Current-Induced Cooling Phenomenon in a Two-Dimensional Electron Gas Under a Magnetic Field

Abstract

We investigate the spatial distribution of temperature induced by a dc current in a two-dimensional electron gas (2DEG) subjected to a perpendicular magnetic field. We numerically calculate the distributions of the electrostatic potential ϕ and the temperature T in a 2DEG enclosed in a square area surrounded by insulated-adiabatic (top and bottom) and isopotential-isothermal (left and right) boundaries (with ϕ _left<ϕ _right and T _left=T _right), using a pair of nonlinear Poisson equations (for ϕ and T) that fully take into account thermoelectric and thermomagnetic phenomena, including the Hall, Nernst, Ettingshausen, and Righi-Leduc effects. We find that, in the vicinity of the left-bottom corner, the temperature becomes lower than the fixed boundary temperature, contrary to the naive expectation that the temperature is raised by the prevalent Joule heating effect. The cooling is attributed to the Ettingshausen effect at the bottom adiabatic boundary, which pumps up the heat away from the bottom boundary. In order to keep the adiabatic condition, downward temperature gradient, hence the cooled area, is developed near the boundary, with the resulting thermal diffusion compensating the upward heat current due to the Ettingshausen effect.

Appendix: Derivation of the Nonlinear Poisson Equations (4) and (6)

In this Appendix, we present, for completeness, the derivation of the nonlinear Poisson equations (4) and (6) from the transport equations (1) and (2) in the steady-state. We basically follow the prescription presented in [26] and [45]. See also [1].

In the steady state (∂/∂t=0), we have, from the continuity equations,

$$ \nabla\cdot\boldsymbol{J} = 0, $$

(33)

and

$$ \nabla\cdot\boldsymbol{J_U} =\nabla\cdot( \boldsymbol{J_Q} + \phi\boldsymbol{J}) = 0. $$

(34)

Taking the divergence of Eq. (1), we have, with the aid of Eq. (33):

(35)

Noting that, under the temperature gradient ∇T, an arbitrary transport coefficient X depends on (x,y) through its temperature dependence, the gradient of X can be expressed as

(36)

We thus can rewrite Eq. (35) as follows:

(37)

Using formulas for vector operation, Eq. (37) becomes

(38)

Using further the Maxwell equation ∇×B=μ J+εμ(d E/dt)=0 in the steady state, omitting the negligibly small term μJ, Eq. (38) becomes

(39)

Similarly, the rotation of Eq. (1) gives

(40)

We thus obtain

(41)

Substituting Eq. (41) into Eq. (39), we obtain the following equation:

(42)

By substituting Eq. (6) (to be derived below) in Eq. (42), we arrive at Eq. (4).

Next, we derive the Poisson equation for T, Eq. (6). We obtain the following equation from the divergence of J _U using Eqs. (33) and (34):

(43)

Using Eq. (36), we have

(44)

Using again formulas for vector operation, we rewrite the above equation as,

(45)

which reduces, with the Maxwell equation ∇×B=0, to

(46)

We substitute Eq. (1) into the above equation and obtain

(47)

By substituting Eq. (41) into Eq. (47), we arrive at Eq. (6).