Nonequilibrium Green's function study of magnetoconductance features and oscillations in clean and disordered nanowires

We explore various aspects of magnetoconductance oscillations in semiconductor nanowires, developing quantum transport models based on the nonequilibrium Green's function formalism. In the clean case, Aharonov-Bohm (AB, $h/e$) oscillations are found to be dominant, contingent upon the surface confinement of electrons in the nanowire. We also numerically study disordered nanowires of finite length, extending the existing literature. By varying the nanowire length and disorder strength, we identify the transition where Al'tshuler-Aronov-Spivak (AAS, $h/2e$) oscillations start dominating, noting the effects of considering an open system. Moreover, we demonstrate how the relative magnitudes of the scattering length and the device dimensions govern the relative dominance of these harmonics with energy, revealing that the AAS oscillations emerge and start dominating from the center of the band, much higher in energy than the conduction band edge. We also show the ways of suppressing the oscillatory components (AB and AAS) to observe the nonoscillatory weak localization corrections, noting the interplay of scattering, incoherence/dephasing, the geometry of electronic distribution, and orientation of magnetic field. This is followed by a study of surface roughness which shows contrasting effects depending on its strength and type, ranging from magnetic depopulation to strong AAS oscillations. Subsequently, we show that dephasing causes a progressive degradation of the higher harmonics, explaining the reemergence of the AB component even in long and disordered nanowires. Lastly, we show that our model qualitatively reproduces the experimental magnetoconductance spectrum of Holloway et al. [Phys. Rev. B 91, 045422 (2015)] reasonably well while demonstrating the necessity of spatial correlations in the disorder potential and dephasing.

Figure 1

Device schematics. (a) The nanowire device and the leads, along with an axial magnetic field. (b) Cubic lattice structure of the nanowire with a surface confining potential. The region marked in red has lower electronic energy than the region marked in blue. This results in a surface confined electronic distribution (in the red region).

Figure 2

The unconfined case. (a) Variation of the transmission from the mean value for each energy, $\delta T(E,\mathrm{\Phi })=T(E,\mathrm{\Phi })-{\langle T(E,\mathrm{\Phi })\rangle }_{\mathrm{\Phi }}$. No AB oscillations are observed. At small fluxes, this resembles the Fock-Darwin spectrum, which quickly aligns with the Landau levels. (b) FFT spectrum of transmission for the same device, which again shows a lack of AB oscillations. Instead we have a broader peak, with frequencies smaller than that of the AB oscillations $[{(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}=1]$.

Figure 3

The surface confined case with $V_0=0.358t$ and $p=2$. (a) Variation of the transmission, from the mean value (with respect to flux) for each energy, with a parabolic radial potential. Diamond shaped structures with a period of $h/e$, synonymous with AB oscillations, can be observed at lower energies. At higher values of flux (roughly over $10{\mathrm{\Phi }}_0$), they start moving up, aligning with Landau levels. (b) FFT spectrum of the transmission, showing a peak at ${(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}=1$, i.e., the AB peak, at low energies. At higher energies, it gives way to lower frequency components.

Figure 4

Transmission traces/cuts at two different energies (chosen for illustration), with (a) a parabolic surface confining potential, and (b) no surface confining potential, corresponding to Figs. 2 and 3. The orange curves are taken at $E-E_c=1.9t$, and the blue curves are taken at $E-E_c=1.2t$, where $E_c$ is the lower band edge. Much better and sustained oscillations, with the AB period, are observed in the surface confined case shown in (a). The sharp steps are a consequence of using zero temperature. In the unconfined case shown in (b), sudden drops in conductance in the unconfined case arise due to the passing of energy bands over the Fermi energy, as they align with the Landau levels. Note that the boundary where this depopulation occurs for the unconfined case follows the same shape as seen in Fig. 2, where the transmission drops to zero.

Figure 5

$T(E,\mathrm{\Phi })-{\langle T(E,\mathrm{\Phi })\rangle }_{\mathrm{\Phi }}$ for $W=1.5t$ and lengths of the disordered section equal to (a) $100a$, (b) $125a$, (c) $150a$, and (d) $175a$, in nanowires with strongly surface confined electrons. For comparison, the strongly confined clean case is shown in Fig. 15. With increasing length, AB oscillations $[{(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}=1]$ weaken while AAS $[{(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}=2]$ oscillations become significant. This leads to a transition point where AAS oscillations start dominating AB. The FFT spectrums shown in Fig. 6 highlight the corresponding harmonic contents.

Figure 6

FFT spectrum of the variation in transmission (Fig. 5) from the mean for each energy, for $W=1.5t$ and lengths of the disordered section equal to (a) $100a$, (b) $125a$, (c) $150a$, and (d) $175a$, in nanowires with strongly surface confined electrons. With increasing length, AB oscillations $[{(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}=1]$ weaken while AAS $[{(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}=2]$ oscillations become significant. This leads to a transition point where AAS oscillations start dominating AB. This is seen more clearly in panels (e) $100a$ and (f) $175a$, where the limiting cases for the AB and the AAS FFT are shown with the corresponding energy averaged AAS magnitude shown in green (see Fig. 7). Here (a) and (e) show the limit where the AB oscillations dominate, while (d) and (f) show the limit where the strength of the AAS oscillations becomes significant and comparable to that of the AB oscillations.

Figure 7

FFT of the variation in transmission from the mean for each energy, averaged over energy, for $W=1.5t$, corresponding to the systems considered in Fig. 6. Note that for this disorder value, the nanowires are in the quantum diffusive/weakly localized regime for all energies. One can observe the transition point (shown by the dashed line) beyond which the AAS starts dominating over the AB oscillations. The increase in the magnitude of the AAS component is an effect of increasing phase coherence length with increasing nanowire length, as we are considering phase coherent transport here. Further, the residual AB component in long nanowires can be attributed to the leads, which are clean extensions of the device, and are subject to the same magnetic field.

Figure 8

$T(E,\mathrm{\Phi })-{\langle T(E,\mathrm{\Phi })\rangle }_{\mathrm{\Phi }}$ for $L=100a$ and a range of disorder strengths (a) $W=0.0t$, (b) $W=1.0t$, (c) $W=1.6t$, and (d) $W=1.9t$. One can observe the transition from the AB dominated regime to the AAS dominated regime on increasing $W$. This is inferred by counting the repeating features with increasing flux at any given energy, which changes from being one in panel (a) to two in panel (d). Also, in (c) the AAS harmonic starts dominating from the center of the band $(E=6t)$ while the AB harmonic is stronger near the band edges (only the upper band edge at $E=10t$ is shown here). This is explained by the variation of the scattering length over the band as shown in Fig. 11 and the following corresponding discussion. Further, the AAS oscillations bear the same phase relation at all energies.

Figure 9

The dominant harmonic contribution as a function of the disorder strength and device length ($\text{diameter}=10a$), as obtained from our numerical results. The boundary between the regions has been smoothly interpolated. The domain yielding a dominant AAS contribution is expected to increase upon using a nanowire with a larger diameter due to a reduction in the scattering length.

Figure 10

Magnetic field perpendicular to axis ($B_{\perp }$). (a) Transmission spectrum, and (b) its FFT, for a nanowire of length $125a$ with $W=1.5t$ and a strongly surface confined distribution. Note that $\mathrm{\Phi }=B_{\perp }\pi R^2$. We observe aperiodic UCFs, which vary much more slowly and randomly than the periodic AB oscillations.

Figure 11

Scattering length [in units of the lattice constant ($a$)] as a function of energy in the band, for (a) $W=1.0t$ and (b) $W=0.15t$ (disorder potential $\in [-W/2,W/2]$). The band edges are located at $2t$ and $10t$. We see that transport is more diffusive in the middle of the band than near the band edges. In order to achieve a transition from the diffusive to the ballistic regime within the band, we must consider a much weaker scattering potential. In (b), we have marked the length $L=150a$, which is the length of the nanowire used in Fig. 12. Note that the dips in the scattering length correspond to the Van Hove singularities, where the associated high density of states enhances scattering.

Figure 12

FFT spectrum of the transmission. (a) For a nanowire of length $150a$, with disorder potential given by $W=0.15t$. A significant AAS component is observed in the diffusive regime near the center of the band ($E=6t$), which subsides as we move higher in energy to the ballistic regime near the band edge ($E=10t$). Only in the range of energies where the scattering length is smaller than the nanowire length (marked in Fig. 11), which is roughly $E\in [3t,9t]$, do we observe a significant AAS component. Outside this range ($E>9t$ in this figure), the AAS component is much smaller than the AB component. (b) For the same nanowire as in (a), but with the on-site energies ($E=5t$ and $E=7t$) alternating on neighboring sites. Apart from the band gap at the center of the band, the AAS oscillations dominate in two separate regions around $E=7t$ and $E=9t$.

Figure 13

$\delta T(E,\mathrm{\Phi })=T(E,\mathrm{\Phi })-{\langle T(E,\mathrm{\Phi })\rangle }_{\mathrm{\Phi }}$. (a) Parallel field: Nanowire of length $175a$, $W=1.5t$, and strong surface confinement. We have introduced a finite temperature by averaging over an energy $\mathrm{interval}=0.31t$. (b) Perpendicular field: Nanowire of length $125a$, $W=1.5t$, and strong surface confinement, corresponding to the system in Fig. 10, with no energy averaging. As expected, we observe that the transmission initially rises with increasing flux, as the phase relationships between the disorder induced localized states are destroyed. Note that the magnitude of the weak localization correction is $\approx 0.1-0.3$, which can get easily subdued by the oscillatory components. Also, upon extending panel (a) to higher flux values, we recover oscillatory behavior which eventually starts dominating. But upon extending (b) we get the same transmission values as shown in Fig. 10 as it is the same system.

Figure 14

Surface roughness without on-site Anderson disorder. (a) $\delta T(E,\mathrm{\Phi })$, and (b) its FFT, of a clean nanowire of length $L=100a$, with surface roughness described by $\zeta =-2$ and ${\mathrm{\Delta }}_R=0.4R_0$. We observe a degradation of the AB component, accompanied by the dominance of the AAS component in general introduced primarily by the hopping disorder (except at $E=7.1t$, which shows up as an odd AB peak in the FFT spectrum, at that energy). The AAS contribution is clearly visible in Fig. 14. Note that we do not observe magnetic depopulation, as seen in Fig. 15. Further, as explained in the text, the skewed hopping disorder reduces the bandwidth of transmission.

Figure 15

Surface roughness without on-site Anderson disorder. (a) Variation of the transmission from the mean value for each energy $\delta T(E,\mathrm{\Phi })$, and (b) its FFT, without SR, in a nanowire with strong surface confinement for comparison. The “discretized” diamonds are due to the presence of a finite number of states. In (c), we see $\delta T(E,\mathrm{\Phi })$, and (d) its FFT, of a clean nanowire of length $L=100a$, but with surface roughness, described by $\zeta =0$ and ${\mathrm{\Delta }}_A=0.4A_0$. Note that in panels (b) and (d), the energy axis goes into the page. Here $E=6t$ and $E=10t$ are the band center and the upper band edge, respectively. A degradation of the sharp features is observed in the transmission spectra as a result of SR, which is manifested as a smoother decay of the corresponding FFT spectrum. The strong low frequency peaks $[{(\mathrm{\Phi }/{\mathrm{\Phi }}_0)}^{-1}<1]$ correspond to the slow drop in transmission due to magnetic depopulation. The magnitudes of the harmonics have been found to be the same. Further, at higher values of flux, the AB oscillation diamonds have degraded.

Figure 16

Dephasing: Clean case. FFT spectrum of the variation in transmission from the mean value for each energy $\delta T(E,\mathrm{\Phi })$, (a) without dephasing and (b) with momentum relaxing dephasing in a clean nanowire of length $L=50a$ with strong surface confinement. The energy axis goes into the page. The difference between (b) and (a) can be observed by comparing the decreased magnitudes of the Fourier components in (b).

Figure 17

Dephasing: Disordered case ($W=0.75t$). (a) Variation of the transmission from the mean value for each energy $\delta T(E,\mathrm{\Phi })$, and (b) its FFT, without dephasing, in a disordered nanowire of length $L=50a$ with strong surface confinement. In (c), we see $\delta T(E,\mathrm{\Phi })$, and (d) its FFT, of the same nanowire, but with momentum relaxing dephasing ($D=0.4^2$). The energy axis goes into the page. A degradation in the amplitudes of the harmonics is observed in the FFT spectrum [from (b) to (d)]. Further, the degradation for the AB peak is smaller than the higher harmonics, increasing its dominance. Consequently, the distinct diamond shaped structures representing the AB oscillations become the dominant feature in the transmission spectrum. Here $E=6t$ and $E=10t$ are the band center and the upper band edge, respectively.

Figure 18

$\delta T(E,\mathrm{\Phi })=T(E,\mathrm{\Phi })-{\langle T(E,\mathrm{\Phi })\rangle }_{\mathrm{\Phi }}$, for each energy for a parabolic transverse potential/weak surface confinement. (a) In the absence of any disorder potential $(W=0)$. (b) In the presence of a spatially uncorrelated disorder with $W=1t$. The angular momentum subbands are observed. (c) In the presence of spatially uncorrelated disorder with $W=1t$ as well as dephasing characterized by $D=0.4^2$, resulting in smoother variations. (d) In the presence of a spatially Gaussian correlated disorder with standard deviation $\sigma =2a$ and $W=1t$. (e) Same as (d), but with dephasing characterized by $D=0.4^2$. Note that in all the panels, the conduction band edge (not shown) lies at $\approx -0.1t$.