Driving Perpendicular Heat Flow: ($p\mathbf{\times }n$)-Type Transverse Thermoelectrics for Microscale and Cryogenic Peltier Cooling

Whereas thermoelectric performance is normally limited by the figure of merit $ZT$, transverse thermoelectrics can achieve arbitrarily large temperature differences in a single leg even with inferior $ZT$ by being geometrically tapered. We introduce a band-engineered transverse thermoelectric with $p$-type Seebeck in one direction and $n$-type orthogonal, resulting in off-diagonal terms that drive heat flow transverse to electrical current. Such materials are advantageous for microscale devices and cryogenic temperatures—exactly the regimes where standard longitudinal thermoelectrics fail. $\mathrm{InAs}/\mathrm{GaSb}$ type II superlattices are shown to have the appropriate band structure for use as a transverse thermoelectric.

Figure 1

The $p\times n$-type transverse thermoelectric has net $p$-Seebeck coefficient along $a$ axis with net $n$-Seebeck along $b$, indicated with the crossed-arrow symbol on upper right. Microscopic electron-hole picture depicts net charge current $J_x$ to the right and net particle or heat current $Q_y$ up.

Figure 2

Optimal temperature profile for longitudinal (dotted line), transverse (solid line), and Ettingshausen (dashed line) thermoelectric cooling for various $ZT$ values assuming a Seebeck component $S_{xx}$ or $S_{xy}=320\mu \mathrm{V}/\mathrm{K}$, as appropriate. At $y=10$ heat flow, $Q_y=0$, and at $y=0$ heat sink is $T_{\mathrm{H}}=300\mathrm{K}$ in panel (a) and 77 K in panel (b), where doped longitudinal thermoelectrics fail.

Figure 3

Temperature $T$ (top) and optimal electric field ${\mathcal{E}}_x$ (bottom) versus normalized sample thickness $y/L$, for rectangular (left) and exponentially tapered geometries (right). Exponential tapering constant $L$ sets length scale. At $y=0$ heat sink is $T_{\mathrm{H}}=300\mathrm{K}$ and at right end point heat flow $Q_{\mathrm{in}}(T_{\mathrm{C}})=0$. (a) For rectangular device, minimum $T_{\mathrm{C}}$ is independent of layer thickness, but thicker layers require weaker electric fields. Inset shows device geometry whereby current in $x$ direction induces heat flow in $-y$. (b) For exponentially tapered device, it can cool to lower temperatures with thicker material. Minimum $T_{\mathrm{C}}$ has two solutions for given sample thickness, one for ${\mathcal{E}}_x<{\mathcal{E}}^{*}$ and one for ${\mathcal{E}}_x>{\mathcal{E}}^{*}$.