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Electrons and Disorder in Solids$

V. F. Gantmakher

Print publication date: 2005

Print ISBN-13: 9780198567561

Published to British Academy Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567561.001.0001

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Electrons and Disorder in Solids
Oxford University Press

Two books about tunneling phenomena, Burstein and Lundquist (1969) and Solymar (1972), published many years ago are still very useful. The latest achievements in this field can be found in the review by Aleiner et al. (2002).

Let two conducting materials (M1 and M2 in Fig. B.1) be in contact with one another via an insulating layer I so thin that it admits electron tunneling. The whole device is called a tunnel junction and the two conducting materials are called electrodes. In equilibrium, the Fermi levels in the electrodes coincide. An electron with energy ε with respect to the Fermi level εF = 0, which was in M1, on the left of the barrier prior to tunneling, after tunneling has the same energy but is located in M2, on the right of this barrier. The equilibrium is dynamic in the sense that there are electron flows in the opposite directions, but the total flow through the barrier I equals zero.

Compare the tunneling considered in this appendix and the hopping conductivity considered in Chapter 4. In both cases, an electron passes under the barrier through a classically inaccessible region. However, the hopping conductivity proceeds due to quantum transitions between localized states, whereas the tunnel current through a junction arises due to quantum transitions between the delocalized states on different sides of the barrier.

In most instances, the resistance of the electrodes is much lower than the resistance of the junction. Therefore the potential difference V applied to the


Fig. B.1. Energy scheme of current flow through a tunnel junction M1–I–M2.

(p.213) electrodes is concentrated within the tunneling gap. The tunnel junction becomes out of balance, and the tunneling current J,
J ( V ) g ( ε e V ) g 1 ( ε ) [ f ( ε e V T ) f ( ε T ) ] d ε ,
starts flowing through the junction (Fig. B.1). The integrand in eqn (B.1) describes the resultant flow of electrons with energy ε from the left to the right caused by the relative shift of the Fermi distribution f(x) = (exp x + 1)−1 on both sides of the barrier by eV. The densities of states g and g 1 enter the integrand as cofactors, because they determine the number of electrons participating in tunneling and the number of states to which tunneling is possible. We omit the proportionality coefficient which should take into account the area of the tunneling contact and the barrier transparency.

Expression (B.1) is considerably simplified if one of the electrodes is a conventional metal with g 1(ε) = const. and the temperature is so low that the Fermi distribution may be considered as a step. Then

J ( V ) g 1 0 e V g ( ε ) d ε .
The addition of an AC voltage V ω sin ωt to the DC voltage V results in the appearance of an alternating current with frequency ω and amplitude J ω proportional to the derivative dJ/dV,
J ( V + V ω sin ω t ) = J ( V ) + d J d V V ω sin ω t .
The above experimental trick, called the modulation method, is rather popular. It follows from eqns (B.2) and (B.3) that
J ω d J d V g ( e V ) .
This allows one to use tunneling measurements for the direct determination of g(ε).

Expression (B.4) and the procedure for obtaining the function g(ε) from the experimental data are somewhat complicated by the finite temperature T ≠0,

J ω d J d V g ( ε ) d ε ( e V ) f ( ε e V T ) ,
but the study of the density of states in the vicinity of the Fermi level with the aid of the current–voltage characteristics of the tunnel junction is possible at finite temperature too.

Note: Tunneling experiments are much more complicated than the scheme considered above. One has to create a reliable and reproducible 10–15Å-thick tunneling break between the electrodes. This experimental problem was solved only in the 1960s. (p.214)


Fig. B.2. Differential conductivity as a function of bias at the tunnel junction Pb–SiO2–Si: B. Circles represent the results of the recalculation of the curve in the field H = 2 kG, which eliminates the effect of the thermal broadening of the distribution function (Massey and Lee 1995). Note the shift of the circles with respect to the solid line.

Tunneling experiments on superconductors allowed one to directly measure superconducting gaps and resulted in the discovery of the Josephson effect. Thus, these experiments considerably influenced the development of low-temperature physics.

Being applied to normal metals, the tunneling method allowed one to reveal the minimum in the density of states at the Fermi level in the spectrum of dirty metals and the Coulomb gap in the spectrum of localized states and also to follow the transformation of the former into the latter.

The revealing of the Coulomb gap in Si: B is illustrated by Fig. B.2. The Pb–SiO2–Si: B structure with one electrode superconducting and the other a classical p-type semiconductor was used. The intricate shape of the curve g(V) in zero magnetic field is explained by the use of superconducting lead as the electrode, whereas Si: B plays the part of a metallic counterelectrode. The corresponding curve depicts the superconducting gap in Pb with the density-of-states maxima on both sides of the curve. As was to be expected, the curve is symmetric with respect to the Fermi level εF = 0. In this case, the measurement of the superconducting gap is, in fact, a calibration experiment. It is an additional demonstration of the method validity. In a field of 2 kG, the superconductivity of lead is completely destroyed, and the density of states in lead becomes energy independent. Now, the lead electrode becomes a counterelectrode, and the minimum on the curve dJ/dV (V) is due to the existence of a parabolic Coulomb gap in Si: B. The 2 kG field is too weak to influence the Coulomb gap. As is seen from the curve, the gap width is about 1 meV ≈10K (i.e., 5K on both sides of the Fermi level).

The curves in Fig. B.2 were obtained at 1.15 K. To obtain these curves at lower temperatures is rather difficult, because of an exponential increase in the bulk (p.215) resistance of Si: B. A decrease in the potential takes place in a considerable part of the electrode and not within the tunneling gap. However, instead of lowering the temperature in the experiment, it is possible to account for it with the aid of mathematics. A simple transformation of eqn (B.5) allows one to extract the function g(ε) (empty circles) from the experimental J ω(V) curve (solid lines in Fig. B.2). It is seen that as long as the temperature is not too high, it only slightly smoothens the function g(ε).

The main impurities in Si: B are acceptors and one may directly apply the model of an impurity band at low doping, within which the Coulomb gap was obtained (see eqn 3.22 in Chapter 3). However, the Coulomb gap is also observed in materials of different types, e.g., in binary amorphous films with one metal and one nonmetal components and ultrathin films. We shall illustrate this with several examples.

Figure B.3 shows the tunneling characteristics of the Ge1−xAux−Al2O3−Al structure. Here, Al is a counterelectrode, and Ge1−xAux is the material to be studied. The set of characteristics demonstrate the evolution of the spectrum with a decrease in the Au concentration x. The minimum of the curve observed at high x values corresponds to the minimum at the Fermi level in the spectrum. It is due to the interaction of diffusing electrons. This effect, which exists in dirty metals due to disorder, was considered in the last section in Chapter 2. However, the model considered there was obtained within the framework of perturbation theory under the assumption that the corrections to the function g(ε) were small. The curves in Fig. B.3 not only confirm the existence of the minimum in the spectrum, but also demonstrate the evolution of the minimum with the approach to the metal–insulator transition. At the lowest Au concentration, x = 0.08, the g(ε) function goes to zero at ε = 0, and, in the vicinity of this point, is described by a parabola. This is a Coulomb gap


Fig. B.3. Tunneling characteristics of the Ge1−xAux−Al2O3−Al structure (McMillan and Mochel 1981).


Note: The model of the impurity band in a partly compensated semiconductor (within which a Coulomb gap was obtained in Chapter 3) cannot be directly applied to Ge1−xAux at low x, because this material does not contain well-defined “donors” and “acceptors,” impurity band, random electric field, etc. Therefore, the Coulomb gap revealed in Ge1−xAux indicates that a Coulomb gap can arise in different classes of random potentials.

Thus, the series of curves in Fig. B.3 demonstrate three important results – the presence of the minimum at the Fermi level in the spectrum of a dirty metal, which arises due to interelectron interactions (Chapter 2); the existence of a Coulomb gap in the spectrum of a strongly disordered insulator (Chapter 3); and the transformation of the former into the latter, which accompanies the metal–insulator transition driven by the change of the concentration of the metal chemical elements in an alloy (Chapter 5).

A similar spectrum structure and evolution were also observed in amorphous Si1−xNbx alloys (Fig. B.4). In the experiments performed on these amorphous alloys, one more interesting feature was observed – the critical vales of concentration determined from the variations of conductivity, x c, and the tunneling, x c ( t ) , , are somewhat different. The curve with concentration x c is indicated by an arrow in Fig. B.4. For this state, the function σ(T) → 0 with T → 0. However, at this concentration, a finite density of states gF) ≠ 0 is observed at the Fermi level. The density of states gF) becomes zero at lower concentration x c ( t ) . .

From the definition of a metal given at the beginning of Chapter 5 it follows that the true critical concentration is x c. The Fermi level of an insulator may


Fig. B.4. Tunneling characteristics of the Si1−xNbx–Al2O3–Al structure (Hertel et al. 1983).

(p.217) have a finite density of states if these states are localized. Therefore, in fact, the conductivity and tunneling measurements do not contradict each other.

At the same time, the discrepancy between x c and x c ( t ) can be interpreted in a different way. Let us give some comments on measurements of the tunneling current. In the consideration which allowed us to write eqn (B.1) for the tunneling current, it was implicitly assumed that electrons in both electrodes were noninteracting quasiparticles. Hence, the total electron energies on the left and on the right of the barrier were equal to E = ∑ εi (the sum of quasiparticles energies) and the tunneling process changed the total energies E on both sides of the barrier by the energy of a tunnelled quasiparticle. The total energy E in the system of interacting particles depends on their number, E = E(n). Then, part of the energy of the electromagnetic field is spent in the change of the total energy of the electron system with the variable number of particles. To avoid any misunderstanding, the density of states g i in eqn (B.1) (which is measured experimentally) is called the tunneling density of states.

Note: Various types of interelectron interactions act differently on the tunneling density of states. As an example, consider the Coulomb interaction of a tunneling electron with all the other electrons. After tunneling, both electrodes acquire spatial charge inhomogeneities, which have to be resolved. The longer the time it takes to resolve them, the more pronounced the inhomogeneity hinders tunneling, manifesting itself as a decrease in the effective density of states. The resorption time τres depends, in particular, on the character of electron motion – ballistic or diffusion. In diffusion motion, this time depends on the electron energy measured from the Fermi level (Chapter 2). Therefore, this effect may result not only in the renormalization of the tunneling density of states but also in the change of energy dependence of the function g(ε).

A similar observation was made in the doped semiconductor Si:B. The detailed measurements illustrated in Fig. B.5 show that here as well gF) goes to zero already in the insulator region where n ≈0.9n c (the critical concentration n c is determined from the dependence of conductivity σ(0) on n). The discrepancy in the determination of the critical concentration is about 10%.

However, the minimum of g(ε) at the Fermi level for the state with n = n c (where n c was determined as the critical concentration from the conductivity measurements) turned out in Fig. B.5 to be broader than for the states with other, both lower and higher, electron concentrations. Therefore, the tunneling measurements also single out the “transport value” n = n c defined above as the true value.

Specific objects of tunneling experiments are ultrathin films. The control parameter in such experiments may be the film thickness, and the quantitative characteristic, the resistivity at a certain fixed temperature. The thicknesses of Be films, whose tunneling characteristics are presented in Fig. B.6, range within 15–20Å. The indicated values of resistivity were measured at 50 mK. Tunneling structures were created by dosed oxidation of films in air and the subsequent deposition of an Ag layer onto the oxidized film. The thinnest Be film had a spectrum with a Coulomb gap, with the density of states in the vicinity (p.218)


Fig. B.5. Differential conductivity of the tunnel junction with a Si:B electrode (Massey and Lee 1996). For a clearer representation, the curves are displaced along the ordinate axis, but the origin of each curve is indicated on the ordinate axis. The correction which eliminates the effect of the thermal broadening of the distribution function is included.


Fig. B.6. Tunneling conductivities of junctions to Be films with different thickness are measured at the temperature T = 50 mK. Since the resistivity at T = 50mK is too high, the measurements for the film with the highest resistance were performed at T = 700 mK (Butko et al. 2000).

of εF varying linearly with the energy, in full accordance with the prediction of eqn (3.23) in Chapter 3. The energy spectra of thicker films showed a narrow minimum in the vicinity of εF. Thus, the ultrathin films, where the thickness determines the effective disorder and the Anderson transition, also have a minimum on the metal side of the transition and the Coulomb gap on the insulator (p.219) side of this transition. The specific feature of the curves in Fig. B.6 is that they represent the spectra of certain two-dimensional system.

Figure B.1 and eqns (B.1)–(B.5) describe the simplest scheme of tunneling experiments in which the finite dimensions of electrodes and the area of the tunnel junction are not taken into consideration. All the specific features of the current–voltage characteristics in this scheme are associated with the density of state of a bulky material. Now, let us assume that one of the electrodes is a small metal grain, say, a sphere of radius a. If an electron tunnels in this grain, the latter becomes charged with an electric field around it. The field energy of a single metal sphere with charge e is

U = e 2 2 C , C = κ a ,
where C is the capacitance of a remote sphere and k is the dielectric constant of the surrounding insulator. At low temperature, TU, the tunneling is possible only if the voltage between the sphere and the bulky electrode exceeds U/e, V = e/C = e/ka > U/e. This voltage threshold is called a Coulomb blockade (cf. eqns (8.6)–(8.8) in Chapter 8).

To observe the tunneling effect, both electrodes should be supplied with contacts. To avoid the enlargement of a small electrode by such a contact, two tunnel junctions should be switched in series with a small metal “island” located between these junctions. A Coulomb blockade is a phenomenon widely studied and used in various nanodevices, where the role of the intermediate electrode is usually played by a single grain. Here, another limiting case will be presented – Coulomb blockade in a system with a very large number of grains. Figure B.7 demonstrates the schematic of the experiment and results


Fig. B.7. Tunneling characteristics of the Al–Sn grains–Al structure at different temperatures. The structure is shown in the inset (Giaever and Zeller 1968). Hatched area is aluminum oxide (insulator).

(p.220) obtained. A slightly oxidized surface of an aluminum film was coated with tin, which formed individual islands with the average size being dependent on the amount of deposited tin. Then the sample was subjected to further oxidation. As a result, a thick aluminum oxide layer was formed between the tin islands, whereas the tin particles were coated with a thin tin oxide layer. Finally, an aluminum layer was deposited to create the upper electrode (inset in Fig. B.7).

The curves shown in Fig. B.7 were obtained in a magnetic field sufficiently intense to destroy superconductivity. As all the electrodes are good metals, the current–voltage characteristics of the structure would be horizontal lines. The drop in conductivity observed at low voltages and becoming more pronounced with lowering of the temperature is a consequence of the smallness in the structure of the intermediate electrode – set of tin grains – in other words, the consequence of a Coulomb blockade.

In the experiments illustrated by Fig. B.7, the grains form a two-dimensional layer. A set of grains may also form a three-dimensional conglomerate or a so-called granular metal, which is considered in Chapter 8. In a granular metal, measurements of the effective density of states performed with the use of tunneling are also possible. They are presented in Chapter 8. There, a granular metal is used as one of the two main electrodes, in full accordance with the scheme in Fig. B.1, and the Coulomb interaction enters the problem differently (see Fig. 8.7 and eqns 8.14 and 8.15).

Considering the further development of the tunneling method and its possibilities, one has to recognize that eqn (B.1) is valid only under the assumption that, in tunneling, the wave vector is not preserved. The change of the component normal to the contact surface is quite natural and is caused by the inhomogeneity of the space along the corresponding direction. The tangential component varies because of the rough junction interface. In principle, tunneling which preserves the tangential component of the wave vector (coherent tunneling) would be possible if one manages to prepare a junction with an atomically smooth interface.

Coherent tunneling was in fact observed in special experiments (Eisenstein et al. 1991) whose schematic and results are shown in Fig. B.8. Tunneling proceeded through a thin barrier between two two-dimensional quantum wells. The wells consisting of two 140 Å thick GaAs layers with a 70 Å thick AlAs barrier in between were prepared by molecular beam epitaxy. Both sides of this structure were coated with Al0.3Ga0.7As layers. The total thickness of this five-layer sandwich was about 50 μm. Indium contacts 1 and 2 were burned into both ends of the plate to provide the electrical contact with both wells filled with a two-dimensional electron gas. The metal gates were deposited onto the upper and lower surfaces (see inset in Fig. B.8). The gates a 1 and a 2 were used to divide the two-dimensional gas in both wells into two parts and, thus, to change the parallel connection of the indium contacts by two two-dimensional wells to in-series connection with a tunneling gap in between. The gates t 1 and t 2 allow one to vary the carrier concentration in both lower and upper wells.

Preparing such a structure, one has to solve two complicated experimental problems, which, in turn, requires a rigorous control of the molecular beam (p.221)


Fig. B.8. Conductivity G 12 between contacts 1 and 2 proportional to the tunnel current J as a function of the voltage at the gate t 2 with fixed voltage at the gate t 1. The inset shows that the “lower electron gas” is separated from contact 2 by the voltage at the gate a 1, whereas the “upper electron gas” is separated from contact 1 by the voltage at the gate a 2 (Eisenstein et al. 1991). The temperature T = 1.5K, the longitudinal voltage v = 0.1 mV.

epitaxy. First, it is necessary to obtain an atomically smooth interface between the GaAs and AlAs layers over quite a large area. Second, the electron densities in both wells should be made approximately equal.

Including contacts 1 and 2 into an external circuit with a negligibly low longitudinal voltage v adjusts the Fermi levels on both sides of the barrier and ensures the fulfillment of the conservation of energy. In a two-dimensional gas, the wave vector is located in the plane and is equal to

k F = ( 2 π n ) 1 / 2 .
Therefore, coherent tunneling in a nonzero magnetic field is possible only if the electron concentrations in both wells are the same.

The curve in Fig. B.8 was obtained at the longitudinal voltage v = 0.1mV and at a certain fixed voltage at the lower gate V t1. This curve demonstrates the dependence of the tunnel current J = vG 1,2 on the voltage V t2 at the upper gate, i.e., on the electron concentration n 2 in the upper well. The maximum of the tunnel current arises at equal electron concentrations in both wells, n 1 = n 2. As was to be expected, with the change of V t1, the maximum is shifted. The relative value of the maximum determines the ratio between coherent and incoherent tunneling.

(p.222) The two steps on the left-hand side of the J(V t2) curve have a natural explanation. The first step is formed when the electron concentration under the gate t 2 in the upper well decreases to such an extent that, under the gate, we have an insulator. Then the area of the tunnel junction decreases by about a factor of 2. The second step is formed at a voltage V t2 such that the electric field due to the gate t 2 divides the electron gas in the lower well and, thus, disrupts the electric circuit between contacts 1 and 2.