Bias-dependent Peltier Coefficient in Bipolar Devices

Temperature stabilization is important in many microelectronic devices due to thermal constraints on device operation and lifetime. The work described here is an investigation of thermoelectric phenomena in bipolar devices, speci£cally the p-n diode. Current injection can modify the Peltier coef£cient at interfaces; this can give rise to thermoelectric cooling or heating depending on device parameters. The bias-dependent Peltier coef£cient is modeled using self-consistent drift-diffusion, and implications for device design are examined. The different regimes of bias for which cooling is achieved are described, as well as the effects of device length, doping, and heterojunction band offset. Extensions of the model are given for applications such as the internal cooling of semiconductor laser diodes. INTRODUCTION Excessive heat can cause deterioration in performance and lifetime for many microelectronic devices, necessitating thermal management in device packaging. This is often accomplished through the use of external thermoelectric (Peltier) coolers. Recently an approach more amenable to integration has been developed which takes advantage of thermionic and thermoelectric effects in III-V semiconductor heterostructures to obtain cooling power densities of several hundred W/cm2 in micron-thick £lms (Shakouri et al., 1997; LaBounty et al., 2000). While the thermoelectric properties of traditional Peltier cooler are well understood, the theory describing the thermoelectric properties of ∗Address all correspondence to this author. the microelectronic devices themselves during operation is less worked out. This is the subject of the current work, speci£cally regarding the p-n diode. PELTIER HEAT EXCHANGE The Peltier effect describes heat exchange which takes place at the junction of two different materials when electrical current ¤ows between them. It is caused by the fact that the average energy that an electron transports can vary from material to material; when crossing between two such materials, a carrier compensates for this energy difference by exchanging heat energy with the surrounding atoms. A material’s thermoelectric Peltier coef£cient Π is related to the average energy transported by its electrical carriers through Etr = Π ∗ q; the amount of heat exchanged for a given current I across a junction is equal to ∆Π∗ I. For a semiconductor, the Peltier coef£cient is given by Π= 1 q ∫ σ(E)(E −EF)(− ∂ feq ∂E )dE ∫ σ(E)(− ∂ feq ∂E )dE (1) where the “differential” conductivity σ(E) gives the contribution of a carrier at energy E to the overall conductivity (Goldsmid, 1986; Rowe, 1995). From this equation it is apparent that Π increases as the carrier distribution becomes more asymmetric with respect to the Fermi level. The Peltier coef£cient thus has an inverse relationship with doping, as decreased doping moves 1 Copyright  2001 by ASME the Fermi level further into the bandgap while the carriers are constrained to stay above the band edge. In the case of a layer of semiconductor material between two metal contacts, current ¤ow causes heat to be extracted at one metal/semiconductor junction and deposited at the other junction. In this case, current is carried by majority carriers which can be either electrons or holes, depending on whether the semiconductor is doped with donors (n-type doping) or acceptors (ptype doping), respectively. Since electrons and holes have opposite charge, carriers in n-type and p-type regions ¤ow in opposite directions for a given direction of current ¤ux, and heat will be extracted/deposited at opposite junctions. If a series of alternating n-type and p-type layers are connected by metal contacts, the heat exchange at the junctions will also alternate between cooling and heating. By fabricating a structure such that the cooling and heating junctions are on opposite sides of a ¤at array, an overall heat ¤ux can be achieved from one side of the array to the other. This method of connecting an array of n-type and p-type materials with metal junctions such that they are electrically in series and thermally in parallel is the traditional scheme for most Peltier coolers. The operation of a traditional Peltier cooler is intrinsically unipolar, since the electrical current which contributes to the thermoelectric effect is carried only by majority carriers. The average energy transported by carriers is constant through each semiconductor region and does not vary with applied voltage bias. P-N DIODE While minority carrier effects are negligible in an n/metal/p device, they are crucial in a device such as a p-n diode, which is made up of a p-type region and an n-type region that are directly connected. In the case of an unbiased diode, a large built-in electric £eld exists at the interface of the n-type and p-type regions; the drift current caused by this £eld balances the diffusion current out of the doped regions. As the diode is forward biased, the potential barrier seen by carriers at the interface is decreased, and a net current develops according to the standard diode equations (Sze, 1981). Since the current through a p-n diode can be carried by injected minority carriers in addition to majority carriers, it is fundamentally different than the case in which the n-type and p-type regions are connected by a metallic layer. The energy transported by minority carriers is strongly in¤uenced by the applied forward voltage bias, due to the fact that the minority carrier concentration in the vicinity of the junction is exponentially related to the bias. The bipolar Peltier coef£cient is made up of both majority and minority carrier components, and the heat exchanged at the junction is no longer simply proportional to the current due to the decrease of the minority carrier Peltier coef£cient with increasing bias. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 -7 -6.5 -6 -5.5 -5 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 -7 -6.5 -6 -5.5 -5 Position (μm) Position (μm) Position (μm) Position (μm) R el at iv e E ne rg y (e V ) A vg . T ra ns po rt E ne rg y (e V ) R el at iv e E ne rg y (e V ) A vg . T ra ns po rt E ne rg y (e V ) (A) (C)