On momentum conservation and thermionic emission cooling

The question of whether relaxing momentum conservation can increase the performance of thermionic cooling device is examined. Both homojunctions and heterojunctions are considered. It is shown that for many cases, a non-conserved lateral momentum model overestimates the current. For the case of heterojunctions with a much heavier effective mass in the barrier and with a low barrier height, however, non-conservation of lateral momentum may increase the current. These results may be simply understood from the general principle that the current is limited by the location, well or barrier, with the smallest number of conducting channels. These results also show that within thermionic emission framework, the possibilities of increasing thermionic cooling by relaxing momentum conservation are limited. More generally, however, when the connection to the source is weak or in the presence of scattering, the situation may be different. Issues that deserve further study are identified.


I. Introduction
Thermionic (TI) cooling is a method of refrigeration with the potential for high cooling power and efficiency. [1][2][3] As depicted in Fig. 1, it is based on thermionic emission over a potential barrier. When carriers with high energy (hot carriers) are injected over the barrier, the carrier distribution in the emitter region is out of equilibrium. To restore equilibrium, cold carriers move up and populate higher energy states by absorbing heat from the lattice. The result is that cooling occurs in the region before the emitter-barrier junction. 4,5 The purpose of this paper is to address the question of whether relaxing momentum conservation at the junction can increase the performance of TI cooling devices, as has been proposed. 6,7 The main differences between TI cooling and the more conventional, thermoelectric (TE) cooling are the carrier transport mechanism and the operating regime. 8 In TI cooling, carrier transport from the well to the barrier is treated as ballistic, as is transport across the barrier (region II in Fig. 1) because the barrier thickness, d, is assumed to be shorter than the carrier mean-free-path, λ. The result is that no joule heating occurs in the channel. 4 In TE cooling, however, transport is assumed to be diffusive, and joule heating is considered in the heat balance. 9 In addition, while TE devices operate in the linear regime with small voltage and temperature differences, TI devices operate in the non-linear regime with high drain bias to eliminate the carrier injection from the drain and maximize the heat current injected from the source. 10 Previous theoretical studies have compared the cooling performances of TI and TE devices. 8,11,12 It has been shown that for the same material, TE cooling is better because it gives higher maximum temperature difference, max T ∆ , than that obtained from TI cooling. 8 This result has been explained in terms of the "material parameter", B, where TE . 11 These studies used two different sets of equations to model TI and TE devices. For TI devices, Richardson's equation 11 or its generalized version with Fermi-Dirac statistics 8 were used, and the Boltzmann transport equation (BTE) 13 was used to model TE devices. Using the Landauer approach, 14,15 however, it is possible to describe both TI and TE devices. In the diffusive limit the Landauer formalism describes TE devices and in the ballistic limit, TI devices. 16  > , which is consistent with the result by Humphrey et al. 12 The physical explanation is that the short d of TI device gives a large heat back-flow that limits max T ∆ .
The top-of-the-barrier model that we describe in the next section is closely related to the Landauer approach.
Although previous theoretical studies show that the cooling performance of TI devices is no better than that of TE devices, it has been suggested that non-conservation of lateral momentum may increase the number of electrons participating in the thermionic emission process and significantly improve the TI cooling performance. 6,7,10 The results to be presented in this paper, however, suggest that significant performance benefits are unlikely to be achieved.
More generally, the results shed light on thermionic emission over barriers, a problem that is relevant to TI cooling but also important in other electronic devices such as Schottky barriers and metal-oxide-semiconductor field-effect transistors (MOSFETs). 17 Our goals in this paper are to study the physics of thermionic emission across homo-and heterojunctions, to examine the concept of non-conserved lateral momentum, and to explore the interesting possibility of increasing TI device cooling performance by relaxing momentum conservation at the junction. The paper is organized as follows. In Sec. II, we compare two approaches to describe carrier injection over the barrier, a top-of-the-barrier model and a thermionic emission model. We also discuss the need for a general model for heterojunctions and introduce the concept of non-conserved lateral momentum. In Sec. III, the general theory of thermionic emission across heterojunctions is reviewed, and results are shown for homo-and heterojunctions. In Sec. IV, a simple physical interpretation is provided to explain the results in Sec. III. In Sec. V, we discuss the underlying physics of injection over a barrier, discuss the validity of the non-conservation lateral momentum model, and identify issues that deserve further attention. Conclusions follow in Sec. VI.

II. Top-of-the-barrier and thermionic emission models
In this section, we compare two different approaches that describe carrier injection over a barrier, the top-of-the-barrier model 18 and the traditional thermionic emission model. 17 Both models assume ballistic transport. In the top-of-the-barrier model, the E-k relation is considered on the barrier as shown in Fig. 2(a), and z k + states (z is the transport direction) are filled according to the source Fermi level, F E . In the thermionic emission model, 17 we focus on the source as shown in Fig. 2 where A is the cross-sectional area of the device, q is the unit charge, B k is the Boltzmann constant, T is the temperature, j F is the Fermi-Dirac integral of order j, 19,20 and , which is the reduced Fermi level in the barrier region.

III. Thermionic emission across heterojunctions
We begin with a review of the general theory of thermionic emission across heterojunctions as presented by Wu and Yang. 21 It is assumed that * m changes abruptly at the junction interface. [21][22][23] Wu and Yang assume that the total energy, E, and the lateral momentum, k ⊥ , are conserved b k k > are injected from the source without considering the occupation of states on the barrier.
Using these three approaches, we examine three cases: 1) homojunction with barrier, 2) heterojunction with no barrier, and 3) heterojunction with barrier. For heterojunctions, we consider two cases: i) * There is, therefore, room for improving the emission current, and the maximum possible current is given by the NCLM model as shown in Fig. 7(b). The TOB model in Fig. 7(c) overestimates the current because it is larger than the maximum that can be supplied by the source, which is given by the NCLM model. In this case, it appears that the proposed increase in TI cooling by relaxing momentum conservation 6,7 could be achieved.
The possible improvement due to non-conservation of lateral momentum when *

IV. Conductance and minimum number of modes
The results in the previous section can be understood with a simple general rule. Given a number of conducting channels (or modes) in the source ( ( ) 1 M E ) and the barrier ( smaller one determines the total conductance. 27 As an example, we consider a 3D heterojunction where the numbers of modes increase linearly with E, and the slope is proportional to * m as 16 k T φ as shown in Fig. 9(b) and 2) as shown in Fig. 9(c). In Fig. 9

V. Discussion
In this section, we examine the underlying physics of the TOB and NCLM models and explore how the maximum emission current can be achieved. From now on, we mainly consider It has been suggested that in a non-planar heterostructure where the translational invariance is broken, the lateral momentum is not conserved, and the emission current may increase. 6,7,29,30 Monte Carlo simulations including inelastic scattering processes 29,30 have shown that adjusting the depth and the width of the zigzagged interface structure enhances the emission by a factor of up to 2. Note, however, that the lateral momentum is conserved at each local interface, so we may interpret this enhancement due to the increased effective area. The current is not directly proportional to the total interface area 31 because the carriers may re-enter the emitter in the zigzagged structure. 29,30 To better understand the physics of carrier emission and explore possibilities to increase the emission current, we need to study the effect of carrier ballisticity on the emission enhancement and optimize the non-planar structures 29,30,32 to maximize the enhancement while not decreasing the carrier mobility. 6 It has been shown that non-conserved lateral momentum is essential to interpret the experimental results of Ballistic Electron Emission Microscopy (BEEM) for non-epitaxial metalsemiconductor interfaces. [33][34][35] In BEEM measurements, carriers with small lateral momentum are predominantly injected. 33 The conserved lateral momentum model does not provide physical interpretation for the experiment where valleys with zero lateral momentum are not preferentially populated as would be expected if lateral momentum were conserved. 33 The observed significant current for the valleys with non-zero lateral momentum indicates that additional lateral momentum is provided by scattering at the non-epitaxial interface. 35 The BEEM measurement results and the theories of non-conservation of lateral momentum used to explain them have motivated the idea that non-conservation of lateral momentum might similarly enhance the emission current and cooling performance of TI devices. 6 The problems are, however, quite different. The critical difference between BEEM experiments and TI devices is that the carrier reservoirs are different. In TI devices, the source should be designed to act as closely as possible to an ideal Landauer reservoir, 36  help in such cases, the maximum current can never exceed the ballistic limit, which is determined by the minimum number of modes as summarized in Table 1.
Finally, we should mention that there are a number of other issues that deserve consideration. We have assumed a ballistic (thermionic emission) model in which all of the scattering occurs in the Landauer contacts. In practice, scattering will occur throughout the structure. In the well region before the barrier, scattering may reduce the current below the thermionic emission value. A similar problem, transport in Schottky barriers, was considered by Bethe 37 and by Berz. 38  the Joule losses, and we do not believe that cooling powers above the ballistic limit discussed in previous sections could be achieved. Fischetti et al. also discussed "downstream" effects 40scattering in the barrier itself and in the well beyond the barrier. Although they are beyond the scope of this paper, more quantitative studies of the effect of scattering on TI devices will be essential to understand the physics and performance limits of such devices.

VI. Conclusions
In this paper, we studied the physics of thermionic emission across homo-and heterojunctions to explore the possibilities to increase the emission current and the cooling performance of TI cooling devices. We showed that the TOB model 18

Appendix A: Mathematical formulations of TE and TI devices
According to the Landauer formalism, 14  In the linear regime where TE devices operate, transport coefficients such as conductance, G, and Seebeck coefficient, S, can be obtained in terms of ( ) 16 The efficiency of TE devices is related to the figure of merit, ZT = 2 S GT K , 9 where K is the thermal conductance, which is the sum of the electronic contribution and the lattice thermal conductance, l K . The relation between ZT and max T ∆ is ZT ≡ max 2 T T ∆ . 8 Then ZT can be calculated using the expressions for S, G, and K from the Landauer formula. 16 For a 3D single parabolic band, for example, we find where Γ is the Gamma function. We can also define the material parameter for TE device, where TI B is the material parameter for TI devices. 11,12 By comparing eqs. (A1) and (A2), we , which is consistent with the previously reported result. 12

Appendix B: Mathematics of the TOB and Wu-Yang Models
In the TOB model, the ballistic I and q I for 3D carriers are calculated as Expressions for 1D and 2D carriers can be obtained in a similar way. We split the CLM model 21 into two cases, * and we note that the results of eq. (B2) are the same as those from eq. (B1). In Fig. 5(a), the hyperbola (A) that maps onto the x y k k − plane with z k = 0 on the barrier is expressed as ( ) , we note that eq. (B4) approaches to eq. (B1), and the model becomes equivalent to the TOB model as shown in Fig. 6(a). As ( ) , however, eq. (B4) is different from the TOB model as shown in Fig. 7(a). In Figs. 6(a) and 7(a), the ellipsoid (A) that maps onto the x y k k − plane with z k = 0 line on the barrier is expressed as eq. (B3). In Fig. 7(a), the x y k k − plane with z k = 0 in the source maps onto a hyperbola (B') on the barrier, which is given as   The TOB model represents an upper limit to the possible current while the NCLM model represents the maximum current that could be supplied by the source, and the minimum of the two determines the total current.