Electron-phonon interaction in disordered conductors: Static and vibrating scattering potentials A. Sergeev and V. Mitin Department of ECE, Wayne State University, Detroit, Michigan 48202 Received 2 July 1999; revised manuscript received 23 September 1999 Employing the Keldysh diagram technique, we calculate the electron-phonon energy relaxation rate in a conductor with the vibrating and static -correlated random electron-scattering potentials. If the scattering potential is completely dragged by phonons, this model yields the Schmid’s result for the inelastic electron- scattering rate  e -ph -1 . At low temperatures the effective interaction decreases due to disorder, and  e -ph -1  T 4 l l is the electron mean-free path. In the presense of the static potential, quantum interference of numerous scattering processes drastically changes the effective electron-phonon interaction. In particular, at low tem- peratures the interaction increases, and  e -ph -1  T 2 / l . Along with an enhancement of the interaction, which is observed in disordered metallic films and semiconducting structures at low temperatures, the suggested model allows us to explain the strong sensitivity of the electron relaxation rate to the microscopic quality of a particular film. I. INTRODUCTION Electron-phonon scattering plays a key role in the descrip- tion of many phenomena, such as electron dephasing, heat removal from hot electrons, superconducting branch imbal- ance relaxation, etc. Although well characterized in clean bulk conductors, the current understanding in disordered and nanoscale systems is limited. Electron scattering from impurities and boundaries de- stroys the single-particle picture of electron-phonon interac- tion. Along with the process of ‘‘pure’’ electron-phonon scattering, which takes place in pure conductors, there is the other basic process, namely, the inelastic electron scattering from vibrating impurities, defects, and boundaries. Together with elastic electron scattering and pure electron-phonon scattering, this mechanism generates a wide variety of inter- ference processes. If the electron-scattering potential impu- rities, defects, and boundaries is completely dragged by phonons, the inelastic electron-impurity scattering may be excluded by a unitary transformation to a frame of reference, which moves with the phonon mode under consideration. Exploiting this transformation, Schmid 1 has found that in the presence of strong disorder ( q T l 1,q T is the wave vector of a thermal phonon the electron-phonon interaction becomes weaker, and the energy relaxation rate  e -ph -1 is of the order of ( q T l )  0 -1  T 4 l (  0 -1  T 3 is the relaxation rate in pure mate- rial. As emphasized in Refs. 2–4, this conclusion is consis- tent with the Pippard’s famous result for the ultrasonic at- tenuation coefficient. It was also demonstrated 2,5 that the correct calculations lead to Schmid’s result and the Pippard formula independent of the reference frame used. Detailed studies show that many experimental results may be understood in the frame of the available theory. 6–8 How- ever, a set of low-temperature observations is in strong dis- agreement with current theoretical understanding of the electron-phonon interaction in disordered conductors. En- hancement of the interaction due to disorder has been found in two-dimensional electron gas 2DEG in GaAs/Al-Ga-As heterojunctions with l =0.3-0.8  m below 0.5 K, 9 and in bulk Ti 1 -x Al x alloys with values of l ranged from 0.26 to 1.1 nm at T =3–10 K. 10 The T 2 dependence of the relaxation rate is widely observed in experiments. In some cases it may be associated with the contribution of transverse phonons in the pure limit q T l 1, predicted by the theory. Recent measurements 11 show that this dependence takes place even in the deep impure limit q T l 0.01, where T 4 dependence is expected. As the current theory is self-consistent, the search for rea- sons behind this discrepancy should turn to the model as- sumptions. Many-body corrections to phonon states were considered in Ref. 12. It has been found that the modification of the electron relaxation rate occurs only under a strong phonon damping. On the electronic side, the main model assumption of the current theory is that the scattering poten- tial is completely dragged by phonons. This assumption may be wrong for structures with heavy defects and tough bound- aries. That is why one can expect the relaxation rate to be modified in nonhomogeneous conductors and nanoscale structures. To study the effects of incomplete drag of scatterers by phonons we consider the model taking into account the vi- brating and static -correlated random-scattering potentials. One of the effects originating from elastic electron scattering is well understood now. The diffusion motion of electrons makes the interaction time longer, which enhances an inter- action. Renormalization of the ‘‘pure’’ electron-phonon ver- tex due to electron-impurity scattering has been calculated in Ref. 13. Here we study all electron-phonon interference pro- cesses, with taking into account the renormalization of all vertices as well as the modification of electron screening due to disorder. The outline of this paper is as follows. Starting with bare vertices of electron scattering from phonons, the static and vibrating potentials, we consider screening of these vertices and build effective vertices of the electron-phonon interac- tion in Sec. II. The energy relaxation rate in a disordered conductor is calculated in Sec. III. Discussion of our main results and comparison with experiment are presented in Sec. IV. PHYSICAL REVIEW B 1 MARCH 2000-I VOLUME 61, NUMBER 9 PRB 61 0163-1829/2000/619/60417/$15.00 6041 ©2000 The American Physical Society