Frictional drag between non-equilibrium electron gases

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Frictional Drag between Non-Equilibrium Electron Gases

X. F. Wang and I. C. da Cunha Lima

Instituto de Física, UERJ, 22.500-013 Rio de Janeiro, RJ, Brazil

Received on 23 April, 2001

Using the balance of forces in the linear regime, it is shown that the ratio between the induced voltage in an open circuit Coulomb drag experiment, and the induced current in closed circuit, equals the internal resistence of the circuit in which the induction occurs.

During the last decade much attention was given to a mechanism of current/voltage induction between two isolated systems (in the sense that momentum transfer via particles exchange is prohibited) made possible through Coulomb interaction between carriers at each of them. Such mechanism has been called Coulomb Drag (CD). The predictions made by Progrebinskii [1] and Price[2], have been confirmed later by Solomon et al [3] and, more recently, by a series of experiments involving quasi-two-dimensional systems in semiconductor double quantum wells [4,5]. Placing two 2D-dimensional electron systems close to each other, and separated by a potential barrier, when an electromotive force is applied in one of them, an induced voltage is observed in the other in an open circuit condition, or an induced current will appear, in the case of closed circuit. In linear regime, a transresistivity is defined as the ratio between the induced electric field and the driving current density. Improvements, like the inclusion of phonons drag, indirect Coulomb interaction by virtual phonons, usage of field theoretical techniques, etc., contributed to the fast development of this subject in recent years.

In general, experiments and calculations are performed in a open circuit set-up, since the existence of a current in the induced circuit makes the calculation much more complicate. In this work we used the Balance Equations formalism [6] to study, in the linear regime of a CD experiment, the relation between induction of voltage and induction of current in open and closed circuit situations. Lets v₁ and v₂ denote the drift velocities of the two systems, T₁ and T₂ their quasi-equilibrium temperatures, n₁ and n₂ their electron densities, and E₁ and E₂ the respective drifting and induced electric fields. The balance of force equations [7]:

where f_i(v) is the frictional force. A detailed calculation leads to an expression of the frictional forces in terms of intra- and inter-conductor relaxation time functions, t₁, t₂, t₁₂, and t₂₁, these last ones corresponding to the transresistive frictional forces f₁₂ and f₂₁. In the linear limit v₁and v₂ are small as compared with natural scales of the systems (like, for instance, the sound velocity), and T₁ = T₂ = T, where T is the temperature of the heat bath. In consequence the above equations can be enormously simplified:

In the case of open circuit ( = 0, 0),

and

since f₁₂ = - f₂₁.

For the case of closed circuit ( = 0, ¹ 0), however,

Next, we calculate the ratio between the induced field in an open circuit set-up and the induced current density in the case of closed circuit. Generally t₂₁ >> t₂,t₁, so, from Eq.(8),

If we apply the same electric field in two cases, i.e. , we have from Eqs.(4) and (7). Comparing this result with Eq.(5), we obtain,

Eq.(10) express the conclusion that the ratio between the induced voltage in open circuit and the induced current in closed circuit equals the resistance of system 2, recovering Ohm's law in the Coulomb drag experiment. It means that in the closed circuit set-up an electromotive force appears in circuit 2, and the resistance of system 2 works as an internal resistance. This result is a mere consequence of the conservation of energy and the balance of forces. Therefore, it is, in fact, not surprising that the Ohm's law applies, since the electromotive force equals the induced voltage in the open circuit set-up.

Most works on Coulomb drag are made in the linear regime, in such a way that the temperature is assumed to be the same for the two subsystems, and equal to that of the heat bath. It is well-known, however, that hot electron effects occur in semiconductor microstructures,[8] where electrons can be easily heated or cooled by very low electric voltage or photon excitations. In non-equilibrium situations, as it is the case in a Coulomb drag experiment, there is no reason to assume, from the very beginning, the two system at the same temperature of the heat bath. Following the suggestion of L. Zheng and A. H. MacDonald[9], we adopt the renormalization of the Coulombic interaction between electron gases 1 and 2, as shown in Fig. 1. For small drift velocities, the drag resistivity becomes[10]:

where (q,w,T) is the (real)immaginary part of the polarization, U₁₂(q,w) is the renormalized interaction potential between gases 1 and 2, and e_T(q,w) is the total dielectric function. In our calculation the real (virtual) phonon contribution comes from the imaginary (real) part of of the electro-phonon coupling matrix, and the contribution of plasmons and electron-phonon collective modes come from the zeroes of the real part of dielectric function. The contributions of the longitudinal acoustic (LA) and the transversal acoustic (TA) phonons via deformation potential and piezoelectric interaction, are fully included. The mean free times, t_^LOph and t_^TOph, are used as fitting parameters for both the LA and the TA branches. The parameters used for GaAs are k_s = 12.9, d₁ = 5.31g/cm³, X = 14eV, m^* = 0.067, v_sl = 5.29×10³m/s, e₁₄ = 1.41×10¹⁹V/m, and v_st = 2.48×10³ for the dielectric constant, density, deformation potential, electron effective mass, longitudinal sound velocity, piezoelectric constant and transversal sound velocity, respectively. The numerical calculation is performed for a structure of two 200Å wide quantum wells, with a 300Å barrier between them.

Fig. 2 shows the drag resistivity R_d/(T₁T₂) in equilibrium cases (solid curves) as a function of the ratio n₁/n₂ of the electron densities at different drag temperatures, T₂ = 0.04, 0.09, 0.16, 0.24 , and 0.44T_F in comparison with the experimental results. The mean free paths of the LA and TA phonons is assumed to be l_ph = 0.01cm. We found that the acoustic phonon enhances the drag resistivity at large n₁/n₂. It can be expected that the inclusion of local field effects will provide good fittings between theoretical and experimental results,[11] except at temperature T₂ = 0.44T_F. The results with bare Coulomb interaction is shown in equilibrium cases by dotted curves, and in non-equilibrium cases with T₁ = 0.9T₂ by dashed curves.

In conclusion, we have studied the momentum transfer rate between two nearby separated electron gases, which are coupled via the Coulomb interaction and phonon mediated interaction. We focused on the non-equilibrium configuration, and a theory was developed to describe the frictional drag force felt by one electron gas as a result of the relative drift of the other, taking into account the possibility of difference on the subsystems temperatures. It was found that a cooler (hotter) driving electron gas greatly enhances (decreases) the frictional force caused by plasmons. This behavior results from the fact the plasmons locate in a region of the w-k space where the correlation function shows a nonlinear frequency dependence, and they modifies little the acoustic phonons force because the latter are important mainly in the region of the energy-momentum space where the correlation function is linear with the frequency.

Acknowledgments

This work was partially supported by FAPERJ and CNPq in Brazil.

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Publication Dates

• Publication in this collection
  26 Nov 2002
• Date of issue
  June 2002

History

• Received
  23 Apr 2001