Since the discovery of conductance quantization in GaAs/AlGaAs quantum wires, enor-mous amount of theoretical works have been contributed to discuss the transport in the presence of impurities. Most of them were numerical calculations, since it is difficult to acquire analytical resolutions. However, similar results and consensuses were deduced about the effect of impurity on the transport. These features are rephrased in the following briefly:
Conductance quantization is degraded.
Conductance of plateaus is lowered.
Additional structures occur due to multiple scattering electron waves.
The fine structures are sensitive to the impurity configurations.
Quasibound states are developed and conductance resonance appears.
Subband energy may be shifted.
In the following, some of the most prominent proposals and numerical works are reviewed in detail. Due to the effects of impurities, the conductance quantization is degraded in the forms of lowered plateau conductance and smearing of steps. Addi-tionally, additional fine structures due to multiple scatterings of electron waves were suggested.[19, 22, 93, 94, 95, 98, 100, 102] Chu et al. considered an isotropic scatterer placed 5 nm away from the center of a 1D channel. [19] Making use of quantum scatter-ing theory and Landauer formula, the conductance as a function of channel width was
Chap. 4 Conductance and spectroscopy of disordered quantum wires 47 calculated. Conductance dips are perceptible for a weak attractive impurity. [Fig. 4.1(a)]
Resonant peaks and plateau smoothening are visible for stronger impurity. [Fig. 4.1(b)]
For a repulsive scatterer, only plateau smoothening was perceptible. The conductance is lower than multiples of 2e2/h of perfect transmission (dashed lines). Similar results were also reported by Bardarson it al. in a more recent report for a spatially extending Guassian-type impurity.[100] Tekman et al. reported more thorough numerical calcula-tions of conductance of QW with different geometries. The influences of surface roughness, quasibound states, and impurities on transport were also investigated by the authors.[98]
(a) (b)
Figure 4.1: Conductance G for an electron waveguide plotted as a function of the width W of the channel for an attractive impurity. (a) Phase shift δ0 = 30◦. (b) δ0 = 60 and 90◦. (After Chu et al., Ref. [19].)
When electrons are scattered by a defect in a quantum channel, there is probability that the incident wave is scattered to higher modes which can be evanescent. Bagwell proposed that the electrons scattered to the evanescent modes are effectively localized since the wavefunction decays spatially rather than propagates. Therefore quasibound states develop accordingly due to the presence of impurity.[20] Linear conductance exhibits peaks and dips reflecting the transmission resonances in the presence of an impurity.
Similar propositions about the quasibound states were also made by Gurvitz et al.[96] and Levinson et al.[99] from alternative calculation approaches and for more general cases of impurities. Tekman et al. demonstrated that the conductance reveals resonant tunneling
peaks when there were quasibound states as shown in Fig. 4.2.[98]
Figure 4.2: Resonant tunneling effects on the conductance of a QPC due to the widening, the potential well, or attractive impurity at the center of the constriction. (After Tekman et al., Ref. [98].)
Vargiamidis et al. considered a Guassian-type impurity in an infinitely long QW. The impurity was modeled as Vi(x, y) = 2m~2γ∗δ(y−yi)e−x2/d2. y is along the transverse direction, while x is along the longitudinal direction.[101] As shown in Fig. 4.3(a), the authors demonstrated that the conductance step shifts to more lower energy as the impurity is displaced away from the center of the QW (from yi/w = 1/12 to 5/12). The results indicated that the subband bottom is altered with respect to the position displacement of impurities. The subband shifting with respect to impurity configurations and positions was also reported in Ref. [105] by Takagaki et al., Ref. [93] by Marel et al., and Ref.
[98] by Tekman et al.. In Ref. [105], the quantum channel was divided in to Nx × Ny = 32× 32 two dimensional grids. Incorporating arbitrarily long range impurities Vimp(r) = Uiexp(−|r − r0|/rd) where Ui was randomly distributed between (−Γ/2, Γ/2) across the lattice sites, the mode matching method and transfer matrix methods gave the conductance of the QPC. As shown in Fig. 4.3(b), the quantization step shifts and the line shape varies for different impurity configurations.
Hitherto, the discussions of impurity are without a priori knowledge of its physical origins. It was sensible that scientists attempt to investigate the influence of physical defects on transport. One of the most intuitive and unavoidable sources of scatterers is the ionized donors in the doping layer separated from 2DEG by a barrier. Adopting Thomas-Fermi approximation and considering randomly distributed donors on the δ-doping layer, Nixon et al. numerically calculated the averaged potential in 2DEG.[21, 22]
Chap. 4 Conductance and spectroscopy of disordered quantum wires 49
(a) (b)
Figure 4.3: (a)G vs. EF through a Gaussian impurity of strength γ = 2.5× 106cm−1 and d=0.7w in a QW of width w, for three different impurity positions (yi/w). (After Vargiamidis et al., Ref. [101].) (b) The conductance of a QPC in the presence of long-range disorder. The dotted line represents the conductance for a perfect sample, while different impurity configurations are assumed for the solid lines. The spatial extension and number of scatterers are characterized by rd/a (After Takagaki et al., Ref. [102].)
The potential is not uniform, and contains fluctuations as shown in Fig. 4.4(a). When the length of the channel was longer than the correlation lengths of fluctuations, the conductance quantization was predicted to break down. Similar arguments were made by Takagaki et al. in Ref.[102]. It is possible that a potential minimum exists inside the QPC due to the potential fluctuations [Fig. 4.4(a)], and conductance resonance stands a chance. As shown in the two lower traces of G(Vg) in Fig. 4.4(b), conductance peaks and dips are visible due to the existence of local potential well. The authors also suggested that the ionized donors redistribute after a thermal cycling and a different potential map would be produced.
The effects of boundary roughness on the transport have also been studied for quite some time.[94, 95, 106, 107] Recently, Csontos et al. modeled the roughness by dividing a QW into a large amount of segments, and applying randomly distributing weak variation of width along the transport direction.[107] The average width variation was 8 nm. With increasing QW length, the step conductance was lowered, conductance dips appeared at the subband energy, and fine conductance oscillations were visible.[Fig. 4.5] The authors suggested that the conductance dips result from the intraband scattering at the subband
Figure 4.4: (a) Gate pattern on surface and density of electrons in 2DEG for a point contact with gap of 0.3 µm and length of 0.2 µm. Contours start from zero and are 4.2× 1014m−2 apart, corresponding to an energy spacing of 1.5 meV. (b) Conductance G as a function of gate voltage Vg for a 0.2 µm QPC. A is in the absence of potential fluctuations. The other curves are for different impurity configurations. (After Nixon et al., Ref. [22].)
thresholds.
Figure 4.5: Conductance calculated at T = 0 K for 100 nm wide quantum wires of various lengths. For each length, three curves correspond to three different roughness distributions. (After Csontos et al., Ref. [107].)