Physics:Thermotunnel cooling

Thermotunneling is a thermoelectric process where electron pass from one surface to another is possible by the two surfaces being very close to each other (few nanometers). By applying low voltage potential across the surfaces, energetic electrons can move from one surface to the other. The energetic electrons are being replaced by electrons with average energy, thus cooling the first surface while the energetic electrons are heating the second surface. The use of low work function materials is required to allow the electrons pass through the gap. The small distance between the two surfaces allows electron transfer to occur at lower energy levels than normally required, as if tunneling through the normal energy barrier. This process is also known as field emission. The thermo-tunneling effect was first observed in 1980 in Al-PbBi tunnel-junctions in which one electrode was heated by laser irradiation.

Description

We are interested in calculating the maximum recoverable power density P_max, the current-voltage characteristic, and the associated efficiency η for a given configuration of an energy thermo-tunneling device. The modeling for thermo-tunneling energy scavenging has already been developed for simple geometrical and physical structure. For cooling applications, two metallic metallic plane electrodes separated by a thin vacuum gap of width [math]\displaystyle{ d }[/math] are needed. One electrode is at the cold temperature [math]\displaystyle{ T_{\rm c} }[/math], and the other at [math]\displaystyle{ T_{\rm h} }[/math]. For simplicity it is common to assume the temperatures of the electrodes are constant. The work functions of electrodes are the only physical parameters relevant for this modeling. In the present model, the work functions Φ_h and Φ_c of the hot and cold electrodes determine the shape of the potential barrier between electrodes. Since the temperatures of the electrodes are different, the energetic repartitions of electrons in each electrode are different too resulting in the hot electrode emitting more electrons than the cold one. The total current density J_tot between the electrodes is defined as the difference between the hot and cold side contributions:

[math]\displaystyle{ \mathbf{J}_{\rm tot}=\mathbf{J}_{\rm tot,hot}-\mathbf{J}_{\rm tot,cold} }[/math]

In the same way, the total thermal power density Q_hc transferred from the hot to the cold electrode, defined as the difference between contributions of hot and cold sides:

[math]\displaystyle{ Q_{\rm h}=Q_{\rm h}-Q_{\rm c} }[/math]

When current density is happening, a bias voltage occurs between the electrodes at different temperatures in response. This bias voltage V_bias counters the effect of the temperature difference. And thus a Fermi-level shift is created between the hot and cold electrodes, E_fh−E_fc=eV_bias. It amplifies electron tunneling from cold to hot side and weakening electron tunneling from hot to cold side. Due to this Fermi level shift between electrodes, the potential barrier between electrodes different for electrons crossing from the hot to the cold side and for electrons crossing from the cold to the hot side.

Contribution of electrons crossing from hot to cold electrode

The potential energy of an electron moving from the hot to the cold electrode in the thin vacuum gap is:

[math]\displaystyle{ {V_{\rm h}(x_{\rm h})=\varPhi_{\rm h}-(eV_{\rm bias}+\varPhi_{\rm h}-\varPhi_{\rm c})\frac{x_{\rm h}}{d}-\frac{e^{2}}{4\pi\varepsilon_{0}}\left[\frac{1}{4x_{\rm h}}+\frac{1}{2}\sum_{n=1}^{\infty}\left(\frac{nd}{n^{2}d^{2}-x_{h}^{2}}-\frac{1}{nd}\right)\right]} }[/math]

Where x_h is the distance from the hot electrode to the electron, [math]\displaystyle{ d }[/math] is the width of the gap, and [math]\displaystyle{ e }[/math] is the elementary charge. Using the classical WKB approximation D_h(E_x) is given by:

[math]\displaystyle{ {D_{\rm h}(E_{x})=\begin{cases} \exp\left\{-\frac{2}{\hbar}\int_{x_{\rm h1}}^{x_{\rm h2}}\sqrt{2m[V_{\rm h}(x_{\rm h})-E_{x}]dx_{\rm h}}\right\} & E_{x}\lt V_{\rm h,max}\\ 1 & \text{otherwise} \end{cases}} }[/math]

where, x_h1 and x_h2 are roots of V_h(x_h)−E_x=0. Current density emitted from hot to cold electrode is then given by:

[math]\displaystyle{ {J_{\rm tot,hot}=e\int_{-\infty}^{\infty}N_{\rm h}(E_{x})D_{\rm h}(E_{x})dE_{x}} }[/math]

Where N_h is the number of electrons per unit area in unit time that can escape from the hot electrode and reach the cold one with their kinetic energy in x direction in infinitesimal range [E_x , E_x+dE_x] and is given by:

[math]\displaystyle{ {N_{\rm h}(E_{x})=\frac{1}{4\pi^{3}\hbar^{3}}\int_{p_{y}p_{z}}f_{\rm h}(E)[1-f_{\rm c}(E+eV_{\rm bias})]dp_{y}p_{z}} }[/math]

The first term in front of the integral is the density of state in the momentum space (p_x p_y p_z), f_h ( and respectively, f_c) represents the Fermi-Dirac distribution for electrons of the hot ( and cold) electrode, E is the total energy of the electron, eV_bias is the Fermi level shift between the two electrodes, and p_y and p_z are electron momentum in in-plane y and z directions in the hot electrode. Thermal power density emitted by the hot electrode on the cold one is given by:

[math]\displaystyle{ {Q_{\rm h}=\int_{-\infty}^{\infty}(k_{\rm B}T_{\rm h}+E_{x})N_{\rm h}(E_{x})D_{\rm h}(E_{x})dE_{x}} }[/math]

In this integral k_BT_h is included to account for the average transverse kinetic energy of electrons which cross the gap from hot to cold side, with k_B being Boltzmann constant.

Contribution of electrons crossing from cold to hot electrode

For an electron moving from the cold to the hot electrode, the potential energy inside the thin vacuum gap is a bit different, and it is considering the Fermi-level shift due to the bias voltage, V_bias between the electrodes. With respect to the Fermi level E_fc of the cold electrode, the potential energy of an electron moving from cold to hot electrode in the thin vacuum gap is:

[math]\displaystyle{ {V_{\rm c}(x_{\rm c})=\varPhi_{\rm c}+(eV_{\rm bias}+\varPhi_{\rm h}-\varPhi_{\rm c})\frac{x_{\rm c}}{d}-\frac{e^{2}}{4\pi\varepsilon_{0}}\left[\frac{1}{4x_{\rm c}}+\frac{1}{2}\sum_{n=1}^{\infty}\left(\frac{nd}{n^{2}d^{2}-x_{\rm h}^{2}}-\frac{1}{nd}\right)\right]} }[/math]

Where x_c is the distance from the electron to cold electrode. Expressions for the current and thermal power densities for the contribution of electrons moving from cold to hot electrode are obtained in a similar way as for the section above. Due to the −eV_bias Fermi level shift with respect to the hot electrode, the probability of crossing the gap for an electron of the cold electrode is redefined as:

[math]\displaystyle{ {D_{\rm c}(E_{x})=\begin{cases} \exp\left\{-\frac{2}{\hbar}\int_{x_{\rm c1}}^{x_{\rm c2}}\sqrt{2m[V_{\rm c}(x_{\rm c})-E_{x}]dx_{\rm c}}\right\} & E_{x}\lt V_{\rm c,max}\\ 1 & \text{otherwise} \end{cases}} }[/math]

E_x is the kinetic energy along the x axis, the normal of electrodes, of an electron from the cold electrode, V_c,max is the maximum of V_c(x_c),x_c1 and x_c2 are the roots of V_c(x_c)−E_x=0. The total current density emitted from cold to hot electrode:

[math]\displaystyle{ {J_{\rm tot,cold}=e\int_{-\infty}^{\infty}N_{\rm c}(E_{x})D_{\rm c}(E_{x})dE_{x}} }[/math]

Where N_c(E_x) is given by:

[math]\displaystyle{ {N_{\rm c}(E_{x})=\frac{1}{4\pi^{3}\hbar^{3}}\int_{p_{y}p_{z}}f_{\rm c}(E)[1-f_{\rm h}(E-eV_{\rm bias})]dp_{y}p_{z}} }[/math]

Thermal power density emitted from the cold electrode to the hot electrode can now be calculated:

[math]\displaystyle{ {Q_{\rm c}=\int_{-\infty}^{\infty}(k_{\rm B}T_{\rm c}+E_{x})N_{\rm c}(E_{x})D_{\rm c}(E_{x})dE_{x}} }[/math]

Potential performance

The theoretical performance of thermo-tunneling devices operating near room temperature has been estimated by GE, Borealis, and Tempronics to be in the range of 50% to 80% of Carnot efficiency. The theoretical performance of thermo-tunneling for room-temperature cooling is equivalent of vapor compression technology, but the design problems are difficult to overcome.