Thermal Enhancement of Interference Effects in Quantum Point Contacts

We study an electron interferometer formed with a quantum point contact and a scanning probe tip in a two-dimensional electron gas. The images giving the conductance as a function of the tip position exhibit fringes spaced by half the Fermi wavelength. For a contact opened at the edges of a quantized conductance plateau, the fringes are enhanced as the temperature $T$ increases and can persist beyond the thermal length $l_T$. This unusual effect is explained by assuming a simplified model: The fringes are mainly given by a contribution which vanishes when $T\to 0$ and has a decay characterized by a $T$-independent scale.

Figure 1

Electron interferometer made of a QPC and a scatterer (red point on the right) induced by a charged tip. The lines with two arrowheads give the main interference paths responsible for the SGM fringes. (a) Saddle-point QPC potential used in model 1: $U_1(X,Y)=(Y/L_Y{)}^2[1-3(X/L_X{)}^2+2|X/L_X{|}^3{]}^{1/4}$ for $|X|\le L_X$ and $U_1=0$ elsewhere. $L_Y=10$ and $L_X=120$ (i.e., long QPC). Model 2 (hard wall QPC not shown) consists in removing for $|X|\le L_X$ the sites of coordinates $|Y|\ge L_Y+3(X/L_X{)}^2$, with $L_Y=6$ and $L_X=20$ but keeping otherwise the site potentials equal to zero. The hopping amplitudes are $t_h=-1$. (b) RLM: 2 semi-infinite square lattices (with zero on-site potential and hopping amplitudes $t_h=-1$) are contacted by hopping terms $t_c$ via a single site $\mathbf{0}$ of energy $V_{\mathbf{0}}$. At a distance $x$ from the contact, there is a tip potential $V\ne 0$.

Figure 2

$\delta g(T)=g(T)-g_0(T)$ as a function of the tip position (in units of ${\lambda }_F/2$). The left figures correspond to $T=0$, while $T\ne 0$ for the right figures. $g_0$ is biased as indicated by the arrow in the insets [giving $g_0(T=0)$ as a function of $E_F$]. (a),(b): QPC opened at the beginning of the first plateau using model 1 with $V=1$ and ${\lambda }_F/2=9.65$. $k_{B}T/E_F=0.01$ for (b) ($2l_T/{\lambda }_F\approx 14.6$). (c),(d): QPC opened at the beginning of the second plateau using model 2 with $V=-2$ and ${\lambda }_F/2=6.7$. $k_{B}T/E_F=0.035$ for (c) ($2l_T/{\lambda }_F\approx 4$).

Figure 3

(a) RLM conductance $g(T)$ as a function of $2x/{\lambda }_F$ for increasing temperatures (from top to bottom). The color points are results obtained by direct numerical simulations, taking from top to bottom $2l_T/{\lambda }_F=\infty$, 8.8, 4.4, and 2.2. The solid lines give the analytical results [Eq. (6)]. $t_c=0.4$, $V=-2$, ${\lambda }_F/2\approx 10$, and $2l_{\Gamma }/{\lambda }_F\approx 4$. (b) As a function of $x/l_{\Gamma }$, rescaled amplitude $\mathcal{A}=A(x/l_{\Gamma },l_T/l_{\Gamma })\mathrm{erfc}(l_T/l_{\Gamma })$ for fixed $l_{\Gamma }$ and increasing temperatures, i.e., ratios $l_T/l_{\Gamma }=1$ (light blue), 0.7 (blue), 0.5 (violet), and 0.3 (red) (solid lines from bottom to top). The dashed and dotted lines give, respectively, the asymptotic behavior $2\exp (-x/l_{\Gamma })$ valid for $x\gg l_T$ and the function $\mathcal{A}(l_T/l_{\Gamma },l_T/l_{\Gamma })$, where $\mathcal{A}$ is maximum. Inset: Maximum $A_{\max }$ of the bare amplitude $A$ as a function of $l_T/l_{\Gamma }$.