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Scanning gate imaging of a disordered quantum point contact
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 193202 (http://iopscience.iop.org/0953-8984/26/19/193202) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.97.90.221 This content was downloaded on 01/05/2014 at 12:37
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 193202 (13pp)
doi:10.1088/0953-8984/26/19/193202
Topical Review
Scanning gate imaging of a disordered quantum point contact N Aoki1, C R da Cunha2, R Akis3, D K Ferry3 and Y Ochiai1 1
Graduate School of Advanced Integration Science, Chiba University, 1–33 Yayoi-cho, Inage-ku, Chiba 263–8522, Japan 2 Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91501–970, Brazil 3 School of Electrical, Computer, and Energy Engineering, and Center for Solid State Electronics Research, Arizona State University, Tempe, AZ 85287–5706, USA E-mail: [email protected] Received 9 November 2013, revised 16 January 2014 Accepted for publication 17 January 2014 Published 25 April 2014 Abstract
Scanning gate microscopy (SGM) is a novel technique that has been used to image characteristic features related to the coherent electron flow in mesoscopic structures. For instance, SGM has successfully been applied to study peculiar electron transport properties that arise due to small levels of disorder in a system. The particular case of an InGaAs quantum well layer in a heterostructure, which is dominated by a quasi-ballistic regime, was analyzed. A quantum point contact fabricated onto this material exhibits conduction fluctuations that are not expected in typical high-mobility heterostructures such as AlGaAs/GaAs. SGM revealed not only interference patterns corresponding to specific conductance fluctuations but also modedependent resistance peaks corresponding to the first and second quantum levels of conductance (2e2/h) at zero magnetic field. On the other hand, clear conductance plateaus originating from the integer quantum Hall effect were observed at high magnetic fields. The physical size of incompressible edge channels was estimated from cross-sectional analysis of these images. Keywords: canning gate microscopy, quantum point contact, disordere, conductance fluctuations, quantum Hall effect S Online supplementary data available from stacks.iop.org/J.PhysCM/26/193202/mmedia (Some figures may appear in colour only in the online journal)
1. Introduction
were observed due to the standing wave formed between the saddle point potential of the QPC and the point potential underneath the tip [6]. More recently, SGM has been applied to study many other systems. For instance, in studies of the coherent quantum structures, fractional conductance steps in a ballistic QPC [7], conductance fluctuations and modal characteristic peaks in a disordered QPC [8], fingerprint of potential fluctuations in nano-wire [9], fractal conductance fluctuations [10] and characteristic electron states in open quantum dot cavities [11, 12], and electron interferences caused by backscattering from a QPC [13] have been imaged. SGM has allowed many phenomena to be visualized such as the current
Scanning gate microscopy (SGM) is a novel tool that has been used to visualize the real space coherent electron flow in mesoscopic devices. In this technique, variations in conduction of a device are monitored while the tip of an atomic force microscope (AFM) is used as a mobile-point-gate electrode scanning an area of interest. The technique was firstly used to image the modal structure of quantum waves that flow through a quantum point contact (QPC) producing integer multiples of 2e2/h (G0) [1–5]. In the course of the study, periodic fringing patterns along the ballistic and coherent current flow paths 0953-8984/14/193202+13$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
flow paths based on the electron focusing effect [14], magnetic field dependent images based on the Aharonov–Bohm effect [15, 16], transition in the quantum Hall (QH) regime [17–19]and conductance steps due to the depopulation of the edge states [20], concentric ring patterns corresponding to Coulomb oscillation by the Coulomb blockade effect in closed quantum dots [21, 22] and an isolated Coulomb island in the quantum Hall regime [23, 24].Theoretical simulations have recently supported quantum transport during SGM observations [25]. The technique has recently been spread to the study of soft materials such as carbon nanotubes [26], organic semiconductors [27], graphene [28, 29], and many others. In this topical review, we will concentrate on the description of our SGM studies dealing with the imaging of coherent electron systems in QPCs fabricated in a slightly disordered InAlAs/InGaAs/InAlAs quantum well. Due to the disorder, this system exhibits a set of patterns in the SGM images which are different from those found in usual QPCs fabricated in high-mobility wafers [13, 30, 31]. Our samples were prepared specifically for performing SGM measurements inside the QPCs that were defined by V-shaped trenches etched through the heterostructure to isolate regions of the InGaAs layer. These were fabricated using electron beam lithography (EBL) and wet etching as shown in figure 1(a). As a consequence of using a narrow gap semiconductor in conjunction with trench-defined structures, our QPCs do not have a saddle point potential at its constriction and this allows us to obtain important images of this region. The details related to the sample preparation and the experimental microscopy techniques used for studying the QPCs will be presented in section 2. The SGM results of the QPCs fabricated on our basic InGaAs material, at zero magnetic field, are discussed in section 3. From these results, we conclude that this material is moderately disordered, either due to the presence of impurities or due to the random potential in the alloy (or both). The QPC shows typical conductance quantization at around 4 K. However, such conductance steps are hidden by conductance fluctuations (CFs) at lower temperatures. Such conductance fluctuations are common in disordered material. Moreover, the CFs are reflected in SGM images as interference patterns. Nevertheless, characteristic peaks depending on specific propagating modes through the QPC are observed when the conductance is settled near G0 (or a few G0). Away from these plateaus, however, we find that the CFs arising from the quantum effects of the disorder dominate the image. These interference patterns, which lead to the CFs, are thought to originate from a background random potential, mentioned above. As the side gate voltage is varied, the channel width changes and this affects the actual random background potential that appears at the Fermi level, and this alters the spatial position of the CF peaks. We will see that a fast-Fourier transform (FFT) analysis reveals that the mean size of the CF patterns is comparable to that found from the same analysis carried out on CF that arise as the magnetic field is varied. This agreement suggests that they correspond to quite similar interference patterns. On the other hand, near pinch-off, the positions of the SGM features are unaltered and
(a)
Vtip: –0.5 V
(b)
V Vsd Weakly depleted point Underneath tip (~100 nm wide)
(c)
Isd: 10 nA
Vg: 0~ – 7.5 V Ref.
Tip LT
A
B
80 mV 18 kHz
Vtip Figure 1. (a) Topographic AFM image around the QPC region of the sample. The scanning area is 2.8 × 2.8 μm2. The depth of the trenches is ~100 nm [8]. Copyright 2005, American Institute of Physics. (b) Schematic view of the 2DEG plane and the configuration during SGM measurements. The channel region is defined by in-plane isolation (black regions). Four terminal resistance measurements were performed by applying Isd of 10 nA. Negative side gate voltage was applied to the 2DEG regions surrounded by trenches to change the transmission of the QPC. The 2DEG underneath the tip was slightly depleted due to the application of a weak negative voltage (Vtip) on the tip. The scanning tip causes a local depletion region and changes the transmission through the QPC. (c) Electric circuit for operation of a piezoresistive cantilever in the SGM observation. The wheatstone bridge holding the tip is isolated by audio-trans from an oscillator (18 kHz). The potential of the tip can be defined by a DC-voltage source. The differential signal (VA–B) in the bridge is detected by a lock-in amp. and the value is used for feedback operation in AFM observation. The broken line indicates a part settled at low temperature.
increase only in intensity (resistance), which serves to unmask the random background potential. That is, near the conductance plateaus, the fact that the background conductance varies 2
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
Resistance [ ]
(a) 400 300
200
100
0
0
1
2
3
4
5
6
7
3.5
0.16
2
Conductance [2e /h]
3
0.12
2.5 2
0.08
1.5
0.04
1
0
0.5 0
2
(b)
-0.04 -8
-7
-6
-5
-4
-3
-2
-1
0
Conductance Fluctuations [2e /h]
Magnetic Field [T] Figure 3. Typical SGM image of the QPC region taken at 280 mK. The image was obtained at around 1G0 and processed to obtain the absolute value of the variations to emphasize the featured characteristics indicated by the arrows (i)–(iii). The dotted lines represent the outer edges of the trenches. The inset indicates a typical SGM image of the GaAs-QPC obtained at the same temperature (scan range 1 × 1 μm2).
2. Sample preparation and imaging technique for SGM studies 2.1. Sample preparation for SGM
Side Gate Voltage [V]
The sample was fabricated with a multi-layer heterostructure composed of (from the top) n-In0.53Al0.47As (5 nm, cap layer)/ n-In0.53Al0.47As (30 nm, doped layer)/ semi-insulating (si)-In0.53Al0.47As (10 nm, spacer layer)/ si-In0.53Ga0.47As (25 nm, the active quantum well)/ si-In0.53Al0.47As (10 nm, spacer layer)/InP (substrate). In this configuration, the quantum well is located between 45 nm and 70 nm below the surface. Electrons transfer from the doped layer into the quantum well, a process known as modulation-doping. As discussed below, measurements show that two levels of the quantum well are occupied at low temperature. The mobility found from the total carrier concentration was 7.4 × 104 cm2 V−1 s−1 and the resulting mean free path (l0) was only 1.2 μm. This small mean free path also indicates the presence of disorder in this material. The active QPC structure was defined by 450 nm wide, and 100 nm deep, trenches fabricated via EBL and wet etching in a solution of phosphoric acid and hydrogen peroxide (H3PO4:H2O2:H2O) for 4 min at 15 °C. This is known to produce smooth sidewalls in a number of semiconductors [32]. Such structures, defined by trench isolation and in-plane gates, were chosen rather than top metal gates because it allows us to scan the entire surface area of the device without disturbing the tip height in non-feedback operation during the SGM scanning. The observed curvature radius of the outer edges of the QPC was 400 nm and the separation that defines the constriction was approximately 600 nm as shown in figure 1(a). This structure permits us to use the two-dimensional electron gas (2DEG) inside the trenches as side gates by biasing the in-plane material as schematically shown in figure 1(b). The trenches are approximately 20 nm deeper than the bottom of the quantum
Figure 2. (a) Shubinikov–de Haas oscillations of the 2DEG at
280 mK. (b) Transmission curve of the QPC at 280 mK and zero magnetic field (red curve) [20]. Reproduced with permission, © American Physical Society. The fluctuating component of the conductance is obtained by subtracting the background from the transmission curve (blue curve). The inset shows the transmission curve at 6 K [8]. Reproduced with permission from [8]. Copyright 2005, American Institute of Physics.
little with gate voltage means that the interference patterns are stabilized and can be analyzed with more ease. The influence of the magnetic field on the SGM images is discussed in section 4. We observe weak localization, which further strengthens the recognition of the random potential and disorder in the material. Such weak localization arises as a result of carrier wave functions, which travel time-reversed paths around a potential peak, and undergo interference with reduced conductance. The magnetic field breaks time reversal symmetry, which reduces this interference effect. In addition, we observe fluctuations for varying weak magnetic fields. When the magnetic field is increased beyond the weak regime, however, the region of wave coherence, leading to CFs, is progressively reduced as the QH regime is reached. In this latter regime, clear conductance steps are observed in the images due to the presence of edge channels propagating through the QPC. The formation and depopulation of edge channels can be visualized by increasingly squeezing the channel with the side gate voltages. The size of compressible and incompressible QH regions can be estimated from crosssectional line profiles of these images. 3
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
Figure 4. Side gate bias dependence of the SGM images (Vg = −0.4 V to −7.2 V) at 280 mK. The scanning area is 2.8 × 2.8 μm2. The dotted
lines indicate the outer topographical edges of the trenches. These images represent the variations of the resistance during the tip scanning. The color bar indicates the resistance change in arbitrary units.
DC sputtering. For low temperature SGM, it is very important to remove the epoxy resin that protects the gold wires on the cantilever using tweezers and applying heat with a soldering iron, since the resin cracks at low temperatures breaking the wires. If this happens, though, the wires can be replaced in a conventional gold wire bonding machine. The cantilever is placed in a conventional Wheatstone bridge as shown in figure 1(c) [33]. One arm is the reference resistor found on the chip, and the other arm is the active cantilever sensor where the tip is connected. The other resistances (500 ohms) on two of four arms are set outside the cryostat. To detect the signal from the cantilever, another lock-in amplifier was employed and a differential measurement is performed to probe the potential difference (VA–B) in the bridge. The internal oscillator of the lock-in amplifier provides a signal of 80 mV, and this excites the bridge at 18 kHz. A time constant of 1 ms is used, which is much less than that used for transport measurements of the sample. The value of the bridge voltage VA–B is used as feedback signal in the AFM operation. In order to decouple the ground of the internal oscillator from the ground of the rest of the circuit, an audio transformer is employed and then the bridge is connected to the secondary side. In this way, a DC tip bias (Vtip) can be applied to the entire bridge without inducing any current into the rest of the circuit. This also keeps the tip bias from causing any heat load in the cryostat.
well, which ensures a complete electrical isolation for the in-plane gates up to −10 V. Making a Schottky barrier to a narrow-band-gap semiconductor such as InAlAs is not a simple task and one usually obtains a very leaky diode. This can be fatal in the case of SGM, since a current can flow from the metallic scanning probe to the surface of the device destroying it especially in a contact-mode operation. In other cases, the probe can make electrical contact with the metallic structures of the device producing the same catastrophe. In order to avoid such accidents, the sample was coated with a 25 nm thick polymethylmethacrylate (PMMA) film. 2.2. Equipment and setting for SGM
For the SGM measurements, a low-temperature SPM head manufactured by Omicron Co. Ltd. (Cryogenic SFM) was installed in a He3 cryostat manufactured by JANIS Co. Ltd. The system has a scanning area of 2.7 × 2.7 μm2 at a base temperature of 280 mK and holds this base temperature for more than 40 h with the pumps for the 1 K pot disconnected. In order to avoid heating and carrier generation from a conventional laser detection system, a piezoresistive cantilever manufactured by SII (NPX1CTP003 for contact mode operation) with a typical spring constant of 4 N m−1 was used. In order to use it for SGM, the probe tip was coated with 15 nm of PtIr20 by 4
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
electron affinity of the material is enough to generate a depletion region. Therefore, small tip biases are enough to cause some level of electron scattering. In our case, it was found that applying negative bias in the range of −0.5 V to −1.0 V is sufficient to give a good signal-to-noise ratio for most measurements. The voltage drop across the device and the current that flows through it are measured simultaneously using two lock-in amplifiers in a four-probe configuration. Quasiconstant current excitation (typically 10 nA) is used at a frequency of 1.7 kHz and integration time of 30 ms in order to achieve an acceptable bandwidth for the measurements. The data is stored in a computer synchronized with the tip position to obtain a map of the conductance across the sample.
(a) 7.2 6.4 5.6 2.8 2.8
2.1 2.1
1.4
1.4
0.7 0 0
0.7
2G0 ~2G
(b) 20
3. SGM image of an InGaAs QPC at zero magnetic field
15
3.1. Basic characteristics of the sample
10 2.8
Studies of the Shubnikov–de Haas magnetoconductance of the 2DEG is shown in figure 2(a). The occupation of two subbands can be found in the quantum well, and these have a carrier concentration of 7.4 × 1015 m−2 and 2.1 × 1015 m−2. Based on the total carrier concentration, the mobility was calculated to be 7.4 × 104 m2 V−1 s−1, a low value that indicates a certain level of disorder in the material. The dependence of the conductance through the QPC with side gate voltage is shown in figure 2(b) for two temperatures. The curve obtained at 6 K (inset in figure 2(b)) shows clear quantized conductance steps near integer values of G0 (no attempt at removing a shunt conductance has been made, and the actual values of the steps can vary with a variety of factors) [34]. At gate voltages above −3 V some conductance steps are missing due to the occupation of the second subband, but below this voltage we believe that only one subband propagates through the QPC. When the temperature is reduced to 280 mK (red curve in figure 2(b)), the plateaus are hidden by the formation of reproducible fluctuations (blue curve in figure 2(b)). Such quantum interference behavior is frequently observed in quasi-ballistic quantum wires due to a random potential caused by impurities [35]. This is further complicated in the case of the InGaAs channel, as there is likely to be an additional background disorder potential due to interface roughness and the random nature of the alloy [36].
2.8
2.1 2.1
1.4
1.4
0.7 0 0
0.7
~1G0
Figure 5. SGM resistance images corresponding to base conductance values of (a) 2G0 and (b) 1G0 at 0.28 K. The dotted lines indicate the outer edges of trenches. Reproduced with permission from [8]. Copyright 2005, American Institute of Physics.
Although contact mode was used for taking topographical images, lift mode was used to perform SGM measurements. In this configuration, the tip is lifted approximately 45 nm above the surface in order to achieve suitable levels of perturbation. The scanning occurred in a plane parallel to the sample surface without feedback but with slope compensation in order to keep the distance from the surface constant (45 nm). The lift mode operation also prevents the tip from altering the distribution of charged impurities within the channel. If SGM is performed during the topographic contact mode imaging, the conductance of the sample changes progressively and irreversibly and then a reproducible SGM image cannot be obtained. Therefore, after placing the tip near the QPC, it should be kept away from the surface using lift mode. Although the height may be corrected some times when the environment (e.g. magnetic field and temperature) was changed or a long time has passed, the calibration should be done as far as possible from the center of the QPC. During SGM imaging, Vtip is kept at −0.5 V, and the side gate bias varies (Vg) between 0 and −7.5 V depending on the measurement. The tip bias and the lift height play an important role as they define the shape and strength of the perturbation. As the tip is pulled away from the surface, the distribution of its electric field widens at the 2DEG region, producing a wider and weaker depletion region. Furthermore, the difference between the work function of the tip and the
3.2. Gate voltage dependence
A typical SGM resistance image obtained at base temperature is shown in figure 3. The SGM observations are performed lifting the tip 40 nm above the surface and applying a small negative bias of typically −0.5 V. This perturbation is big enough to cause a maximum change (ΔG) of 0.4G0 in the transmission. The image is completely reproducible at the same side gate bias and is characterized by three features [8]. (i) A large feature on both sides of the QPC, which is attributed to an electrostatic coupling between the tip and the in-plane gates and is visible even at high temperatures: when the probe hovers right above the in-plane gates, its potential perturbs the potential of 5
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
(a)
0.4G0
Resistance [k ]
(b) 600
0.3G0
(c)
0.2G0
500
0.2G0
0.3G0 0.4G0
400 300 200 100 0
0
500 1000 1500 2000 2500
X [nm] Figure 6. (a) SGM resistance images obtained near pinch-off corresponding to the background potential at base conductance values of (left) 0.4G0, (middle) 0.3G0 and (right) 0.2G0. The dotted lines indicated the outer edges of the trenches. (b) The cross-section along the direction of the channel length for 0.4G0 (blue), 0.3G0 (green) and 0.2G0 (red). Reproduced with permission from [8]. Copyright 2005, American Institute of Physics. (c) Schematic view of the potential profile just before pinch-off (upper to lower).
the gates since some depletion occurs. Consequently, the width of the channel is varied, usually increasing its conductance. Although there are no fine fluctuations in this area, there exist some wavy patterns, which correspond to CFs caused by the side gate bias as shown in the transmission curve of figure 2(b). (ii) A fine undulating pattern within the QPC: the typical extent of the structure is approximately 100–200 nm in diameter and
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Scanning gate imaging of a disordered quantum point contact
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 193202 (http://iopscience.iop.org/0953-8984/26/19/193202) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.97.90.221 This content was downloaded on 01/05/2014 at 12:37
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 193202 (13pp)
doi:10.1088/0953-8984/26/19/193202
Topical Review
Scanning gate imaging of a disordered quantum point contact N Aoki1, C R da Cunha2, R Akis3, D K Ferry3 and Y Ochiai1 1
Graduate School of Advanced Integration Science, Chiba University, 1–33 Yayoi-cho, Inage-ku, Chiba 263–8522, Japan 2 Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91501–970, Brazil 3 School of Electrical, Computer, and Energy Engineering, and Center for Solid State Electronics Research, Arizona State University, Tempe, AZ 85287–5706, USA E-mail: [email protected] Received 9 November 2013, revised 16 January 2014 Accepted for publication 17 January 2014 Published 25 April 2014 Abstract
Scanning gate microscopy (SGM) is a novel technique that has been used to image characteristic features related to the coherent electron flow in mesoscopic structures. For instance, SGM has successfully been applied to study peculiar electron transport properties that arise due to small levels of disorder in a system. The particular case of an InGaAs quantum well layer in a heterostructure, which is dominated by a quasi-ballistic regime, was analyzed. A quantum point contact fabricated onto this material exhibits conduction fluctuations that are not expected in typical high-mobility heterostructures such as AlGaAs/GaAs. SGM revealed not only interference patterns corresponding to specific conductance fluctuations but also modedependent resistance peaks corresponding to the first and second quantum levels of conductance (2e2/h) at zero magnetic field. On the other hand, clear conductance plateaus originating from the integer quantum Hall effect were observed at high magnetic fields. The physical size of incompressible edge channels was estimated from cross-sectional analysis of these images. Keywords: canning gate microscopy, quantum point contact, disordere, conductance fluctuations, quantum Hall effect S Online supplementary data available from stacks.iop.org/J.PhysCM/26/193202/mmedia (Some figures may appear in colour only in the online journal)
1. Introduction
were observed due to the standing wave formed between the saddle point potential of the QPC and the point potential underneath the tip [6]. More recently, SGM has been applied to study many other systems. For instance, in studies of the coherent quantum structures, fractional conductance steps in a ballistic QPC [7], conductance fluctuations and modal characteristic peaks in a disordered QPC [8], fingerprint of potential fluctuations in nano-wire [9], fractal conductance fluctuations [10] and characteristic electron states in open quantum dot cavities [11, 12], and electron interferences caused by backscattering from a QPC [13] have been imaged. SGM has allowed many phenomena to be visualized such as the current
Scanning gate microscopy (SGM) is a novel tool that has been used to visualize the real space coherent electron flow in mesoscopic devices. In this technique, variations in conduction of a device are monitored while the tip of an atomic force microscope (AFM) is used as a mobile-point-gate electrode scanning an area of interest. The technique was firstly used to image the modal structure of quantum waves that flow through a quantum point contact (QPC) producing integer multiples of 2e2/h (G0) [1–5]. In the course of the study, periodic fringing patterns along the ballistic and coherent current flow paths 0953-8984/14/193202+13$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
flow paths based on the electron focusing effect [14], magnetic field dependent images based on the Aharonov–Bohm effect [15, 16], transition in the quantum Hall (QH) regime [17–19]and conductance steps due to the depopulation of the edge states [20], concentric ring patterns corresponding to Coulomb oscillation by the Coulomb blockade effect in closed quantum dots [21, 22] and an isolated Coulomb island in the quantum Hall regime [23, 24].Theoretical simulations have recently supported quantum transport during SGM observations [25]. The technique has recently been spread to the study of soft materials such as carbon nanotubes [26], organic semiconductors [27], graphene [28, 29], and many others. In this topical review, we will concentrate on the description of our SGM studies dealing with the imaging of coherent electron systems in QPCs fabricated in a slightly disordered InAlAs/InGaAs/InAlAs quantum well. Due to the disorder, this system exhibits a set of patterns in the SGM images which are different from those found in usual QPCs fabricated in high-mobility wafers [13, 30, 31]. Our samples were prepared specifically for performing SGM measurements inside the QPCs that were defined by V-shaped trenches etched through the heterostructure to isolate regions of the InGaAs layer. These were fabricated using electron beam lithography (EBL) and wet etching as shown in figure 1(a). As a consequence of using a narrow gap semiconductor in conjunction with trench-defined structures, our QPCs do not have a saddle point potential at its constriction and this allows us to obtain important images of this region. The details related to the sample preparation and the experimental microscopy techniques used for studying the QPCs will be presented in section 2. The SGM results of the QPCs fabricated on our basic InGaAs material, at zero magnetic field, are discussed in section 3. From these results, we conclude that this material is moderately disordered, either due to the presence of impurities or due to the random potential in the alloy (or both). The QPC shows typical conductance quantization at around 4 K. However, such conductance steps are hidden by conductance fluctuations (CFs) at lower temperatures. Such conductance fluctuations are common in disordered material. Moreover, the CFs are reflected in SGM images as interference patterns. Nevertheless, characteristic peaks depending on specific propagating modes through the QPC are observed when the conductance is settled near G0 (or a few G0). Away from these plateaus, however, we find that the CFs arising from the quantum effects of the disorder dominate the image. These interference patterns, which lead to the CFs, are thought to originate from a background random potential, mentioned above. As the side gate voltage is varied, the channel width changes and this affects the actual random background potential that appears at the Fermi level, and this alters the spatial position of the CF peaks. We will see that a fast-Fourier transform (FFT) analysis reveals that the mean size of the CF patterns is comparable to that found from the same analysis carried out on CF that arise as the magnetic field is varied. This agreement suggests that they correspond to quite similar interference patterns. On the other hand, near pinch-off, the positions of the SGM features are unaltered and
(a)
Vtip: –0.5 V
(b)
V Vsd Weakly depleted point Underneath tip (~100 nm wide)
(c)
Isd: 10 nA
Vg: 0~ – 7.5 V Ref.
Tip LT
A
B
80 mV 18 kHz
Vtip Figure 1. (a) Topographic AFM image around the QPC region of the sample. The scanning area is 2.8 × 2.8 μm2. The depth of the trenches is ~100 nm [8]. Copyright 2005, American Institute of Physics. (b) Schematic view of the 2DEG plane and the configuration during SGM measurements. The channel region is defined by in-plane isolation (black regions). Four terminal resistance measurements were performed by applying Isd of 10 nA. Negative side gate voltage was applied to the 2DEG regions surrounded by trenches to change the transmission of the QPC. The 2DEG underneath the tip was slightly depleted due to the application of a weak negative voltage (Vtip) on the tip. The scanning tip causes a local depletion region and changes the transmission through the QPC. (c) Electric circuit for operation of a piezoresistive cantilever in the SGM observation. The wheatstone bridge holding the tip is isolated by audio-trans from an oscillator (18 kHz). The potential of the tip can be defined by a DC-voltage source. The differential signal (VA–B) in the bridge is detected by a lock-in amp. and the value is used for feedback operation in AFM observation. The broken line indicates a part settled at low temperature.
increase only in intensity (resistance), which serves to unmask the random background potential. That is, near the conductance plateaus, the fact that the background conductance varies 2
Topical Review
J. Phys.: Condens. Matter 26 (2014) 193202
Resistance [ ]
(a) 400 300
200
100
0
0
1
2
3
4
5
6
7
3.5
0.16
2
Conductance [2e /h]
3
0.12
2.5 2
0.08
1.5
0.04
1
0
0.5 0
2
(b)
-0.04 -8
-7
-6
-5
-4
-3
-2
-1
0
Conductance Fluctuations [2e /h]
Magnetic Field [T] Figure 3. Typical SGM image of the QPC region taken at 280 mK. The image was obtained at around 1G0 and processed to obtain the absolute value of the variations to emphasize the featured characteristics indicated by the arrows (i)–(iii). The dotted lines represent the outer edges of the trenches. The inset indicates a typical SGM image of the GaAs-QPC obtained at the same temperature (scan range 1 × 1 μm2).
2. Sample preparation and imaging technique for SGM studies 2.1. Sample preparation for SGM
Side Gate Voltage [V]
The sample was fabricated with a multi-layer heterostructure composed of (from the top) n-In0.53Al0.47As (5 nm, cap layer)/ n-In0.53Al0.47As (30 nm, doped layer)/ semi-insulating (si)-In0.53Al0.47As (10 nm, spacer layer)/ si-In0.53Ga0.47As (25 nm, the active quantum well)/ si-In0.53Al0.47As (10 nm, spacer layer)/InP (substrate). In this configuration, the quantum well is located between 45 nm and 70 nm below the surface. Electrons transfer from the doped layer into the quantum well, a process known as modulation-doping. As discussed below, measurements show that two levels of the quantum well are occupied at low temperature. The mobility found from the total carrier concentration was 7.4 × 104 cm2 V−1 s−1 and the resulting mean free path (l0) was only 1.2 μm. This small mean free path also indicates the presence of disorder in this material. The active QPC structure was defined by 450 nm wide, and 100 nm deep, trenches fabricated via EBL and wet etching in a solution of phosphoric acid and hydrogen peroxide (H3PO4:H2O2:H2O) for 4 min at 15 °C. This is known to produce smooth sidewalls in a number of semiconductors [32]. Such structures, defined by trench isolation and in-plane gates, were chosen rather than top metal gates because it allows us to scan the entire surface area of the device without disturbing the tip height in non-feedback operation during the SGM scanning. The observed curvature radius of the outer edges of the QPC was 400 nm and the separation that defines the constriction was approximately 600 nm as shown in figure 1(a). This structure permits us to use the two-dimensional electron gas (2DEG) inside the trenches as side gates by biasing the in-plane material as schematically shown in figure 1(b). The trenches are approximately 20 nm deeper than the bottom of the quantum
Figure 2. (a) Shubinikov–de Haas oscillations of the 2DEG at
280 mK. (b) Transmission curve of the QPC at 280 mK and zero magnetic field (red curve) [20]. Reproduced with permission, © American Physical Society. The fluctuating component of the conductance is obtained by subtracting the background from the transmission curve (blue curve). The inset shows the transmission curve at 6 K [8]. Reproduced with permission from [8]. Copyright 2005, American Institute of Physics.
little with gate voltage means that the interference patterns are stabilized and can be analyzed with more ease. The influence of the magnetic field on the SGM images is discussed in section 4. We observe weak localization, which further strengthens the recognition of the random potential and disorder in the material. Such weak localization arises as a result of carrier wave functions, which travel time-reversed paths around a potential peak, and undergo interference with reduced conductance. The magnetic field breaks time reversal symmetry, which reduces this interference effect. In addition, we observe fluctuations for varying weak magnetic fields. When the magnetic field is increased beyond the weak regime, however, the region of wave coherence, leading to CFs, is progressively reduced as the QH regime is reached. In this latter regime, clear conductance steps are observed in the images due to the presence of edge channels propagating through the QPC. The formation and depopulation of edge channels can be visualized by increasingly squeezing the channel with the side gate voltages. The size of compressible and incompressible QH regions can be estimated from crosssectional line profiles of these images. 3
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Figure 4. Side gate bias dependence of the SGM images (Vg = −0.4 V to −7.2 V) at 280 mK. The scanning area is 2.8 × 2.8 μm2. The dotted
lines indicate the outer topographical edges of the trenches. These images represent the variations of the resistance during the tip scanning. The color bar indicates the resistance change in arbitrary units.
DC sputtering. For low temperature SGM, it is very important to remove the epoxy resin that protects the gold wires on the cantilever using tweezers and applying heat with a soldering iron, since the resin cracks at low temperatures breaking the wires. If this happens, though, the wires can be replaced in a conventional gold wire bonding machine. The cantilever is placed in a conventional Wheatstone bridge as shown in figure 1(c) [33]. One arm is the reference resistor found on the chip, and the other arm is the active cantilever sensor where the tip is connected. The other resistances (500 ohms) on two of four arms are set outside the cryostat. To detect the signal from the cantilever, another lock-in amplifier was employed and a differential measurement is performed to probe the potential difference (VA–B) in the bridge. The internal oscillator of the lock-in amplifier provides a signal of 80 mV, and this excites the bridge at 18 kHz. A time constant of 1 ms is used, which is much less than that used for transport measurements of the sample. The value of the bridge voltage VA–B is used as feedback signal in the AFM operation. In order to decouple the ground of the internal oscillator from the ground of the rest of the circuit, an audio transformer is employed and then the bridge is connected to the secondary side. In this way, a DC tip bias (Vtip) can be applied to the entire bridge without inducing any current into the rest of the circuit. This also keeps the tip bias from causing any heat load in the cryostat.
well, which ensures a complete electrical isolation for the in-plane gates up to −10 V. Making a Schottky barrier to a narrow-band-gap semiconductor such as InAlAs is not a simple task and one usually obtains a very leaky diode. This can be fatal in the case of SGM, since a current can flow from the metallic scanning probe to the surface of the device destroying it especially in a contact-mode operation. In other cases, the probe can make electrical contact with the metallic structures of the device producing the same catastrophe. In order to avoid such accidents, the sample was coated with a 25 nm thick polymethylmethacrylate (PMMA) film. 2.2. Equipment and setting for SGM
For the SGM measurements, a low-temperature SPM head manufactured by Omicron Co. Ltd. (Cryogenic SFM) was installed in a He3 cryostat manufactured by JANIS Co. Ltd. The system has a scanning area of 2.7 × 2.7 μm2 at a base temperature of 280 mK and holds this base temperature for more than 40 h with the pumps for the 1 K pot disconnected. In order to avoid heating and carrier generation from a conventional laser detection system, a piezoresistive cantilever manufactured by SII (NPX1CTP003 for contact mode operation) with a typical spring constant of 4 N m−1 was used. In order to use it for SGM, the probe tip was coated with 15 nm of PtIr20 by 4
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electron affinity of the material is enough to generate a depletion region. Therefore, small tip biases are enough to cause some level of electron scattering. In our case, it was found that applying negative bias in the range of −0.5 V to −1.0 V is sufficient to give a good signal-to-noise ratio for most measurements. The voltage drop across the device and the current that flows through it are measured simultaneously using two lock-in amplifiers in a four-probe configuration. Quasiconstant current excitation (typically 10 nA) is used at a frequency of 1.7 kHz and integration time of 30 ms in order to achieve an acceptable bandwidth for the measurements. The data is stored in a computer synchronized with the tip position to obtain a map of the conductance across the sample.
(a) 7.2 6.4 5.6 2.8 2.8
2.1 2.1
1.4
1.4
0.7 0 0
0.7
2G0 ~2G
(b) 20
3. SGM image of an InGaAs QPC at zero magnetic field
15
3.1. Basic characteristics of the sample
10 2.8
Studies of the Shubnikov–de Haas magnetoconductance of the 2DEG is shown in figure 2(a). The occupation of two subbands can be found in the quantum well, and these have a carrier concentration of 7.4 × 1015 m−2 and 2.1 × 1015 m−2. Based on the total carrier concentration, the mobility was calculated to be 7.4 × 104 m2 V−1 s−1, a low value that indicates a certain level of disorder in the material. The dependence of the conductance through the QPC with side gate voltage is shown in figure 2(b) for two temperatures. The curve obtained at 6 K (inset in figure 2(b)) shows clear quantized conductance steps near integer values of G0 (no attempt at removing a shunt conductance has been made, and the actual values of the steps can vary with a variety of factors) [34]. At gate voltages above −3 V some conductance steps are missing due to the occupation of the second subband, but below this voltage we believe that only one subband propagates through the QPC. When the temperature is reduced to 280 mK (red curve in figure 2(b)), the plateaus are hidden by the formation of reproducible fluctuations (blue curve in figure 2(b)). Such quantum interference behavior is frequently observed in quasi-ballistic quantum wires due to a random potential caused by impurities [35]. This is further complicated in the case of the InGaAs channel, as there is likely to be an additional background disorder potential due to interface roughness and the random nature of the alloy [36].
2.8
2.1 2.1
1.4
1.4
0.7 0 0
0.7
~1G0
Figure 5. SGM resistance images corresponding to base conductance values of (a) 2G0 and (b) 1G0 at 0.28 K. The dotted lines indicate the outer edges of trenches. Reproduced with permission from [8]. Copyright 2005, American Institute of Physics.
Although contact mode was used for taking topographical images, lift mode was used to perform SGM measurements. In this configuration, the tip is lifted approximately 45 nm above the surface in order to achieve suitable levels of perturbation. The scanning occurred in a plane parallel to the sample surface without feedback but with slope compensation in order to keep the distance from the surface constant (45 nm). The lift mode operation also prevents the tip from altering the distribution of charged impurities within the channel. If SGM is performed during the topographic contact mode imaging, the conductance of the sample changes progressively and irreversibly and then a reproducible SGM image cannot be obtained. Therefore, after placing the tip near the QPC, it should be kept away from the surface using lift mode. Although the height may be corrected some times when the environment (e.g. magnetic field and temperature) was changed or a long time has passed, the calibration should be done as far as possible from the center of the QPC. During SGM imaging, Vtip is kept at −0.5 V, and the side gate bias varies (Vg) between 0 and −7.5 V depending on the measurement. The tip bias and the lift height play an important role as they define the shape and strength of the perturbation. As the tip is pulled away from the surface, the distribution of its electric field widens at the 2DEG region, producing a wider and weaker depletion region. Furthermore, the difference between the work function of the tip and the
3.2. Gate voltage dependence
A typical SGM resistance image obtained at base temperature is shown in figure 3. The SGM observations are performed lifting the tip 40 nm above the surface and applying a small negative bias of typically −0.5 V. This perturbation is big enough to cause a maximum change (ΔG) of 0.4G0 in the transmission. The image is completely reproducible at the same side gate bias and is characterized by three features [8]. (i) A large feature on both sides of the QPC, which is attributed to an electrostatic coupling between the tip and the in-plane gates and is visible even at high temperatures: when the probe hovers right above the in-plane gates, its potential perturbs the potential of 5
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(a)
0.4G0
Resistance [k ]
(b) 600
0.3G0
(c)
0.2G0
500
0.2G0
0.3G0 0.4G0
400 300 200 100 0
0
500 1000 1500 2000 2500
X [nm] Figure 6. (a) SGM resistance images obtained near pinch-off corresponding to the background potential at base conductance values of (left) 0.4G0, (middle) 0.3G0 and (right) 0.2G0. The dotted lines indicated the outer edges of the trenches. (b) The cross-section along the direction of the channel length for 0.4G0 (blue), 0.3G0 (green) and 0.2G0 (red). Reproduced with permission from [8]. Copyright 2005, American Institute of Physics. (c) Schematic view of the potential profile just before pinch-off (upper to lower).
the gates since some depletion occurs. Consequently, the width of the channel is varied, usually increasing its conductance. Although there are no fine fluctuations in this area, there exist some wavy patterns, which correspond to CFs caused by the side gate bias as shown in the transmission curve of figure 2(b). (ii) A fine undulating pattern within the QPC: the typical extent of the structure is approximately 100–200 nm in diameter and
