Three-particle collisions in quantum wires: Corrections to thermopower and conductance

We consider the effect of electron-electron interaction on the electron transport through a finite length single-mode quantum wire with reflectionless contacts. The two-particle scattering events cannot alter the electric current and therefore we study the effect of three-particle collisions. Within the Boltzmann equation framework, we calculate corrections to the thermopower and conductance to the leading order in the interaction and in the length of wire $L$. We check explicitly that the three-particle collision rate is identically zero in the case of several integrable interaction potentials. In the general (nonintegrable) case, we find a positive contribution to the thermopower to leading order in $L$. The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore, the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.

Figure 1

A schematic picture of two metallic gates depleting the underlying two-dimensional electron gas and thereby forming a short 1D quantum wire of length $L$. This fabrication method has the advantage of producing reflectionless contacts to the leads (Ref. 3), so that the boundary conditions of the distribution function are given by the Fermi function of the reservoirs. We define the thermopower as $S=V∕{\phantom{\mid }\Delta T\mid }_{I=0}$, i.e., the voltage $V$ required to counteract a current due to the temperature difference $\Delta T$.

Figure 2

(a) The dominant three-particle scattering process at low temperature in a single energy band. (b) The three-particle scattering process perturbing the initial distributions shown with warm left-moving electrons $(k<0)$ and cold right-moving electrons $(k>0)$. Due to the temperature difference of the initial distributions, the scattering process creating left movers dominates compared to the opposite scattering and therefore it gives a positive correction to the thermopower.

Figure 3

A visualization of the connected three-particle scattering matrix element [Eq. (30)], where three particles interchange their momenta and energy. This matrix element enters the scattering rate via the generalized Fermi golden rule [Eq. (23)]. (a) The basic three-particle interaction consisting of two interaction lines and a free propagation [see Eq. (30)]. (b) Picture of the exchange processes times the basic interaction needed to form the matrix element [Eq. (30)].

Figure 4

(Color online) (a) The integration region $\mathcal{A}$ for the integral [Eq. (B13)] to calculate the current due to interactions. The two boundaries for the integration area close to the origin stemming from the signs of $k_2$ and $k_{2^{\prime }}$ are indicated. (b) The $\Xi (Q_1,Q_3)=\xi (k_F{Q}_1,k_F{Q}_3)∕{\epsilon }_F$ function, a dimensionless version of $\xi (q_1,q_3)$ [Eq. (B9)], important in the calculation using the method of steepest decent.