About the Frequency-Dependence of Electrical Characteristics of Quantum Devices

The frequency-dependence of electrical characteristics of quantum device components was researched. There were two types of nanostructures: quantum wire and junction nanostructures between two quantum wires with different cross sections. It is shown that conductivity of the first nanostructure is decreased with growth of the frequency and conductivity of the second nanostructure is increased with growth of the frequency.


Introduction
Interest to electrical characteristics of quantum wires is caused by new physical effects which are observed in one-dimensional conductors [1,2] and also by prospects of high-frequency applications of devices based on quantum wires [3,4].
The consecutive analysis of frequency-dependence of quantum device characteristics can be conducted in the framework of a mu ltiphase model of charge transport [4,5]. The model was successfully applied in order to calculate the characteristics of resonant-tunneling diodes and devices based on quantum wires [6,7].
In this paper, it is shown that frequency-dependence of one-dimensional electronic gas conductivity is determined by the frequency properties of the hydrodynamic velocity of electrons.
In quantum devices, the junction between quantum wires with d ifferent cross-sections can be used as a source of nonequilib riu m electrons [4,7]. Nonequilibriu m effects lead to specific dependence of such junction conductivity from frequency of external signal.
(10) where k -is the Bo ltzmann constant, T -is the absolute temperature, which in this article is a constant.
(12) In exp ression (12), the first co mposed has a linear dependence on the microscopic flow j(t,r,λ) and the second composed has a quadratic dependence of this value.
(13) Here, summation on λ 1 was replaced by equivalent summation on λ and designated as F l (t,r,λ) = F(t,r,λ), F r (t,r,λ) = F(t,r,λ 1 ). The distribution functions within brackets in the formula (13) differ only by the values of chemical potentials at the local chemical equilibriu m when F l (t,r,λ) = F r (t,r,λ) the flo w density is zero. Thus, expression (13) describes a flow generated by local chemical nonequilib riu m of charge particles. The well-known formu la for a tunnel current [8] follows fro m the ratio (13).
The length of chemical potentials of different electrons relaxation to local chemical equilibriu m is defined by the formula L rel (λ) = (ħτ(λ)/ m*) 1/2 . (15) Size of L rel is different for different materials. It is equal to about 10 nanometers for Si, 24 nanometers for GaAs and 72 nanometers for InSb. Where structures are larger than L rel it is possible to conclude that the electrons are in a state of local chemical equilibriu m.
Fro m the considered formulas, it follows that the current in electronic devices is created by two factors: deviations of electronic gas fro m the local chemical equilibriu m and at nonzero values of hydrodynamic velocity of electrons. According to formu la (1), the nonzero values v(t,r,λ) are caused by gradients of chemical potentials F(t,r,λ), that are deviations from chemical equilibriu m of electronic gas in various spatial points. Thus, it is possible to conclude that the electronic current is the consequence of the nonequilibriu m phenomena in electronic gas.
(16) Substitution of expression (16) to the formu la (14) results to the Ohm's law in the differential form and to well-known formulas for mobility and conductivity of electronic gas.
Let's assume that the potential difference changing in time under the harmonious law with cyclic frequency ω is applied to a spatially ho mogeneous sample. In this case, the gradient of chemical potential in the sample may be presented as ∇F = eE 0 cos(ωt).
(17) where E 0 -is a certain constant field.
(19) For mesoscopic structures, the momentu m relaxation time is about 10 -13 s and the factor ωτ is necessary to take into account when the frequencies are more than 1 THz. Co mparing expressions (13) and (16), we can see that increase of frequency results in decrease of conductivity (growth of resistance) of electronic gas.
In the development of exp ression (18), no assumptions about quantum dimensions of electronic gas were made. It means that formu las (18) and (19) are fair for quantum wires, which represent one-dimensional conductors.
(24) As shown in [4], the value F 0 is connected with the voltage drop in the device by the formula V ≈ -F 0 /e. Thus, to within boundary effects [4], the rat io (24) defines the current-voltage characteristic (CVC) of junction between quantum wires with different thickness.
In a one-dimensional appro ximation, when the total electron flow is constant for CVC o f junction, we shall receive the formu la V = J(ħ/e 2 )/2γdn + .
(25) Here J = eI = const (26) -is current density through the junction, d is the effective width of the junction. Value r j = (ħ/e 2 )/2γdn + (27) -is a specific resistance of the junction between two quantum wires with different cross-sections (dimension of r j -is Ω*cm 2 ).  Figure 4 shows a typical CVC of such a junction. Exponential gro wth of current density demonstrated in the direct (right) part of the CVC is caused by exponential growth in the nu mber of "left electrons" under conditions of voltage increasing and a corresponding decrease in potential barriers for electrons (see Figure 5). Similarly, under conditions of negative voltage, the potential barrier for electrons increases (see Figure 6), and the current does not practically flow through the junction.  (27), the expression occurs V=J 0 r j (cos(ωt)+ωτ 0 (α + /n + )sin(ωt))/(1+(ωτ 0 (α + /n + )) 2 ). (29) Non-stationary effects become essential at ω > ω 0 = 1/τ 0 = 4γkT/ħ. (30) An increase of frequency results in a decrease in junction resistance. Within the limit ω >> ω 0 the estimation is fair V ≈ J 0 r j sin(ωt))/ωτ 0 (α + /n + )).
(31) Vo ltage appears to be phase-shifted by π/2 in relation to current and effective specific resistance of the junction tends to zero with growth of frequency. At room temperature, the specified effects become essential if the frequencies exceed 1 THz. If temperature is decreased, the threshold determined by ratio (30) is decreased linearly. If temperature equals to 3 0 K, then frequency effects need to be considered at 10 GHz already.

Conclusions
In this paper, it is shown that the frequency of an external signal ω influences conductivity of quantum wires and devices based on them.
Conductivity of the conducting channel of a quantum wire is decreased linearly with the growth of ω.
For the junction between two conducting channels of quantum wires with different cross-sections, the inverse relationship is a representative one. Conductivity of such a junction is increased linearly with growth of the frequency, and resistance tends to zero if frequency tends to infinity. Similar frequency-dependence should be characteristic for a junction between the contact area and the conducting channel of a quantum wire.