Nonequilibrium electron and phonon transport and ... - Semantic Scholar
Thermal Challenges in Next Generation Electronic Systems, Joshi & Garimella (eds) © 2002 Millpress, Rotterdam, ISBN 90-77017-03-8
Nonequilibrium electron and phonon transport and energy conversion in heterostructures Taofang Zeng Mechanical Engineering Department, North Carolina State University, Raleigh, NC, USA
Gang Chen Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA, USA
Keywords: nonequilibrium, transport, heterostructures, energy conversion
ABSTRACT: We establish a unified model dealing with the transport of electron and phonon in double heterojunction structures with the coexistence of three nonequilibrium processes: (1) nonequilibrium among electrons, (2) nonequilibrium among phonons, and (3) nonequilibrium between electrons and phonons. Using this model, we investigate the energy conversion efficiency based on concurrent thermoelectric and thermionic effects on electrons and size effects on electrons and phonons. It is found that heterostructures can have an equivalent figure of merit higher than the corresponding bulk materials. 1 INTRODUCTION Size effects and large deviation from equilibrium often characterize transport of electrons and phonons in nanostructures. In very thin films with a thickness less than one electron mean free path (MFP), for example, electrons going through the film experience little or no scattering and the electric current is controlled by the barrier heights of the heterojunction, as described by the Richardson’s thermionic emission theory [1]. Cooling and power generation based on such thermionic processes in heterostructures and superlattices have been studied [2-5]. On the other hand, scattering dominates the transport process of electrons flowing in thick films and bulk materials. Electrons and phonons are at quasi-equilibrium states and their transport is diffusive, as described by the thermoelectric theory [6]. Enhanced thermoelectric energy conversion along thin films based on quantum effects on electron energy states but diffusive quasiequilibrium electron transport has been predicted and confirmed [7,8]. In addition, phonon size effects in thin films and superlattices are also under consideration as a means to enhance thermoelectric and thermionic energy conversion efficiency [9-11]. Some studies on nonequilibrium transport of electrons and phonons in thin films have been reported based on hot carrier assumption [12]. In these studies, however, the subsystems of electrons and phonons individually are still at quasi-equilibrium. In the ballistic limit, the quasi-equilibrium assumption for electrons as a subsystem or for phonons as a subsystem is clearly not valid. Although electronic thermoelectric and thermionic effects and phonon size effects in heterostructures all have been evaluated individually as potential mechanisms for improving the efficiency of solid-state coolers and power generators, a general theory that could encompass the effects of nonequilibrium among electrons, among phonons, and between electrons and phonons, has been missing. In this study we fill in this gap and establish a unified model including all these nonequilibrium processes such that the relative importance of size effects on electrons and phonons, and thermionic and thermoelectric energy conversion processes can be evaluated. Our model treats the nonequilibrium transport of nonequilibrium electron and phonons subsystems and their interactions. It is based on the Boltzmann transport equations (BTEs) for nonequilibrium electrons and phonons, and their nonequilibrium coupling. The BTE is appropriate under the particle description of electrons and phonons, which means that the film thickness must be larger than the thermal de Broglie wavelength of electrons and phonons. Approximate solutions of BTE are obtained, leading to the distribution of the electron and phonon temperatures and Fermi level inside heterostructures. We emphasize that the temperature and Fermi level cannot be understood in the conventional sense of an equilibrium quan103
tity since we are dealing with nonequilibrium processes. Rather, they are considered as a measure of the local electron and phonon energy for temperature and the local particle number for the Fermi level. The coexistence of discontinuities in the electron and phonon temperatures, as well as the Fermi levels at the interfaces are demonstrated for the first time, as a consequence of nonequilibrium transport when the film thickness is much smaller than the electron and phonon mean free path (MFP). The coexistence of thermoelectric and thermionic effects may increase the power factor. Phonon size effects can enhance the equivalent figure of merit of heterostructures. 2 PHYSICAL MODEL Figure 1 shows the objective in this study. A semiconducting thin film (II) is sandwiched within another semiconductor or metal, which are used as substrates I and III maintained at uniform temperature Tc and Th respectively. The two substrates act as two reservoirs. This is a very simplified schematic, whereas it allows us to focus on the main physical insights on nonequilibrium propensities inside the film and at the interfaces. We further limit our attention to the films with a thickness larger than the tunneling length and the coherent length or matter wavelength, so that we can neglect the quantum effects. In this regime, the BTE is applicable for describing nonequilibrium transport of nonequilibrium electrons and phonons. We begin with electrons: ν x ∂f / ∂x + Fx ∂f / ∂p x = (∂f / ∂t ) c , where f is the distribution function, νx the electron velocity component in the x direction, Fx the local electric force, and px the electron momentum component in the x direction. The right hand side represents the scattering. In this work, we will use the relaxation time approximation, (df/dt)c=-(f-f0)/ι, where f0 is the Fermi-Dirac distribution for electrons, and ι the relaxation time of electrons. This relaxation time approximation gives us a way to calculate the effective temperature and Fermi level, which are embedded in f0. Since the major concern here is the deviation far from equilibrium in space, we assume that the deviation from equilibrium in the k-space is small such that ∂f / ∂p x ≈ ∂f 0 / ∂p x . The assumption can be justified by assuming quick adaptation of electrons to the new electronic structure of the thin film [13]. In the BTE, Fx = -qεε, where ε is the electric field, dž = dE c / dx for electrons. By neglecting the space charge and assuming a Boltzmann distribution at the interface for the electrons, which can be rationalized, for example, with doping inside the film and a reasonable barrier height, we can obtain the following solution to the BTE:
f + (ξ, µ ) = f + (0, µ ) • e − ξ / µ + ∫0ξS 0 (ξ1 )e
( )
− ( ξ − ξ1 ) / µ
dξ1 , (+ means positive direction, or 0< µ 1, τh-τ>>1; ξ>>1, and Nonequilibrium electron and phonon transport and energy conversion in heterostructures
105
ξh-ξ>>1. Under these assumptions the influence of the boundaries becomes negligible and Eqs. (2)-(4) can be simplified into the ordinary diffusion form, J ( x ) = e / C • H ( x ) , Q e ( x ) = 1 / C • R ( x ) , Q ph ( x ) = −1 / 3 • C P ν ph dTph / dx .
(6)
The boundary conditions are derived from the Deissler’s jump boundary conditions [16] as follows: J 1 = 1 / 2H x (0) , J 2 = −1 / 2H x (ξ h ) , P1 = 1 / 2R x (0) , P2 = −1 / 2R x (ξ h ) , I1 − I(0) = 1 / 2Q ph (0) , I 2 − I(τ h ) = −1 / 2Q ph (τ h ) ,
(7)
0) (İ + ∇E f / q ) + L(xx1) (− ∇Te ) and R x ( x ) = L(xx1) (İ + ∇E f / q ) + L(xx2) (− ∇Te ) are standard where H x = L(xx thermoelectric expressions, except in the current case, the electron temperature and its gradient are unknown and must be solved with the boundary terms. Based on the parabolic band approximation and the Fermi-Dirac statistics, the transport coefficients can be expressed as ∞
(
)
2
α) = q 2 τ(8m * )1 / 2 (K B Te ) 3 / 2+ α / 3π 2 h 3 • ∫ x 3 / 2 ( x − ς) α e x −ς / e x −ς + 1 dx , where ς = E f / K B Te is the L(xx 0
dimensionless Fermi level. 3 RESULTS AND DISCUSSION We now focusing on the open-circuit condition, J(x) = 0, and derive the corresponding Seebeck coefficient. By introducing the approximation of (Th-Tc)/Tc
Nonequilibrium electron and phonon transport and energy conversion in heterostructures Taofang Zeng Mechanical Engineering Department, North Carolina State University, Raleigh, NC, USA
Gang Chen Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA, USA
Keywords: nonequilibrium, transport, heterostructures, energy conversion
ABSTRACT: We establish a unified model dealing with the transport of electron and phonon in double heterojunction structures with the coexistence of three nonequilibrium processes: (1) nonequilibrium among electrons, (2) nonequilibrium among phonons, and (3) nonequilibrium between electrons and phonons. Using this model, we investigate the energy conversion efficiency based on concurrent thermoelectric and thermionic effects on electrons and size effects on electrons and phonons. It is found that heterostructures can have an equivalent figure of merit higher than the corresponding bulk materials. 1 INTRODUCTION Size effects and large deviation from equilibrium often characterize transport of electrons and phonons in nanostructures. In very thin films with a thickness less than one electron mean free path (MFP), for example, electrons going through the film experience little or no scattering and the electric current is controlled by the barrier heights of the heterojunction, as described by the Richardson’s thermionic emission theory [1]. Cooling and power generation based on such thermionic processes in heterostructures and superlattices have been studied [2-5]. On the other hand, scattering dominates the transport process of electrons flowing in thick films and bulk materials. Electrons and phonons are at quasi-equilibrium states and their transport is diffusive, as described by the thermoelectric theory [6]. Enhanced thermoelectric energy conversion along thin films based on quantum effects on electron energy states but diffusive quasiequilibrium electron transport has been predicted and confirmed [7,8]. In addition, phonon size effects in thin films and superlattices are also under consideration as a means to enhance thermoelectric and thermionic energy conversion efficiency [9-11]. Some studies on nonequilibrium transport of electrons and phonons in thin films have been reported based on hot carrier assumption [12]. In these studies, however, the subsystems of electrons and phonons individually are still at quasi-equilibrium. In the ballistic limit, the quasi-equilibrium assumption for electrons as a subsystem or for phonons as a subsystem is clearly not valid. Although electronic thermoelectric and thermionic effects and phonon size effects in heterostructures all have been evaluated individually as potential mechanisms for improving the efficiency of solid-state coolers and power generators, a general theory that could encompass the effects of nonequilibrium among electrons, among phonons, and between electrons and phonons, has been missing. In this study we fill in this gap and establish a unified model including all these nonequilibrium processes such that the relative importance of size effects on electrons and phonons, and thermionic and thermoelectric energy conversion processes can be evaluated. Our model treats the nonequilibrium transport of nonequilibrium electron and phonons subsystems and their interactions. It is based on the Boltzmann transport equations (BTEs) for nonequilibrium electrons and phonons, and their nonequilibrium coupling. The BTE is appropriate under the particle description of electrons and phonons, which means that the film thickness must be larger than the thermal de Broglie wavelength of electrons and phonons. Approximate solutions of BTE are obtained, leading to the distribution of the electron and phonon temperatures and Fermi level inside heterostructures. We emphasize that the temperature and Fermi level cannot be understood in the conventional sense of an equilibrium quan103
tity since we are dealing with nonequilibrium processes. Rather, they are considered as a measure of the local electron and phonon energy for temperature and the local particle number for the Fermi level. The coexistence of discontinuities in the electron and phonon temperatures, as well as the Fermi levels at the interfaces are demonstrated for the first time, as a consequence of nonequilibrium transport when the film thickness is much smaller than the electron and phonon mean free path (MFP). The coexistence of thermoelectric and thermionic effects may increase the power factor. Phonon size effects can enhance the equivalent figure of merit of heterostructures. 2 PHYSICAL MODEL Figure 1 shows the objective in this study. A semiconducting thin film (II) is sandwiched within another semiconductor or metal, which are used as substrates I and III maintained at uniform temperature Tc and Th respectively. The two substrates act as two reservoirs. This is a very simplified schematic, whereas it allows us to focus on the main physical insights on nonequilibrium propensities inside the film and at the interfaces. We further limit our attention to the films with a thickness larger than the tunneling length and the coherent length or matter wavelength, so that we can neglect the quantum effects. In this regime, the BTE is applicable for describing nonequilibrium transport of nonequilibrium electrons and phonons. We begin with electrons: ν x ∂f / ∂x + Fx ∂f / ∂p x = (∂f / ∂t ) c , where f is the distribution function, νx the electron velocity component in the x direction, Fx the local electric force, and px the electron momentum component in the x direction. The right hand side represents the scattering. In this work, we will use the relaxation time approximation, (df/dt)c=-(f-f0)/ι, where f0 is the Fermi-Dirac distribution for electrons, and ι the relaxation time of electrons. This relaxation time approximation gives us a way to calculate the effective temperature and Fermi level, which are embedded in f0. Since the major concern here is the deviation far from equilibrium in space, we assume that the deviation from equilibrium in the k-space is small such that ∂f / ∂p x ≈ ∂f 0 / ∂p x . The assumption can be justified by assuming quick adaptation of electrons to the new electronic structure of the thin film [13]. In the BTE, Fx = -qεε, where ε is the electric field, dž = dE c / dx for electrons. By neglecting the space charge and assuming a Boltzmann distribution at the interface for the electrons, which can be rationalized, for example, with doping inside the film and a reasonable barrier height, we can obtain the following solution to the BTE:
f + (ξ, µ ) = f + (0, µ ) • e − ξ / µ + ∫0ξS 0 (ξ1 )e
( )
− ( ξ − ξ1 ) / µ
dξ1 , (+ means positive direction, or 0< µ 1, τh-τ>>1; ξ>>1, and Nonequilibrium electron and phonon transport and energy conversion in heterostructures
105
ξh-ξ>>1. Under these assumptions the influence of the boundaries becomes negligible and Eqs. (2)-(4) can be simplified into the ordinary diffusion form, J ( x ) = e / C • H ( x ) , Q e ( x ) = 1 / C • R ( x ) , Q ph ( x ) = −1 / 3 • C P ν ph dTph / dx .
(6)
The boundary conditions are derived from the Deissler’s jump boundary conditions [16] as follows: J 1 = 1 / 2H x (0) , J 2 = −1 / 2H x (ξ h ) , P1 = 1 / 2R x (0) , P2 = −1 / 2R x (ξ h ) , I1 − I(0) = 1 / 2Q ph (0) , I 2 − I(τ h ) = −1 / 2Q ph (τ h ) ,
(7)
0) (İ + ∇E f / q ) + L(xx1) (− ∇Te ) and R x ( x ) = L(xx1) (İ + ∇E f / q ) + L(xx2) (− ∇Te ) are standard where H x = L(xx thermoelectric expressions, except in the current case, the electron temperature and its gradient are unknown and must be solved with the boundary terms. Based on the parabolic band approximation and the Fermi-Dirac statistics, the transport coefficients can be expressed as ∞
(
)
2
α) = q 2 τ(8m * )1 / 2 (K B Te ) 3 / 2+ α / 3π 2 h 3 • ∫ x 3 / 2 ( x − ς) α e x −ς / e x −ς + 1 dx , where ς = E f / K B Te is the L(xx 0
dimensionless Fermi level. 3 RESULTS AND DISCUSSION We now focusing on the open-circuit condition, J(x) = 0, and derive the corresponding Seebeck coefficient. By introducing the approximation of (Th-Tc)/Tc
