Scanning gate microscopy simulations for quantum rings: Effective potential of the tip and conductance maps

We simulate electron flow through a semiconductor quantum ring perturbed by a charged tip of a scanning microscope. We describe the interaction of the tip with the electron gas solving the density functional theory equations for up to several hundred electrons forming the background potential for the current flow at the Fermi level. The screening of the repulsive tip potential involves an appearance of the Friedel oscillations of the electron density. The effective potential of the tip turns out to be anisotropic and close to a Lorentzian along the channel. The Lorentzian width along the channel is comparable to the distance between the tip and the electron gas. The width is insensitive to the charge of the tip and the electron density. We discuss the conductance maps as calculated in the Landauer approach including the case when the tip is outside the ring. We discuss both the case of weak perturbation introduced by the tip in the context of extraction of the local density of states as well as the case of strong tip-electron-gas interaction, which modifies the potential landscape within the structure. For strong perturbation we find that the repulsive tip introduces radial fringes of conductance within the ring and concentric ones outside the ring. The radial ones correspond to interrupted current circulation around the ring and are insensitive to the external magnetic field while the external concentric fringes evolve in external magnetic field due to an interplay of the electrostatic and magnetic Aharonov-Bohm effects.

Figure 1

(a) Geometry of the model system. Within the white (gray) area the confinement potential is $W=0$ ($W=4$ eV). The channels are assumed $t=96$ nm wide. The inner and outer radii of the ring are taken equal to 128 and 224 nm, respectively. The computational box is 1920 nm long. (b) The plane of confinement of the electron gas is located $z_d=12.5$ nm below the sheet of positive charge of ionized donor impurities. The scanning tip is located $z_t=50$ nm above electron gas. (c) For calculations of the electrostatic potential within the original computational box ${\Omega }_0$ we copy the electron and donor densities to adjacent cells above (${\Omega }_1$) and below ${\Omega }_2$ the original one.

Figure 2

(Color online) Dispersion relation obtained for a straight channel of width 96 nm with background electrostatic potential obtained in the asymptotic region by the DFT scheme for 320 electrons within the computational box ($B=0$). Fermi wave vectors are indicated by the vertical lines. Insets show the wave functions across the channel for the three lowest subbands.

Figure 3

Density of the electron gas within the channels (a),(e),(i),(m) (color scale given in units of cm${}^{-2}$), the potential $V_q$ due to the electron gas and the positive background of ionized donor impurities at the channel level (b),(f),(j),(n) (color scale given in meV), the Coulomb potential of the tip $V_t$ (c),(g),(k),(o) and the total potential $T=W+V_q+V_t$ (d),(h),(l),(p) (common color scale given in meV for $V_t$ and $V_q$). In the first row of plots (a)–(h) the “gas” is formed by 44 electrons contained within the computational box and $Q_{\mathrm{tip}}=-4e$, while in the second row (i)–(p) 134 electrons form the gas and $Q_{\mathrm{tip}}=-8e$. In panels at left (a)–(d),(i)–(l) the tip is above the right arm of the ring [$x=144$ nm $y=1048$ nm; see (c) and (k)]. In panels at right (e)–(h),(m)–(p) the tip is above ($x=266$ nm, $y=1048$ nm);- see the pink dot in (e) and (m).

Figure 4

Results for the electron gas confined in a straight channel of width 96 nm containing $N$ electrons per computational box of length 1920 nm. (a) The repulsive tip potential $V_t$, the potential induced by the tip $V_q$ (solid lines), and the total response potential $T$. Gray and black lines were obtained for potential range parameters $\lambda =500$ nm (gray) and $\lambda =1000$ nm (black). The dashed line shows the response potential calculated for the tip placed above $x=0,y=1340$ nm. (b) The electron charge density for the repulsive tip placed above the center of the computational box for various number of electron forming the gas. $P$ stands for the number of subbands at the Fermi level determined according to the approach described in Sec. 2d. (c) A zoom of the fragment of (b) that is marked by the dashed rectangle. The tics on the plot shows half the Fermi wavelength $0.5{\lambda }_F=2\pi /k_F$ for the Fermi wave vector determined from the dispersion relation (see Sec. 2d) for the lowest subband $k_F=k\left(1\right)$ (for numerical values see text). (d) Charge density for repulsive $Q_{\mathrm{tip}}=-16e$ and attractive $Q_{\mathrm{tip}}=+16e$ tip potential for two screening length parameters $\lambda =500$ nm (dashed lines) and $\lambda =1000$ nm (solid lines). (e) Same as (b) but for attractive tip potential $Q_{\mathrm{tip}}=+16e$. (f) The solid lines show the total potential $T$ corresponding to electron charge densities of (b). The tip potential $V_t$ is plotted with the dotted line. The response potential for the attractive tip potential are plotted with the dashed lines.

Figure 5

(Color online) Same as Fig. 4c for $N=102$. The density presents a regular Friedel oscillation only at the axis of the channel ($x=0$). The densities off the axis of the channel ($x=40$ nm and $x=48$ nm) are also plotted.

Figure 6

Normalized potential of the tip [see Eq. (22) and the text below] for (a) $N=246$ and various $Q_{\mathrm{tip}}$. The dotted line shows the Lorentzian curve with $c=920$ nm and $d=40$ nm; (b) for fixed $Q_{\mathrm{tip}}$ and various $N$. The tip is localized above the axis of the channel and the potential plots are given along the channel. The actual potential is not isotropic.

Figure 7

Conductance maps for $N=44$ electrons per computational box (single subband $P=1$ transport) for $Q_{\mathrm{tip}}=+16e$ (a), $Q_{\mathrm{tip}}=+8e$ (b), $Q_{\mathrm{tip}}=+4e$ (c), $Q_{\mathrm{tip}}=-4e$ (d), $Q_{\mathrm{tip}}=-8e$ (e) for $B=0$.

Figure 8

Amplitude of the Fermi-level wave functions for parameters of Fig. 7 and tip positions corresponding to $G\simeq 0$, for (a)–(c) $Q_{\mathrm{tip}}=+16e$ [see Figs. 7a and 7b], and 7d, 7e, 7f $Q_{\mathrm{tip}}=-8e$ [see Figs. 7g and 7h. The tip positions are marked with the cross. In (c) and (d) the tip is outside the plots at points $(-350,960)$ nm and $(-416,1120)$ nm, respectively.

Figure 9

(a) [(b)] same as Fig. 7b [Fig. 7h] only for $B=0.0212$ T, corresponding to half of the flux quantum threading a one-dimensional ring with radius 176 nm.

Figure 10

Conductance maps for various values of $Q_{\mathrm{tip}}$ at $B=0$ for potential background of $N=134$ electrons within computational box for which two subbands $P=2$ appear at the Fermi level.

Figure 11

Conductance maps for 134 electrons per computational box (as in Fig. 10) only for $B=0.0212$ T with $Q_{\mathrm{tip}}=+16e$ (a) and $Q_{\mathrm{tip}}=-24e$ (b).

Figure 12

$B=0$ conductance maps for various values of $Q_{\mathrm{tip}}$. Results were obtained for 560 electrons per computational box, for which $P=4$ subbands appear at the Fermi level. (a),(b) and (d)–(f) have common $G$ scales.

Figure 13

Same as Fig. 12 only for $B=0.0212$ T for $Q_{\mathrm{tip}}=24e$ (a) and $-64e$ (b).

Figure 14

(a) Local density of states (LDS) at the Fermi level obtained for $N=44$ electrons within the computational box in the absence of the tip. (b)–(k) Conductance maps simulated for the Lorentzian ansatz of the tip potential [Eq. (23)] for $V_l=0.06$ meV (b)–(d), $V_l=-0.06$ meV (g)–(i), $V_l=0.46$ meV (e)–(f) and $V_l=-0.46$ meV (j),(k). The range of the Lorentzian is taken equal to $d=10$ nm (b),(e),(g),(j), 40 nm (c),(f),(h),(k), and 80 nm (d),(i).

Figure 15

Conductance maps for $N=44$ electrons per computational box (single band transport) for $Q_{\mathrm{tip}}=-e$ (a) and $Q_{\mathrm{tip}}=+e$ (b). For other values of $Q_{\mathrm{tip}}$, see Fig. 7.

Figure 16

(a) Local density of states (LDS) at the Fermi level obtained for $N=560$ electrons within the computational box in the absence of the tip. (b)–(i) Conductance maps simulated for the Lorentzian ansatz of the tip potential [Eq. (23)] for $V_l=0.92$ meV (b),(d),(f),(h) and $V_l=-0.92$ meV (c),(e),(g),(i) with $d=2$ nm (b),(c), 10 nm (d),(e), 20 nm (f),(g), and 40 nm (h),(i).

Figure 17

Conductance maps for $N=560$ electrons per computational box (single band transport) for $Q_{\mathrm{tip}}=-2e$ (a) and $Q_{\mathrm{tip}}=+2e$ (b). For larger values of $|Q_{\mathrm{tip}}|$, see Fig. 12