Nanomaterials Characterization by lowtemperature Scanning Probe Microscopy
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1 Nanomaterials Characterization by lowtemperature Scanning Probe Microscopy Stefan Heun NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore Piazza San Silvestro 12, Pisa, Italy SGM Group
2 Inner structure of edge channels in the quantum Hall regime probed by scanning gate microscopy Stefan Heun NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore Piazza San Silvestro 12, Pisa, Italy SGM Group
3 Outline Introduction to quantum Hall physics Basics of Scanning Gate Microscopy Inner structure of edge channels in the quantum Hall regime by SGM
4 Outline Introduction to quantum Hall physics Basics of Scanning Gate Microscopy Inner structure of edge channels in the quantum Hall regime by SGM Further reading: Klaus von Klitzing: The quantized Hall effect, Nobel lecture, 1985 ( R. J. Haug: Edge-state transport and its experimental consequences in high magnetic fields, Semicond. Sci. Technol. 8 (1993) 131.
5 2 Dimensional Electron System see also: Horst L. Stoermer, Nobel Lecture, December 8, 1998
6 Hall bar geometry Measurement of longitudinal and transversal resistance K. von Klitzing et al.: Physik Journal 4 (2005) No.6 p.37.
7 Drude model F m dv dt e E v B mv (=0; steady state) E 0 E 0 x y c c j y j x j x j y with c 2 eb z ne, 0, m m j nev E Transverse current is zero: jy 0 It follows: x xx const. jx 0 xy E E j y x 1 Bz ne B z Ashcroft / Mermin: Solid State Physics.
8 Hall effect Drude model: R xy = proportional to B R xx = costant in B K. von Klitzing et al.: Physik Journal 4 (2005) No.6 p.37.
9 Quantum Hall Effect Behaviour observed at low temperature in high mobility samples in high B ( c 1): Plateaux in R xy Minima in R xx Quantum Hall Effect: deviations from Drude model observed around B = nh / ne n: filling factor K. von Klitzing et al.: Physik Journal 4 (2005) No.6 p.37.
10 The Nobel prize in Physics 1985: integer QHE 1998: fractional QHE
11 Landau Levels Single particle Hamiltonian for an electron in an external magnetic field: Landau-Gauge: For It follows: r 0 V A E H 0, Bx,0 1 2m 2 p ea V r with the wavefunction can be written as: B ik y x, y e y x 2 p 2m R. J. Haug, Semicond. Sci. Technol. 8 (1993) Be z x 2 2 x m x x x c 0
12 Landau Levels E This is the equation of a harmonic oscillator with angular frequency center of cyclotron orbit at 2 p 2m x 2 2 x m x x x E j x 2 0 k y B 1 2 c c j 0 and magnetic length 1 2 B c eb eb 25nm m B( T ) Drift velocity v y de j dk y 0 Landau Levels R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.
13 R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131. Landau Levels Degeneracy of each Landau Level: n deg The number of filled Landau Levels for a given carrier concentration n s is then: ns nsh n n eb n: filling factor eb h deg B
14 Landau Levels At the edges of the sample, a boundary confinement can be added by a small V(x) which does not depend on y. E j 2 k j 2 V k y c 1 y B B. I. Halperin, Phys. Rev. B 25, 2185 (1982).
15 Introduction to edge channels y x x see also: R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.
16 Introduction to edge channels y x x see also: R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.
17 Introduction to edge channels edge: 1D one-way conductor, skipping orbits bulk: insulator y x x see also: R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.
18 Introduction to edge channels edge: 1D one-way conductor, skipping orbits bulk: insulator y x x Quantization of energy levels (Landau levels) Gap for excitation in the bulk Transport via edge states see also: R. J. Haug, Semicond. Sci. Technol. 8 (1993) 131.
19 Edge states incompressibilities Example Semiconductor compressibility: ch n s incompressible: ch n s VB n* CB n* independent of B n K. von Klitzing et al.: Physik Journal 4 (2005) No.6 p.37.
20 Edge states compressibility: ch n s incompressible: ch n s D. B. Chklovskii et al.: PRB 46 (1992) 4026.
21 Edge states compressibility: ch n s incompressible: ch n s D. B. Chklovskii et al.: PRB 46 (1992) 4026.
22 Non-interacting vs. interacting picture The self consistent potential due to e-e interactions modifies the edge structure For any realistic potential the density goes smoothly to zero. Alternating compressible and incompressible stripes arise at the sample edge Incompressible stripes: The electron density is constant The potential has a jump Compressible stripes: The electron density has a jump The potential is constant D. B. Chklovskii et al.: PRB 46 (1992) SGM Group
23 Non-interacting vs. interacting picture The self consistent potential due to e-e interactions modifies the edge structure For any realistic potential the density goes smoothly to zero. Alternating compressible and incompressible stripes arise at the sample edge Each of these edge channels Incompressible stripes: The electron density is constant The potential has a jump carries I Compressible stripes: The electron density has a jump The potential is constant e h e 2 hv M. Büttiker, Phys. Rev. B 38, 9375 (1988) D. B. Chklovskii et al.: PRB 46 (1992) SGM Group
24 How to probe edge channels? S. Roddaro et al., Phys. Rev. Lett. 90 (2003)
25 How to probe edge channels? S. Roddaro et al., Phys. Rev. Lett. 90 (2003)
26 How to probe edge channels? S. Roddaro et al., Phys. Rev. Lett. 90 (2003)
27 How to probe edge channels? V Vg t V r 0 I(V) S. Roddaro et al., Phys. Rev. Lett. 90 (2003)
28 How to probe edge channels? V Vg t Current conservation: t + r = 1 V r 0 I(V) we induce backscattering by reducing this distance Quantum Point Contact (QPC) S. Roddaro et al., Phys. Rev. Lett. 90 (2003)
29 Hot topics The quantum Hall Effect as an Electrical Resistance Standard R K = Ω
30 Hot topics The quantum Hall Effect as an Electrical Resistance Standard Quantum Hall Interferometry Y. Ji et al., Nature 422 (2003) 415.
31 Hot topics The quantum Hall Effect as an Electrical Resistance Standard Quantum Hall Interferometry Quantum Computing
32 Hot topics The quantum Hall Effect as an Electrical Resistance Standard Quantum Hall Interferometry Quantum Computing Basic Research
33 Outline Introduction to quantum Hall physics Basics of Scanning Gate Microscopy Inner structure of edge channels in the quantum Hall regime by SGM Further reading: M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard: Imaging coherent electron flow from a quantum point contact, Science 289 (2000) M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gossard: Coherent branched flow in a two-dimensional electron gas, Nature 410 (2001) 183.
34 Scanning Gate Microscopy AFM with conductive tip Tip at negatively bias (local gate - locally depletes the 2DEG), no current flows SGM performed in constant height mode (10-50 nm above surface), no strain M. A. Topinka et al.: Science 289 (2000) 2323.
35 Scanning Gate Microscopy AFM with conductive tip Tip at negatively bias (local gate - locally depletes the 2DEG), no current flows SGM performed in constant height mode (10-50 nm above surface), no strain A. Iagallo et al.: Nano Reserach, in press.
36 Coherent branched flow of electrons G: 0.00e 2 /h 0.25e 2 /h M. A. Topinka et al., Nature 410 (2001) 183.
37 Tip-induced backscattering N. Paradiso et al., Physica E 42 (2010) 1038.
38 Tip-induced backscattering N. Paradiso et al., Physica E 42 (2010) 1038.
39 Coherent branched flow of electrons G: 0.00e 2 /h 0.25e 2 /h M. A. Topinka et al., Nature 410 (2001) 183.
40 The lab in Pisa Setup: AFM non-optical detection scheme (tuning fork) With vibration and noise isolation system 3 He insert (cold finger base temp. :300 mk) 9 T cryomagnet SGM performed in constant height mode (10-50 nm above surface), no strain Tip at negative bias (moveable gate locally depletes the 2DEG) source-drain current Pioneering work by: M. A. Topinka et al.: Science 289 (2000) SGM Group
41 Oct 2007: First cooldown
42 AFM Head Positioner (range in xyz > 6 mm) allow to locate features on the sample with µm precision Scan range: 42µm x 42µm x RT 8.5µm x 8.5µm x 300 mk Temperature measurement (RuO 2 ) close to sample Drift < 1 nm / h Temperature stability Delta T / T < 5% for hours, even at max. B field Noise in z at 300 mk: 1nm
43 Tuning Fork
44 Sample holder for transport measurements Base mounted on AFM scanner Contact via pogo pins Chip carrier holds sample
45 Sample holder mounted on scanner
46 Tip sample geometry
47 Outline Introduction to quantum Hall physics Basics of Scanning Gate Microscopy Inner structure of edge channels in the quantum Hall regime by SGM Further reading: N. Paradiso et al.: Spatially resolved analysis of edge-channel equilibration in quantum Hall circuits, Phys. Rev. B 83 (2011) N. Paradisoet al.: Selective control of edge-channel trajectories by scanning gate microscopy, Physica E 42 (2010) 1038.
48 Hall-bar samples 1900 m x 300 m N. Paradiso et al., Physica E 42 (2010) 1038.
49 Hall-bar samples N. Paradiso et al., Physica E 42 (2010) 1038.
50 Hall-bar samples N. Paradiso et al., Physica E 42 (2010) 1038.
51 Hall-bar samples N. Paradiso et al., Physica E 42 (2010) 1038.
52 Hall-bar samples N. Paradiso et al., Physica E 42 (2010) 1038.
53 Conductance quantization in QPCs 1D confinement In 1D systems the current is carried by a finite number of modes (arising from confined subbands). Each mode contributes two quantum of conductance. 2e 2 /h H. van Houten and C. Beenakker: Physics Today, July 1996, p. 22 SGM Group
54 Conductance quantization in QPCs 1D confinement In 1D systems the current is carried by a finite number of modes (arising from confined subbands). Each mode contributes two quantum of conductance. 2e 2 /h First we fix the mode number (QPC setpoint), then we start scanning the biased tip at a fixed height. SGM Group
55 Conductance quantization in QPCs 1D confinement In 1D systems the current is carried by a finite number of modes (arising from confined subbands). Each mode contributes two quantum of conductance. The biased tip creates a depletion spot that we use to backscatter the electrons passing through the constriction source-drain current 2DEG First we fix the mode number (QPC setpoint), then we start scanning the biased tip at a fixed height. The split gates define a constriction by depleting the 2DEG underneath SGM Group
56 QPC at 3rd plateau 3 rd plateau G = e 2 /h 0.00 G = 0 600nm SGM Group
57 QPC at 2nd plateau 2 nd plateau G = e 2 /h 0.00 G = 0 600nm SGM Group
58 QPC at 1st plateau 1 st plateau G = e 2 /h 0.00 G = 0 600nm SGM Group
59 Branched flow and interference fringes 400nm 100nm QPC conductance G = 6 e 2 /h (3 rd plateau) Tip voltage V tip = -5 V, height h tip = 10 nm see also M. A. Topinka et al., Nature 410 (2001) 183. Fringe periodicity: l F /2=20 nm SGM Group
60 SGM in the Quantum Hall regime Tip voltage V tip = -5 V, height = 30 nm N. Paradiso et al., Physica E 42 (2010) 1038.
61 Magnetoresistance N. Paradiso et al., Physica E 42 (2010) 1038.
62 Selective control of edge channel trajectories by SGM SGM technique: we select individual channels from the edge of a quantized 2DEG, we send them to the constriction and make them backscatter with the biased SGM tip. N. Paradiso et al., Physica E 42 (2010) e e 2 /h 2 /h nm /h 0.0 e 2 /h Bulk filling factor n=4 B = 3.04 T 2 spin-degenerate edge channels gate-region filling factors g 1 = g 2 = 0 SGM Group
63 How we probe incompressible stripes conductance (e 2 /h) Self-consistent potential 1 0 Landau levels inside the constriction tip position (nm) tip induced potential ħω c tip position SGM Group
64 How we probe incompressible stripes conductance (e 2 /h) tip position (nm) backscattering tip position SGM Group
65 How we probe incompressible stripes conductance (e 2 /h) tip position (nm) tip position SGM Group
66 How we probe incompressible stripes conductance (e 2 /h) Energy gap: ħω=5.7 mev Plateau width: 60 nm Incompr. stripe width: 30nm tip position (nm) backscattering tip position SGM Group
67 Can we exploit the non-trivial edge structure? The picture of a QH device emerging from the experiments A bus of compressible and incompressible channels Is it possible to study and image the microscopic details of the inter-channel interactions? SGM Group
68 Studying inter-channel equilibration Two co-propagating edge channels originating from two ohmic contacts at different potential g=1 n=2 d SGM Group
69 Studying inter-channel equilibration devices with fixed interaction length d: elusive determination of the microscopic details of the equilibration mechanisms g=1 n=2 d SGM Group
70 The oppurtunity of the Scanning Gate Microscopy Our technique allows to selectively control the channel trajectory Our idea: exploit the mobile depletion spot induced by the SGM to continuously tune d SGM tip N. Paradiso et al., PRB 83 (2011) n bulk = 4 two spin degenerate edges n=0 n=2 n=4 SGM Group
71 The oppurtunity of the Scanning Gate Microscopy Our technique allows to selectively control the channel trajectory Our idea: exploit the mobile depletion spot induced by the SGM to continuously tune d SGM tip N. Paradiso et al., PRB 83 (2011) n bulk = 4 two spin degenerate edges n=0 n=2 n=4 SGM Group
72 The oppurtunity of the Scanning Gate Microscopy Our technique allows to selectively control the channel trajectory Our idea: exploit the mobile depletion spot induced by the SGM to continuously tune d N. Paradiso et al., PRB 83 (2011) Each channel carries 2e 2 /h units of conductance SGM tip n bulk = 4 two spin degenerate edges n=0 n=2 n=4 SGM Group
73 The oppurtunity of the Scanning Gate Microscopy Our technique allows to selectively control the channel trajectory Our idea: exploit the mobile depletion spot induced by the SGM to continuously tune d N. Paradiso et al., PRB 83 (2011) SGM Group
74 The oppurtunity of the Scanning Gate Microscopy Our technique allows to selectively control the channel trajectory Our idea: exploit the mobile depletion spot induced by the SGM to continuously tune d No eq. Full eq. A 2 e 2 /h 1 e 2 /h B 0 e 2 /h 1 e 2 /h N. Paradiso et al., PRB 83 (2011) SGM Group
75 Experimental setup edge-selector gate transmitted component reflected component tip source bias V=V AC + V DC SEM micrograph of the device Scheme of the electronic setup SGM Group
76 Imaging the inter-channel equilibration The profiles of G B (d) along the trajectory show a strict dependance on the local details Imaging the edge channel equilibration [e 2 /h] 1 0 Source bias: V AC =50V, V DC =0mV SGM Group
77 Imaging the inter-channel equilibration SGM scan at zero magnetic field We can directly image the potential induced by the most important defects by means of a scan at zero magnetic field correlation found The profiles of G B (d) along the trajectory show a strict dependance on the local details Imaging the edge channel equilibration [e 2 /h] 1 0 Source bias: V AC =50V, V DC =0mV SGM Group
78 Tight binding simulations differential conductance G B (e 2 /h) Pictorial model for the disorder potential tip potential big impurities potential Simulations made by the theoretical group of Scuola Normale Superiore (Pisa, Italy) D. Venturelli, F. Taddei, V. Giovannetti and R.Fazio 0.40 background potential Experimental data Tight binding simulations scattering centers 0.20 [e 2 /h] 1 SGM map of the inter-channel equilibration in another device N. Paradiso et al., PRB 83 (2011) position (m) SGM Group
79 Nonlinear regime The backscattered current is a function of the local imbalance V(x) that depends on the specific scattering process. SGM Group
80 Two mechanisms for the inter-channel scattering For low bias the only relevant mechanism is the elastic scattering induced by impurities, which determines an ohmic behavior (linear I-V) At high bias (Δμ ħω c ) vertical transition with photon emission are enabled (threshold and saturation) SGM Group
81 Impact of the electron heating Electron heating due to injection of hot carriers: The relaxation of hot carriers induces a dramatic temperature increase. This is why the transition is smoothened and the threshold voltage reduced for high d SGM Group
82 Conclusions We explore the use of Scanning Gate Microscopy to selectively control edge channel trajectories Control of the edge channel trajectory allows us to study their structure We built size-tunable QH circuits to directly image the equilibration between imbalanced co-propagating edges The comparison with the SGM scan at zero magnetic field reveals a correlation between the local potential and steps in the G B (d) curve Shift of the threshold voltage for the onset of photon emission is explained by a simple model for the electron heating SGM Group
83 Coworkers N. Paradiso S. Roddaro G. Biasiol (IOM) L. Sorba F. Beltram D. Venturelli F. Taddei V. Giovannetti R. Fazio
84 Funding
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