivity curves and in Sec. 4.2 temperature dependence of differential conductance curves. Then we compare our results with available theoretical predictions concerning disordered conductors: In Sec. 4.3, a comparison with the conventional EEI effect, originally developed for weakly disordered homogeneous conductors by Altshuler and coworkers [9, 12, 13], is made. Meanwhile, we try to discuss our observations in terms of the recent theory of granular metals, formulated by Efetov and Tschersich [26] and Beloborodov and coworkers [25, 44], in both Sec 4.4 logarithmic temperature dependence of conductivity and Sec. 4.5 differential conductance curves and tunneling density of states.

4.1 Temperature Dependence of Conductivity Curves

Figure 4.1 (a) shows the variation in normalized resistivity, ρ(T)/ρ(280 K), with temperature for the four Cr electrodes of our tunnel junctions. Except for the junction A, the resistivities of all other samples monotonically increase with decreasing temperature.

However, the amounts of the resistivity rise are much smaller than what would be expected for samples falling deep on the insulating side, where resistivity should show

- 37 -

ρ ∝ exp(√T₀/T) dependence and rapidly increase with decreasing temperature. This result immediately reflects that the intergrain tunneling conductances g in our Cr electrodes must

ρ ∝ exp(√T₀/T) dependence and rapidly increase with decreasing temperature. This result immediately reflects that the intergrain tunneling conductances g in our Cr electrodes must

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