Quantum coherence at low temperatures in mesoscopic systems: Effect of disorder

We study the disorder dependence of the phase coherence time of quasi-one-dimensional wires and two-dimensional (2D) Hall bars fabricated from a high mobility GaAs/AlGaAs heterostructure. Using an original ion implantation technique, we can tune the intrinsic disorder felt by the 2D electron gas and continuously vary the system from the semiballistic regime to the localized one. In the diffusive regime, the phase coherence time follows a power law as a function of diffusion coefficient as expected in the Fermi-liquid theory, without any sign of low-temperature saturation. Surprisingly, in the semiballistic regime, it becomes independent of the diffusion coefficient. In the strongly localized regime we find a diverging phase coherence time with decreasing temperature, however, with a smaller exponent compared to the weakly localized regime.

Figure 1

(Color online) Scanning electron microscopy (SEM) image of the sample. The dark and white parts represent the mesas and electrodes, respectively. The voltage probes for the 1000 nm wide wires as well as the ground and current bias are added in the figure.

Figure 2

(Color online) Schematic of a FIB microscope placed on the GaAs wafer. The inset shows an SRIM simulation (see Ref. 26) of the implanted ion concentration as a function of depth at a dose of ${10}^9{\text{cm}}^{-2}$ and at an energy of 100 keV. The ions are predominantly implanted 50 nm above the 2DEG.

Figure 3

(Color online) Diffusion coefficient as a function of ion dose for ${\text{Ga}}^{+}$ and ${\text{Mn}}^{+}$.

Figure 4

Schematic of our electric circuit. A ratio transformer is used to subtract the background resistance and to extract the small WL signal above 1 K.

Figure 5

(Color online) Magnetoresistance curves of 1000 and 1500 nm wide wires at $T=36\text{mK}$ and $D=290{\text{cm}}^2/\text{s}$.

Figure 6

(Color online) WL curves of (a) 1000 and (b) 1500 nm wide wires at $D=290{\text{cm}}^2/\text{s}$ and $170{\text{cm}}^2/\text{s}$, respectively. The conductance here is divided by $e^2/h$. The broken lines are the best fits of Eq. (6). The insets in (a) and (b) show the magnetoconductance at $T=140\text{mK}$ in larger field ranges.

Figure 7

(Color online) Phase coherence length of 1000 and 1500 nm wide wires as a function of $T$ at $D=290{\text{cm}}^2/\text{s}$. The solid lines are the best fits with Eq. (8).

Figure 8

(Color online) Magnetoconductance curves of a Hall bar at $D=46{\text{cm}}^2/\text{s}$ at different temperatures. The conductance is normalized by $e^2/h$. The broken lines are the best fits to Eq. (7). The fitted curves deviate from the experimental data at around $B_c$. The inset shows a closeup view of the low-field part of the magnetoconductance at low temperatures.

Figure 9

(Color online) Phase coherence length of the Hall bar at $D=46{\text{cm}}^2/\text{s}$ as a function of $T$. The solid line is the best fit with Eq. (9).

Figure 10

(Color online) Magnetoresistance curves of 1000 and 1500 nm wide wires at $T=33\text{mK}$ and $D=3100{\text{cm}}^2/\text{s}$.

Figure 11

(Color online) Magnetoresistance curves of 1000 nm wide wires at three different diffusion coefficients; (a) $D=3100$, (b) 1400, and (c) $600{\text{cm}}^2/\text{s}$. The broken line shows the maximum position of the small bump, i.e., $B_{\text{max}}$.

Figure 12

(Color online) WL curves of (a) 1500, (b) 1000, and (c) 600 nm wide wires at $D=3500{\text{cm}}^2/\text{s}$, $3100{\text{cm}}^2/\text{s}$, and $2400{\text{cm}}^2/\text{s}$, respectively. The conductance is normalized by $e^2/h$. The broken lines are the best fits to Eq. (10).

Figure 13

(Color online) Phase coherence length of 1500, 1000, and 600 nm wide wires as a function of $T$ at $D=3500{\text{cm}}^2/\text{s}$. The solid lines are the best fits with Eq. (8).

Figure 14

(Color online) Phase coherence length $L_{\varphi }$ of three different wires as a function of $T$ at several different diffusion coefficients. The open and solid symbols correspond to the semiballistic regime and the diffusive one, respectively. The solid lines are the best fits to Eq. (8).

Figure 15

(Color online) The diffusion coefficient dependence of the phase coherence length $L_{\varphi }$ at $T=60\text{mK}$ of the three different wires. The broken and solid lines show $D^{2/3}$ and $D^{1/2}$ laws, respectively.

Figure 16

(Color online) Phase coherence time ${\tau }_{\varphi }$ of 1500 nm wide wires as a function of $T$ at different diffusion coefficients.

Figure 17

(Color online) The experimental coefficient $a_{\text{exp}}$ of Eq. (8) scaled by ${\alpha }_{\text{AAK}}$ as a function of $D$. The dashed line represents $D^{-1/3}$.

Figure 18

(Color online) Residual resistance of the wires as a function of $D$ obtained from the Hall bar. The broken lines (blue and red) show $R_{\text{res}}\propto 1/D{w}_{\text{eff}}$ for $w_{\text{eff}}=1130\text{nm}$ and 630 nm, respectively.

Figure 19

(Color online) (a) An example to extract $1/{\tau }_{\varphi }^0$ proposed by GZ (Ref. 60). The solid line shows the best fit of the AAK formula, Eq. (1). The broken line is a linear fit of the dephasing rate $1/{\tau }_{\varphi }\left(T\right)$. The intercept represents $1/{\tau }_{\varphi }^0$. (b) Dephasing rate $1/{\tau }_{\varphi }$ at different $D$ as a function of $T$ on a linear scale $\left(w=1500\text{nm}\right)$. The broken lines have the same meaning as in (a).

Figure 20

(Color online) ${\tau }_{\varphi }^0$ for all the investigated samples. The dotted and dashed-dotted lines show $D^2$ and $D$ laws, respectively.

Figure 21

(Color online) Resistance variations in 1000 (top) and 1500 nm wide wires (bottom) as a function of $T$ at $B=500\text{G}$ and 300 G, respectively. The broken lines are the best fits of Eq. (13). The real electron temperature below 40 mK is corrected by the $1/\sqrt{T}$ law (see the arrow). In the inset of the top figure, the resistance of 1000 nm wide wires is plotted as a function of $1/\sqrt{T}$ to extract $R_{\text{res}}$.

Figure 22

(Color online) Resistance variation in the Hall bar at $D=46{\text{cm}}^2/\text{s}$ as a function of $T$. The broken line shows a $\text{ln}\left(T\right)$ law.

Figure 23

(Color online) Resistance of 1500 nm wide wires as a function of $1/\sqrt{T}$ at $D=3500$ (left axis) and $130{\text{cm}}^2/\text{s}$ (right axis). The resistance at $D=130{\text{cm}}^2/\text{s}$ is measured at $B=300\text{G}$ $\left(B/B^{\ast }=0.07\right)$ while the resistance at $D=3500{\text{cm}}^2/\text{s}$ is measured at $B=20$ ($B/B^{\ast }=0.12$; black dots) and 170 G ($B/B^{\ast }=1$; red curve). The broken lines are linear fits to extract $R_{\text{res}}$. The temperature below 40 mK has already been corrected for both diffusion coefficients.

Figure 24

(Color online) Resistance variations in 1500 nm wide wires as a function of $T$ for three different diffusion constants. The dashed-dotted lines are the best fits to Eq. (15). Inset: resistance variation in the 1500 nm wide wires at $D=3100{\text{cm}}^2/\text{s}$. The broken line (black) shows the $1/\sqrt{T}$ law whereas the dotted line (green) shows the $\text{ln}\left(T\right)$ dependence. The dashed-dotted line (blue) is again the best fit to Eq. (15). $T^{\ast }$ indicates the temperature where $T^{\ast }=\hslash /\left({\tau }_e{k}_B\right)$.

Figure 25

(Color online) The experimental prefactors $A_{\text{exp}}$ (left axis) and $B_{\text{exp}}$ (right axis) of the AA correction as a function of $D$. The broken line shows a $D^{1/2}$ law.

Figure 26

(Color online) $R_{xx}$ and $R_{xy}$ at $D=8{\text{cm}}^2/\text{s}$ and $T=100\text{mK}$. The Hall resistance $R_{xy}$ is normalized by $h/e^2$.

Figure 27

(Color online) (a) Magnetoresistance curves of the Hall bar at $D=8{\text{cm}}^2/\text{s}$ at several different temperatures. (b) $R_{xx}$ at $B=0$ and 2000 G as a function of temperature. The broken and dashed-dotted lines represent $\text{ln}\left(T\right)$ and 2D variable range hopping laws, respectively.

Figure 28

(Color online) (a) Magnetoconductance curves of the Hall bar at $D=8{\text{cm}}^2/\text{s}$ at several different temperatures. The conductance is divided by $e^2/h$. The broken lines show the best fits of Eq. (7). The inset shows a closeup view of the low-field part of the magnetoconductance at low temperatures. (b) Temperature dependence of $L_{\varphi }$ in the strongly localized regime (open symbols). For comparison, $L_{\varphi }$ in the weakly localized regime (closed symbols) is also plotted. (c) Temperature dependence of ${\sigma }_{xx}$ in the representation corresponding to the self-consistent theory, Eq. (17).

Figure 29

(Color online) Phase coherence length $L_{\varphi }$ of Hall bars at $T=60\text{mK}$ as a function of $D$. The broken and dashed-dotted lines represent the 2D localization length, Eq. (16), and theoretically expected phase coherence length $L_{\varphi }^{2\text{D}}$ based on Eq. (4), respectively.

Figure 30

(Color online) [(a)–(c)] Diffusion coefficient dependence of the theoretical 1D localization length, Eq. (18), and phase coherence length expected in the AAK theory at $T=25\text{mK}$; (a) $w_{\text{eff}}=1130$, (b) 630, and (c) 320 nm. The closed symbols represent the diffusion coefficient where the $R\left(T\right)$ and $L_{\varphi }$ measurements have been performed. [(d)–(f)] Experimental data of $R\left(T\right)$ (red solid lines) and $L_{\varphi }$ (blue open symbols) as a function of $T$; (d) $w_{\text{eff}}=1130$, (e) 630, and (f) 320 nm. The solid lines for $L_{\varphi }\left(T\right)$ are the best fits of Eq. (8). For the dashed-dotted lines, we have changed the exponent of $L_{\varphi }\left(T\right)$ at the low-temperature part from $-1/3$ (AAK) to $-0.29$ $\left(w_{\text{eff}}=1130\text{nm}\right)$, $-0.26$ $\left(w_{\text{eff}}=630\text{nm}\right)$, or $-0.24$ $\left(w_{\text{eff}}=320\text{nm}\right)$ in order to get better fitting precisions down to lower temperatures. The broken, dotted, dashed-dotted lines for $R\left(T\right)$ show the best fits of $1/\sqrt{T}$, $\text{exp}\left(T_0/T\right)$, and $\text{exp}\left[{\left(T_M/T\right)}^{1/2}\right]$ laws, respectively. The vertical dotted lines in (e) and (f) represent the temperatures below which $L_{\varphi }\left(T\right)$ and $R\left(T\right)$ deviate from the AAK theory, Eq. (8), and from the activation law Eq. (19), respectively.