Thermoelectric corrections to quantum voltage measurement

A generalization of Büttiker's voltage probe concept for nonzero temperatures is an open third terminal of a quantum thermoelectric circuit. An explicit analytic expression for the thermoelectric correction to an ideal quantum voltage measurement in linear response is derived and interpreted in terms of local Peltier cooling/heating within the nonequilibrium system. The thermoelectric correction is found to be large (up to $\pm 24\%$ of the peak voltage) in a prototypical ballistic quantum conductor (graphene nanoribbon). The effects of measurement nonideality are also investigated. Our findings have important implications for precision local electrical measurements.

Figure 1

The calculated response of a voltage probe scanned 3 Å above the plane of a zigzag graphene nanoribbon. (top) The voltage distribution calculated using Engquist and Anderson's theory [1] [see Eq. (8)]. This theory neglects thermoelectric effects. The peak voltage for this system is 47.6 mV. (bottom) The thermoelectric correction $\Delta V_p$ to the probe voltage, calculated using Eqs. (10, 11, 12, 13), reaching a maximum value for this system of 11.5 mV. Calculations are performed at ${\mu }_0-{\mu }_{\mathrm{Dirac}}=-57.5\text{meV},{\mu }_2-{\mu }_1=0.1\text{eV},T_1=T_2=T_0=300\text{K}$, and ${\kappa }_{p0}=0$.

Figure 2

(top) The heat current $I_p^{\left(1\right)}$ flowing into the probe when it is held at the ambient temperature $T_0=300\text{K}$, calculated from Eq. (9). (bottom) The temperature $T_p$ of the probe when it is in both electrical equilibrium and thermal equilibrium with the nonequilibrium electron system in the graphene nanoribbon, calculated from Eq. (12).

Figure 3

Thermoelectric circuit diagram for the three-terminal source-drain-probe circuit. According to Eqs. (9) and (12), there is Peltier heating (cooling) of the probe if $({\mathcal{L}}_{p1}^{\left(1\right)}{\mathcal{L}}_{p2}^{\left(0\right)}-{\mathcal{L}}_{p2}^{\left(1\right)}{\mathcal{L}}_{p1}^{\left(0\right)})\left({\mu }_1-{\mu }_2\right)>0$ $\left(<0\right)$. Equation (12) also includes a direct thermal coupling ${\kappa }_{p0}$ of the probe to the environment (not shown in the diagram).

Figure 4

(top) The thermoelectric correction $\Delta V_p$ to the probe voltage and (bottom) the deviation of the probe temperature from ambient temperature for the same system as that shown in Figs. 1 and 2, but with a finite thermal coupling ${\kappa }_{p0}={\kappa }_0$ of the probe to the environment, where ${\kappa }_0=({\pi }^2/3)k_B^2{T}_0/h$ (0.284 nW/K at 300K) is the thermal conductance quantum. Equations (10) and (12) indicate that the thermoelectric corrections for larger values of ${\kappa }_{p0}$ scale as ${\kappa }_{p0}^{-1}$.