Current-voltage characteristics of quantum-point contacts in the closed-channel regime: Transforming the bias voltage into an energy scale

We investigate the $I\left(V\right)$ characteristics (current versus bias voltage) of side-gated quantum-point contacts, defined in $\mathrm{GaAs}∕{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ heterostructures. These point contacts are operated in the closed-channel regime, that is, at fixed gate voltages below zero-bias pinch-off for conductance. Our analysis is based on a single scaling factor, extracted from the experimental $I\left(V\right)$ characteristics. For both polarities, this scaling factor transforms the change of bias voltage into a change of electron energy. The latter is determined with respect to the top of the potential barrier of the contact. Such a built-in energy-voltage calibration allows us to distinguish between the different contributions to the electron transport across the pinched-off contact due to thermal activation or quantum tunneling. The first involves the height of the barrier, and the latter also its length. In the model that we are using the channel length remains the only adjustable parameter since the barrier height can be experimentally determined. For short $(\sim 0.06\mathrm{\mu }\mathrm{m})$ contacts, the $I\left(V\right)$-derived lengths agree rather well with those estimated from the geometrical layout, whereas nominally long $(\sim 1.2\mathrm{\mu }\mathrm{m})$ contacts are typically found to consist of very short $(\sim 0.2\mathrm{\mu }\mathrm{m})$ barriers. We have mapped the height of the barrier as a function of the gate voltage, and found that its behavior differs strongly from that extrapolated using conventional bias spectroscopy in the open-channel regime above conductance pinch-off.

Figure 1

(Color online) Schematic variation of (a) the potential barrier $E_1\left(x\right)$ as well as (b) the width $w\left(x\right)$ along the QPC channel. The following parameters were used: $w_0=12\mathrm{nm}$ and $L=100\mathrm{nm}$. Thick solid lines describe the QPC at zero bias as reference and the biased QPC without self-gating. The negative $(-)$ and positive $(+)$ bias voltages were chosen to set the minimum barrier height at $E={ϵ}_F+\mathrm{\Delta }E=10\mathrm{meV}$. The thin solid lines show the barrier height and width when the self-gating is included in the calculations. For this example, self-gating is assumed to change the channel width by $30\mathrm{nm}∕\mathrm{V}$.

Figure 2

(Color online) Calculated curvature parameter ${\omega }_x$, barrier height $E_{10}$, as well as the $\alpha$ parameter as function of the gate-voltage-dependent barrier height $E_{10}(V=0)$ of the unbiased QPC. The negative $(-)$ and the positive $(+)$ bias voltage were chosen to set the minimum barrier height at $E=10\mathrm{mV}$, as in Fig. 1. (a) Curvature parameter ${\omega }_x$ normalized with respect to its zero-bias value. Without self-gating, the results for negative and positive polarity coincide (thick solid line), while they differ when self-gating is included (thin solid lines). (b) Barrier height $E_{10}$ normalized to the height at zero bias as well as the $\alpha$ parameter. Without self-gating (thick solid lines) and with self-gating (thin solid lines). These parameters have been derived as follows. For each barrier height at zero bias $E_{10}(V=0)$ the critical voltages $V_{+∕-}$ are first determined as for the point contact in Fig. 1. Then the curvature parameters are calculated from the total potentials, and $E_{10}$ as well as $\alpha$ from $V_{+∕-}$ as described in the text. When the effect of self-gating is included (assumed here to change the channel width by $30\mathrm{nm}∕\mathrm{V}$ as in Fig. 1), the curvature parameter for positive polarity diverges above about $55\mathrm{meV}$, and both $E_{10}$ and $\alpha$ cannot be determined for higher barriers. The dashed lines in (a) and (b) were obtained when a weakened self-gating at large barriers was assumed, according to Fig. 14 below.

Figure 3

Scanning-electron micrograph of (a) the short and (b) the long QPC. The white horizontal bars define a length of $1\mathrm{\mu }\mathrm{m}$. The bright thin lines mark the trench edges where material has been etched away to deplete the 2DEG below. The bias voltage $V$ is applied to the source contact (S), and the current $I$ is measured at the drain contact (D). Electrodes on top and bottom serve as side gates (G).

Figure 4

(Color online) Schematic width and potential distribution for the first subband $E_1$ along the channel of (a) the short and (b) the long QPC. For both samples a minimum width of $10\mathrm{nm}$ was assumed.

Figure 5

(Color online) Typical normalized conductance $G∕G_0$ versus gate voltage $V_g$ at the indicated temperatures for (a) a short and (b) a long QPC. The curves are displaced vertically in steps of 0.5. These are the raw data, that is the $\sim 600\mathrm{\Omega }$ serial resistances of the contact pads and 2DEG are not subtracted. The so-called 0.7 anomaly of the short QPC as well as the position of the Coulomb-blockade anomaly of the long QPC are marked.

Figure 6

(Color online) (a) Current $I$ versus gate voltage $V_g$ of the long QPC at $T=315\mathrm{mK}$ [see Fig. 5b]. The point contact was biased from $V=-0.5$ to $+0.5\mathrm{mV}$ in steps of $0.1\mathrm{mV}$. The arrow marks the position of the Coulomb blockade anomaly, shown in detail in (b).

Figure 7

(Color online) Coulomb-blockade anomaly of the long QPC. (a) Normalized zero-bias conductance $G∕G_0$ versus gate voltage $V_g$ at the indicated temperatures. (b) Width (open symbols) and height (closed symbols) of the Coulomb peak versus temperature. The solid lines vary as $T$ and $1∕T$, respectively. There is no saturation at the lowest temperature.

Figure 8

(Color online) Typical characteristics of current $I$ versus bias voltage $V$. (a) The short QPC at $T=3.3\mathrm{K}$ and at gate voltages from $-500$ to $-50\mathrm{mV}$ in steps of $50\mathrm{mV}$. (b) The long QPC at $0.3\mathrm{K}$ and gate voltages from $-40$ to $100\mathrm{mV}$ in steps of $20\mathrm{mV}$. Arrows mark $V_{-}$ and $V_{+}$ for the indicated gate voltages.

Figure 9

(Color online) Magnitude of the current $I$ versus change of energy $\delta E$. (a) The short QPC at $T=14.8,12.9,10.4,8.4,5.5$, and $0.32\mathrm{K}$ (from left to right). The asymmetry parameter was $\alpha =0.33$, and the gate voltage was fixed at $V_g=-100\mathrm{mV}$. (b) The long QPC at $T=6.75,5.94,4.57,3.15,1.82$, and $0.315\mathrm{K}$ (from left to right). For this contact $\alpha =0.235$ and $V_g=100\mathrm{mV}$. For both contacts traces for positive (thick) and negative (thin solid lines) polarity are almost identical.

Figure 10

(Color online) Effective temperature $T_{eff}$ versus temperature $T$ of (a) the short QPC and (b) the long QPC (two different colours for two series of measurements). Open and solid symbols represent results for negative and positive polarities, respectively. Solid lines are guides to the eye with $T_{eff}=\sqrt{T^2+(6\mathrm{K}{)}^2}$ and $T_{eff}=\sqrt{(1.4T{)}^2+(2.5\mathrm{K}{)}^2}$, respectively. The offset by a factor of 1.4 is discussed in the text. The straight solid lines indicate $T_{eff}=T$.

Figure 11

(Color online) Effective temperature $T_{eff}$ versus gate voltage $V_g$ of (a) the short and (b) the long QPC. Thick solid lines indicate the sample temperature slightly above $T=0.3\mathrm{K}$. Open and solid symbols represent results for negative and positive polarities, respectively. Zero-bias conductance pinch-off is at about $100$ and $650\mathrm{mV}$ for (a) and (b), respectively. The thin lines through the data points show how $T_0=T_{eff}$ should vary if the dependence of $E_{10}$ on $V_g$ is taken into account.

Figure 12

(Color online) (a) Barrier height $E_{10}$ of the short QPC versus gate voltage $V_g$. The $\mathrm{\Delta }E\approx 4\mathrm{meV}$ offset as well as the Fermi energy ${ϵ}_F=7\mathrm{meV}$ have been taken into account while determining the height in the closed-channel regime (open symbols). Estimated barrier heights in the open-channel regime are included as well (closed symbols). The solid line shows how the barrier height would change if the hard-wall potential and a linear variation of the depletion width with gate voltage was assumed. The inset shows the original $V_{+}$ (top) and $V_{-}$ (bottom) data versus gate voltage in mV units. (b) Derivative of $E_{10}\left(V_g\right)$ from (a). The transition to the first conductance plateau has its center at around $220\mathrm{mV}$, indicated by the arrow. The solid lines fit tentatively the linear as well as the constant part. The inset shows in detail how to estimate the offset $\mathrm{\Delta }E$.

Figure 13

(Color online) (a) Barrier height $E_{10}$ of the long QPC versus gate voltage $V_g$. The $\mathrm{\Delta }E\approx 2.7\mathrm{meV}$ offset as well as the Fermi energy ${ϵ}_F=11\mathrm{meV}$ have been taken into account while determining the height in the closed-channel regime (open symbols). Estimated barrier heights in the open-channel regime are included (closed symbols). The solid line shows how the barrier height would change if the hard-wall potential and a linear variation of the depletion width with gate voltage was assumed. The inset shows the original $V_{+}$ (top) and $V_{-}$ (bottom) data versus gate voltage in mV units. (b) Derivative of $E_{10}\left(V_g\right)$ from (a). The transition to the first conductance plateau has its center at around $640\mathrm{mV}$, indicated by the arrow. The two solid lines are guides to the eye. The inset shows in detail how to estimate the offset $\mathrm{\Delta }E$.

Figure 14

(Color online) The depletion width per gate voltage $dw∕d{V}_g$ versus barrier height $E_{10}$. Squares and circles are for the short and the long QPC, respectively. Open symbols are the data in the closed-channel regime; closed symbols are estimates from the position of the conductance steps of the open-channel contacts. In the ideal case one would expect a constant $dw∕d{V}_g$, while at large barrier heights the data follow $dw∕d{V}_g\propto E^{-3∕2}$ as indicated by the solid lines for the long QPC.

Figure 15

(Color online) (a) Magnitude of the current $I$ versus change of energy $\delta E$ of the long QPC for a second stable configuration. Temperatures are $T=10.3,11.9,5.43,3.80$, and $0.315\mathrm{K}$ (from left to right). Thin (thick) solid lines represent negative (positive) polarity. (b) Effective temperature $T_{eff}$ versus temperature $T$. Open (closed) symbols denote negative (positive) polarity. The solid line is a guide to the eye with $T_{eff}=\sqrt{(1.2T{)}^2+(6.5\mathrm{K}{)}^2}$. The offset at high temperatures is discussed in the text. The straight line is $T_{eff}=T$.

Figure 16

(Color online) Effective temperature $T_{eff}$ versus temperature $T$ of one short and two long QPCs. Open and closed symbols denote negative and positive polarity, respectively. The straight solid lines are $T_{eff}=T$. The lines through the data points discussed in the text are guides to the eye.