Band-Engineered Local Cooling in Nanoscale Junctions

The stability and performance of nanoscale junctions are closely related to the local effective temperature. The local effective temperature is mainly caused by the competition between heating and cooling processes in inelastic electron-phonon scat- tering. Local cooling occurs when the rate of energy in cooling exceeds that in heating. Previous research has been done using either specific potential configuration or an adatom to achieve local cooling. We propose an engineer-able local-cooling mechanism in asymmetric two-terminal tunneling junctions, in which one electrode is made of metal, whereas the other is made of a selectable bad-metal, such as heavily-doped polysilicon. The width of energy window of the selectable material, defined as the width covering all possible energy states counting from the conduction band minimum, can be engineered through doping. Interestingly, we have shown that substantial local cooling can be achieved at room temperature when the width of energy window of the low-density electrode is comparable to the energy of the phonon. The unusual local cooling is caused by the narrowed width of energy window, which obstructs the inelastic scattering for heating.

framework of density functional theory in scattering approaches, where and ; effective single-particle wavefunctions describes electrons incident from the left and right electrodes; ; and is the annihilation operators of electrons incident from the left (right) reservoir, satisfying the anti-commutation relations, where or .
The expectation value of the product of electron creation and annihilation operator at thermal equilibrium is given by, where the statistics of electrons coming from the left (right) electrodes are determined by the equilibrium Fermi-Dirac distribution function in the left (right) reservoir. The Lippmann-Schwinger equation allows the wave functions of the entire system to satisfy the same continuum normalization condition, To consider the electron-vibration interactions, we start with a more general Hamiltonian including ions and the interaction between electrons and ions, where is the electronic part of the Hamiltonian, where is the electron mass, is the momentum of the -th electron, and is the position of the -th electron.
is the ionic part of Hamiltonian, where is the mass of the -th ion, is the momentum of -th ion, is the position of the -th ion, and is the interaction between the -th and -th ions. describes the interaction between the -th electron and -th ions, To consider the vibronic coupling, we start by considering small ionic oscillations, where is a small deviation of position away from the equilibrium position for the -th ion represented in the Cartesian coordinate system. Therefore, where the oscillatory part of ions is (11) Ion oscillations can be mapped into a set of independent simple harmonic oscillators via normal coordinates , i.e., . The oscillatory part of ions is diagonalized and has the form of (12) where is the frequency of the -th normal mode. One can introduce a canonical transformation, and , which transforms to (13) where one has . Using and , the oscillatory part Hamiltonian can be second quantized by phonon creation and annihilation operators and become a set of independent simple harmonic oscillators, Next, we expand the electron-vibration interactions in terms of lowest order in small oscillation, , where we consider the interaction between electron and ion at nonlocal pseudopotential level, where is the derivative with respect to the position of the -th ion in ,and directions. By using and placing back into , we obtain (16) where we place back the mass of the -th ions . The orthonormal conditions for the canonical transformation between normal coordinates and Cartesian coordinates: . The vibronic coupling can be second quantized by applying the field operator (17) Finally,the many-body Hamiltonian of the system under consideration becomes , is the electronic part of the Hamiltonian under adiabatic approximations, and is the ionic part of the Hamiltonian, which can be casted into a set of independent simple harmonic oscillators via canonical transformation. is the part of the Hamiltonian for electron-vibration interactions that has the following form: The coupling constant between electrons and the vibration of the -th atom in ( , , ) component can be calculated as (19) where is the nonlocal pseudopotential, which represents the interaction between electrons and the -th ion.
The normal modes of the single Au atom connected to electrodes ( ) and the canonical transformation matrix ( ) are calculated with Gaussian09 code by freezing the positions of atoms located in the electrode region. Three normal modes (two degenerate transverse modes with energies meV; one longitudinal mode with energy meV) are associated to the degrees of freedom of the single atom, as shown in Figure S1. Feynman diagrams of the first-order electron-vibration scattering mechanisms. Diagrams (1) to (4) shows the cooling processes, and diagrams (5) to (8) show the heating processes.
The rate of energy absorbed (emitted) by the anchored nano-structures due to incident electrons from the electrode and scattered to the electrode via a vibrational mode is denoted by . The total thermal power generated in the junction is the sum of contribution from eight first-order scattering processes (four heating and four cooling processes) and from all the vibrational modes shown in Figure S2, where the power for each process is estimated using the Fermi golden rule for all modes: where the statistic average of state includes electron states of the left and right electrodes and local phonon states in the scattering region. Considering the eight scattering processes, one obtains (22) where for heating processes and for cooling processes which excite (relax) the normal mode vibration.
is for cooling and is for heating processes.
When the system comes to thermal equilibrium, the rate of thermal energy, which is generated by the heating processes, balances the energy rate that is absorbed by the local processes. Thus, the temperature in the center scattering ( ) can be obtained by solving . The current can be calculated from the wave function and can be further reduced to Landauer's formula (23) where is the transmission function, is the Seebeck coefficient, and is the induced temperature difference. Similarly, the heat current can be obtained via (24) where the electric thermal conductance. Following from our previous work in Ref [1], after expanding the Seebeck coefficient and electric thermal conductance in the lowest order in the temperature, we can obtain Now we can find the ratio between the electric current and the heat current, namely (28) where L is the Lorentz number .