Figs. 4.7(a) and 4.7(b) show the linear conductance of two unintentionally disordered QWs. Traces of G(Vg) are plotted against the lateral shifting voltage ∆Vg and are offset in turn in 50 mV steps. The plateau conductance is much lower than the expected integer multiples of G0 in both devices indicating that electrons are partially backscattered. As a consequence, more steps are present in a smaller conductance range. For instance, more than eight steps are visible in the range of 0<G<5G0 on the ∆Vg =0 trace for sample A in Fig. 4.7(a). The higher subband conductance shows slight distortions. Sample B is much more disordered with lower quantization conductance, more distortions, and more missing steps. One thing worth being noticed is that the traces are expected to be evenly spaced if the subband energies remained unchanged with respect to the position of the impurities, i.e. the pinch-off voltage should be unchanged. However, in both figures, traces are closer to each other at the positive ∆Vg side but looser at the other. In a clean QW (discussed in the latter paragraphs), the conductance traces are more evenly spaced. This indicates that the threshold energies of subbands vary with the position of impurities, or effectively with the impurity configurations. In addition, a couple of conductance resonances are observable for G < G0 for sample B in Fig. 4.7(b). The results are in congruence with the prediction that the quasi-bound states in a QW given rise by the impurities lead to conductance resonance and alter the subband threshold voltage.[19, 98, 99, 96, 100, 101]
The source-drain bias spectroscopy was investigated by sweeping a dc source-drain voltage Vsd across a confined QW and measuring the dynamic conductance G = dI/dVsd. Fig. 4.7(c) shows the spectroscopy, G(Vsd) against a series of Vg, of sample A at ∆Vg =-0.6 V. It demonstrates a series of single peak centered at Vsd=0 for G < G0, which resembles the ZBA of QWs. However, this structure is not the same as the ZBA of QWs since the single peak evolves into double splitting peaks by shifting the QW to the other side at ∆Vg = +0.6 V as shown in Fig. 4.7(d). On the contrary, splitting of the ZBA was not observed in a clean QW against various ∆Vg as shown in Figs. 4.10(b)-4.10(c). The energy difference of the splitting in Fig. 4.7(d), ϵd, is not a constant. ϵd decreases with decreasing Vg for 0.72G0 < G < G0, and begins to increase for G . 0.72G0. The peak
Chap. 4 Conductance and spectroscopy of disordered quantum wires 53
Figure 4.7: G(Vg) and soure-drain bias spectorscopies for two unintentionally disordered QWs at 0.3 K. (a) Sample A. G(Vg) versus the differential gate voltage. ∆Vg=+0.6 (leftmost, labeled by a star) ∼-0.6 V(rightmost, labeled by a diamond) in 0.2 V steps.
The traces are offset in turn by 0.05 V. Inset: Micrograph of the disordered QWs A and B.
The white scale bar equals a length of 1 µm. (b) Sample B. G(Vg) versus ∆Vg for the same conditions, and traces are offset by the same manner. (c) Source-drain bias spectroscopy of sample A for ∆Vg=-0.6 V. Vg=-900∼-969 mV in 3 mV steps. (d) Source-drain bias spectroscopy of sample A for ∆Vg=+0.6 V. Vg=-900∼-960 mV in 3 mV steps.
splitting was generally observable in the disordered QWs, and is believed not to be an inherent feature of QWs, such as the ZBA.
In the presence of impurities, electrons have the probability to be scattered into non-propagating evanescent modes. The mechanism effectively creates localized electrons close to the impurities. Quasibound states would be developed and conductance resonances are expected.[20, 96] Fig. 4.8 presents results of a disordered QW that has strong tunneling resonances. Here, a different voltage biasing approach was adopted. Vg was applied on the upper gate and Vg+ ∆ ˜Vg was applied on the lower one. Weak conductance oscillations are discernible along the conductance trace of ∆ ˜Vg =0 in Fig. 4.8(a), besides the five quantized plateaus. A strong resonant peak emerges by shifting the QW to ∆ ˜Vg ≤ −0.1 V, while the peak conductance tends to pin at 0.5G0. The superimposing resonances are stronger for the negative ∆ ˜Vg. The effect of resonances on the source-drain bias spectroscopy is presented in Figs. 4.8(c)-4.8(e) for ∆ ˜Vg = +0.3, 0, and−0.3 V. Surprisingly, The ZBA of QWs is seen in all three spectroscopies for G < G0, coexisting with the splitting peaks due
to the resonant states. The concurrence of the ZBA and resonant peaks forms a triple-peak structure which was also noticed by Sfigakis et al..[52] E.g. two strong satellite triple-peaks appear at the sides of the ZBA at Vsd ∼ ±0.8 mV on the dot-labeled curve of Vg=-383 mV in Fig. 4.8(d). The amplitude and width of the ZBA are affected by the resonant peaks.
The satellite peaks appear at larger|Vsd|, e.g. at ∼1.9 mV in Fig. 4.8(c) of ∆ ˜Vg=+0.3 V, and at ∼1.8 mV in Fig. 4.8(e) of ∆ ˜Vg=-0.3V. Comparing Fig. 4.8(d) with Figs. 4.8(c) and 4.8(e), the ZBA is weaker and suppressed more rapidly with decreasing Vg at ∆ ˜Vg=0 in Fig. 4.8(d) . The result indicates that the existence of the resonant peaks due to the resonant states nearby the Vsd=0 suppresses the ZBA. The evolution of the split peaks with respect to Vg, e.g. as the dashed lines in Fig. 4.8(d), resembles the diamond structure of the tunneling spectroscopy of zero-dimensional states. Similar evolution of the satellite peaks is also observable in Fig. 4.8(e) for either G < G0, or G0 < G < 1.75G0 for the higher subband.
Interestingly, the triple-peak structure appears also at the higher conductance regime for 1.55G0 < G < 1.75G0 at ∆ ˜Vg=-0.3V. The central peak is weaker compared with the strong single peak observed in the same range in the clean QW in Fig. 4.9(d). The appearance of the central peak at such high conductance regime implies that the ZBA is not unique for low conductance regime on which most groups focused. Notice that groups of crossed traces appearing in all three graphs indicates strong conductance resonances in finite Vsd regime. Consider a QW in the presence of resonant states induced by impurities as shown in Fig. 4.8(b). Aligning of a resonant level with the chemical potential of the source or drain increases the tunneling probability, as well as the differential conductance G = dVdI
sd . Finite Vsd would be required to align the chemical potential with a resonant state when the QW is not initially in resonance, resulting in the evolutions of splitting peaks with respect to Vg in the spectroscopies and conductance resonances at finite Vsd. Although this is an oversimplified scenario, it provides a qualitatively appropriate expla-nation to our results.
Linear conductance, G(Vg), of a quasi-zero in length QW in two separate cooling pro-cesses are shown in Figs. 4.9(a)-4.9(b). Both traces reveal non-oscillating conductance
Chap. 4 Conductance and spectroscopy of disordered quantum wires 55 con-ductance oscillations at 0.3 K. ∆ ˜Vg=+0.5 (leftmost)∼-0.5 V (rightmost) in 0.1 V steps.
Inset: Micrograph of the device. The scale bar has a length of 0.4 µm. (b) Scenario for the observed split peaks due to the presence of resonant levels in a QW. The thick lines stand for the last two subbands of a QW; the dotted lines stand for the resonant levels induced by impurities. (c)-(e): source-drain bias spectroscopies for various ∆ ˜Vg. (c) +0.3 V, (d) 0 V, and (e) -0.3 V. Dashed lines are visual guides for the evolution of conductance peaks as functions of Vg and Vsd.
and clear quantization steps without distortions, indicating that the sample is low disor-dered. The plateaus can be aligned exactly to the integer multiples of G0 by subtracting a serial resistance of about 300 Ω from the data, such as in [67] by Thomas et al. We present the raw data here since the series resistance does not affect our results. Fig. 4.9(c) shows the spectroscopy in the first cooling operation. The dynamic conductance exhibits clearly a series of single peak centered at Vsd=0 for G < G0 referring to as ZBA. The ZBA is reproducible after thermal cycling, as shown in Fig. 4.9(d) for the second cooling process. The half plateaus and the 0.7 anomaly at finite biases are visible as bunches of curves (opposed to crossing traces due to resonant states) at G∼1.5G0 and G∼0.8G0, for
Vsd >1.4 mV and Vsd >0.7 mV, respectively.
Figure 4.9: Zero-bias differential conductance G vs. split gate voltage G(Vg), (a)-(b), and source-drain bias spectroscopies, (c)-(d), in two separate cooldowns at 0.3 K of a clean quantum wire. Traces of G(Vsd) in (c) and (d) are for discrete consequential split gate voltages in 4 mV steps. (c) Vg=-1.450∼-1.558 V. (Top to bottom) (d) Vg=-1.720∼ -1.944 V. Inset in (a): Micrograph of the device. The white scale bar indicates a length of 0.5 µm.
Since 2DEG is generally not free from disorder[113], lateral shifting was applied to fur-ther scrutinize the quality of the sample. Fig. 4.10(a) shows conductance traces against a series of ∆Vg. Similarly, the traces are offset in turn by 50 mV step. For ∆Vg=0, the trace demonstrates more than fourteen clear quantization steps. The traces are much uniformly spaced compared with Figs. 4.7(a) and 4.7(b). It implies that the pinch-off voltages are the same for different ∆Vg. A slight conductance oscillation appears around the sixth and seventh steps for ∆Vg =+0.2 and +0.4 V. Except this small disturbance, the conductance quantization remains unaltered with respect to the lateral shifting, confirming that the device is quite clean. Figs. 4.10(b)-4.10(c) show the spectroscopies of the QW at two opposite positions, ∆Vg= +0.6 and -0.6 V. Both figures demonstrate ZBA for G < G0. The ZBA is slightly weaker in Fig. 4.10(b), however, there is no additional structure, such as satellite peaks, compared with the disordered QWs. The varying ZBA character-istics are understandable. It is known that the charactercharacter-istics of the ZBA are sensitive to electron scattering, which is affected by the local electric field, density, or physical
Chap. 4 Conductance and spectroscopy of disordered quantum wires 57 geometry of a device.[114] For a pair of ideally symmetric split gates, one would expect the ZBA to be identical for these two ∆Vg. However, we believe that the deviation is due to the asymmetry of the realistic gate edge, which is beyond the experimental resolution.
The random potential fluctuation due to ionized donors is another possibility that causes asymmetric local field.[22] Nevertheless, the result shows that the ZBA is a rather robust feature of low-disordered QWs.
Figure 4.10: G(Vg) of the clean quantum wire against differential gate voltage ∆Vg. From left to right, ∆Vg=+1.0∼-1.0 V in 0.2 V steps. The traces are offset in turn by 0.05 V for clarity. (b)-(c): source-drain bias spectroscopies for ∆Vg= -0.6 and +0.6 V respectively.
(b) Vg=-1.846∼-1.934V in 4 mV steps. (c)Vg=-1.848∼-1.964V in 4 mV steps.
To further confirm the influence of impurities on the transport in QWs, a ‘standalone’
(opposed to the closely packed samples) pair of 0.3 µm split gates intentionally imposed with disorder is also studied. As shown in the inset of Fig. 4.11(a), the QW contains a small local widening. The electric potential generated by the applied gate voltage is expected to be lower in the widened area. Electrons traveling across the QW would be backscattered due to mode mismatch resulting in effectively an attractive impurity in a QW. We estimate the additional depletion region besides the lithographic area to be 75∼100 nm from other experiments. The device is not expected to be developing a quantum dot, since the dent is also about 100 nm in length. The argument is confirmed by the absence of charging effect for G < G0. In Figs. 4.11(a) and 4.11(b) of two
cooling processes, only the first quantized plateau sustains in the linear conductance. For G > G0, conductance plateaus were distorted and missing. The fine structures of linear conductance are different between thermal cyclings. The traces in Fig. 4.11(b) have several resonances which are weak in Fig. 4.11(a).
The spectroscopy appears to be altering between the thermal cycling and presenting features of disordered QWs. In Fig. 4.11(c), double splitting-peaks are observable for G < G0 and the energy difference of splitting decreases with decreasing Vg. The ZBA is not observable. On the contrary, the single peak structure is visible for 0.84G0 < G < G0
in Fig. 4.11(d) for the second cooling operation. The conductance shoulder visible at
∼ 0.74G0 at zero-bias rises with increasing Vsd, and becomes visible as a resonant peak for Vsd ? 0.88 mV (crossings of traces). The deviations between two cooldowns are comprehensible by the fact that the ionized donors redistribute between thermal cycles, resulting in the different averaged potential fluctuations [22] which alter the impurity arrangement, and correspondingly the interference between the backscattered electrons.
-1.4 -1.2 -1.0 -0.8
Figure 4.11: G(Vg), (a)-(b), and source drain bias spectroscopy ,(c)-(d), in two separate cooldowns of a QW with a small local widening at 0.3 K. Inset in (a): micrograph of the device with the white scale bar indicating a length of 0.5 µm. (c) Vg=-1.140∼-1.248 V in 4 mV steps. (d) Vg=-0.780∼ -0.845 V in 3 mV steps.
The ZBA is a peculiar phenomenon of QWs. One dimensional Kondo model, which was based on the assumption of a spin dependent localized potential resulting in a spontaneous
Chap. 4 Conductance and spectroscopy of disordered quantum wires 59 magnetic impurity, was proposed to explain the experimental findings.[37, 48, 49, 50, 51] According to the 1D Kondo model, ZBA was expected to split in parallel magnetic fields while the splitting width was about the Zeeman energy. More recently, however, several groups reported more intricate features of the ZBA.[52, 53, 54] In some devices, splitting occurs in magnetic fields, but two split peaks can resolve back into a single peak by laterally shifting the QW. In some other devices, splitting even occurs in zero magnetic fields.[53, 54] Chen et al. claimed that disorder is related with the splitting of the ZBA in magnetic fields.[53] Sarkozy et al. suggested that resonant backscatterings and length resonances could result in split ZBA.[54, 115] As we demonstrated here, impurities presenting in QWs indeed lead to these complicated features.
4.5 Conclusions
In conclusion, ZBA is robust in clean QWs against lateral shifting and thermal cycling.
On the other hand, impurities can cause resonances leading to complicated source-drain spectroscopies. These resonant features are sensitive to thermal cycling and lateral shifting of QWs revealing the random nature of impurities in ballistic QWs. The double split peaks due to resonant levels affect the characteristics of ZBA. The results indicate that cleanness is crucial for studying the intrinsic behaviors of ZBA in quasi-one dimensional systems.
Time dependent electric fields
generated DC currents in a large gate-defined open dot
5.1 Introduction
In my PH.D research, efforts were also devoted to realizing quantum charge pump in an quantum dot (QD). Although whether genuine quantum charge pumping was real-ized is still controversial, interesting result is found in our investigation. A charge pump in general generates a dc current by driving a system with two out of phase ac electric fields by analogy with a water pump.[116] The classical charge pump in a closed QD has been realized for quite a long time.[117] In a closed QD where the potential bar-riers between the leads and the dot are oscillating out of phase in the same frequency, electrons can transport discretely. A finite external bias is required to develop poten-tial difference between the source and drain so as to provide current. Classical charge pump was also realized in a one dimensional semiconductor wire recently, while a travel-ing wave-like potential was generated to drive the electrons without bias.[118] Besides the classical charge pump, pumping charge quantum mechanically was of particular interests since the current is expected to be dissipationless, i.e. the power consumption would be
60
Chap. 5 Time dependent electric fields generated DC currents in a large gate-defined open dot 61 infinitesimal.[119] The experimental report of adiabatic quantum pump in an open QD has intrigued huge amount of theoretical works.[120] The wavefunction of an open QD configured in e2/h corresponding to one transmission mode in the source and drain leads for each were periodically deformed. However, it turned out that the results were confused with another classical mechanism, rectification, which is basically a spurious electrical cir-cuitry effect.[121] Dicarlo et al. claimed that at high enough frequencies photovoltaic effect which is also a quantum mechanical effect dominates. On the other hand, at low frequencies it is rectification that told the story.[122, 123] Rectification current is induced from the coupling between time-varying conductance and chemical potentials of reservoirs of a QD. To the best of our knowledge, experiments are still scarce and quantum charge pump has not yet been truly accomplished for the time this dissertation is prepared.
Technically speaking, charge pump in mesoscopic devices converts time-dependent signals to dc. For two dimensional systems, H¨ohberger et al. and M¨uller et al. used interdigitated ratchets fabricated on top of a large 2DEG to produce dc currents.[124, 125]
These techniques may be useful, since such transformations may serve the operations of power rectification and signal processing in integrated circuits[126] or signal modulation in telecommunications[127]. Therefore, it is important to understand comprehensively those interfering effects before further practical and industrial applications in nanoscale semiconductor devices.
In this chapter, we present the performance of generated dc current Idc by two fast oscillating electric fields in an open dot for two different electrical configurations. In the first setup, the ac voltages are applied on two side gates, resembling the typical charge pump; in the other, one ac voltage is relocated to the source lead, alike the condition of rectification. Both setups demonstrate a sinusoidal dependence of Idc on the phase difference between the two ac voltages, ∆ϕ. Additionally, Idc is proportional to the product of the voltage amplitudes. However, the dependences on frequency, coupling strength, and magnetic field in the two setups show opposite features. The results indicate that the currents are generated by different mechanisms, but not any circuitry effects for the pumping-type current.