1. Y. Miyahara, A. Roy-Gobeil
Revealing Energy Level Structure of Individual Quantum Dots by Single-Electron Sensitive Electrostatic Force Spectroscopy
Peter Grütter, FRSC
Y. Miyahara, A. Roy-Gobeil
L. Cockins, A. Clerk
McGill University
Montreal (Canada)
30 minutes, invited talk APS 2017 (New Orleans)
P. Grutter

2. Outline/aim Introduction How to measure electron energy levels
How to determine shell structure of qdot
Summary & conclusion

3. Coulomb Blockade Rings

4. Coulomb blockade energy e2/C

8. Experimental configuration
Charge sensing by oscillating AFM tip (frequency mode AFM)
Tip is a moveable gate
Force detection by AFM (tip act as an electrometer)
Single barrier tunneling (to/from sample ONLY)
PNAS 107, 9496 (2010)

9. ç
back electrode

10. back electrode

15. Cockins et al., PNAS 107, 9496 (2010)
Note:
clean decoupling of conservative and dissipative forces is necessary!
Labuda et al., PRB 84, (2011) OR optical excitation

17. P. Grutter, McGill University
Subtle: tunneling rate assumed to be fast compared to field oscillations (Keldysh approximation).
There are many interesting analogies between our work on quantum dots and Albert Stolow’s work at NRC on the tunnel ionization of atoms and /or molecules in strong optical fields.
There are three fundamental approximations in the quasi-static theory of optical strong field ionization. They are: (1) an adiabatic approximation where the electronic response is assumed fast as compared to the time-varying applied field. (2) the Keldysh approximation that the tunnelling rate is also fast compared to the field oscillation; (2) the single active electron approximation that effective only one electron responds (tunnel ionizes) in the strong field.
P. Grutter, McGill University

18. Non-Degenerate Energy Levels
Using Linear Response Theory:
*NOTE: The A is housing some extra factors (omega^2/k) in order to streamline the equation… therefore it is not exactly the same as in the paper!! And I can call A the coupling strength but DON’T give an equation for it.-
Tunneling
Rate
Coupling Strength E
Fermi
Function
dissipation
Res.
Freq.

19. T=4.2K a=0.25nm a= 0.04 (DE=e aVB) EC= 29.5 meV DE=11 meV
Cockins et al., PNAS 107, 9496 (2010)

20. How can we measure the shell structure?

21. Temperature-dependent shifts of Coulomb blockade peaks predicted for degenerate single particle levels by Beenakker, Phys Rev B 44, 1646 (1991)
Cockins et al., PNAS 107, 9496 (2010)

22. Large oscillation amplitude spectra: strong coupling regime
Bennett et al., Phys. Rev. Lett. 104, (2010)

24. Drive amplitude changes coupling strength!
Peaks 1 (a) and 2 (b) Peaks 3-6: Exp (c) and Theory (d)
Experiment (solid) Theory (dashed)
Shell-filling can be determined by skewed peaks
Bennett et al., Phys. Rev. Lett. 104, (2010)

25. How can we measure the shell structure?
T- dependence of peak position [weak coupling regime]
Line shape (asymmetry) [strong coupling regime]
3. Energy dependence of tunneling rate
How can one measure the tunneling rate?

26. Lets look at weak coupling regime a bit more in detail
Due to the finite tunneling rate of electrons, the probability of having an extra electron on the dot at time t, <P(t)>, is out of phase with respect to the cantilever motion <z>. This results in a back-action force that changes both the force sensors resonance frequency, w and its dissipation, g , with magnitudes:
(in weak coupling regime)

27. What does G(E) look like?
ΔE/kB T
single non-degenerate level
continuous density of states with r=1
10, 8, 6, kBT
energy level spacing
Tunneling rate G= Gon + Goff depend on DOS(dot)

28. Tunneling rate measurements on InAs qdots
Shell structure and fill factors can directly be determined by measuring G(E)
Nano Lett. 15, 2324 (2015)
Degenerate states:
Two-fold degenerate level:
Red: shell filling n=0
Green: shell filling n=1
Four-fold degenerate level:
Orange: n = 0

29. Engineering tunneling barrier is crucial
Line shapes (weak coupling, single non-degenerate level):
Experimentally, in order to achieve sufficient signal to-noise, it is ideal to operate in a regime where the cantilever's response is evenly split between w and g.
This condition is met when the ratio between the effective tunneling rate G between the dot and the back electrode matches the mechanical resonance frequency w of the cantilever.

30. DOS of 5 nm Au particle (77K) C11 alkane thiol barrier
EC = 16 meV
Template stripped ultra flat gold
C11 alkane thiol barrier
5 nm Au particles,
measured at 77K
C11 alkane thiol barrier
Nano Lett. 15, 2324 (2015)

31. DOS of 5 nm Au particle (77K)
EC = 16 meV
Tunneling rate data (blue) superimposed with a fit to the analytical expression for a continuous density of states (dashed line)
Nano Lett. 15, 2324 (2015)

32. Bias spectroscopy above a 3nm Au dot @ RT in UHV
Engineering tunneling barriers: ML NaCl on Fe(001)
Bias spectroscopy above a 3nm Au RT in UHV
Measured EC = 137 meV. Au particle must be truncated sphere (FE model)
Truncated sphere
Nanotechnology 23, (2012)

33. Capacitance of a truncated sphere
Parameter
Value
Height nm
Volume
7 nm³
Capacitance
583 zF (10-21)
Hemisphere in vacuum r = 3.5 nm
Half-space filled with NaCl
337 zF
976 zF
C = 630 zF (6 ML) C = 596 zF (7 ML)
4 um
A gold crystal at 1000 ºC J. Cryst. Growth 50, 571 (1980); Acta Metal. 28, 1789 (1980)
TEM of our Au nanoparticles

34. What information can be extracted quantitatively?
Coulomb blockade energy
Eigen state energy (energy levels)
Shell structure
Excited state energies

35. Excited State Spectroscopy
Increase in dissipation occurs when the electron can tunnel into a higher shell.
s-shell peak: D D
An increase in dissipation occurs when the electron can tunnel into a higher shell.
Nano Lett. 12, 709 (2012)
Phys. Rev. Lett. 104, (2010) 35

36. What information can be extracted quantitatively?
Coulomb blockade energy
Eigen state energy (energy levels)
Shell structure
Excited state energies
Stability diagrams (coupling between dots)
Capacitive coupling (‘weak’ coupling)
Tunneling coupling (‘strong’ coupling)

37. Stability diagrams via imaging1 2 3 4 5 6 1 2 3 4 5 6
Topography
Frequency Shift
Dissipation

38. Imaging mode: spatial axis (x,y,z) = energy axis
Imaged at fixed z.
20nm
First scalebar 0.85Hz
Scanning changes the lever arm (or voltage divider) a
-8.0V
-9.0V
+6.8V
-8.0V

40. Imaging Coupled Quantum dots: stability diagram
20 nm
Test if lever arm a is linear, then
Convert spatial axis to energy axis

41. Coupled quantum dots Coherent or correlated? Weaker coupling
Example of 3 coupled QDs:
20nm
20nm
Stronger coupling

43. Note the large variability of confinement potentials!

44. Effect of environment on confinement potential
InAs structure appears to contain one QD.
While scanning suddenly we observe 2 QDs!
20nm
First scalebar 0.85Hz
Scan dir’n
-8.0V
-9.0V
+6.8V
-8.0V

45. Summary: what can one measure/learn?
Coulomb blockade energy
Eigen state energy levels & excitation energy via Amplitude shape (strong coupling regime),
Peak position as f(Temperature) or
Energy dependence of tunneling rate
Coupling between dots (Coulomb and tunneling)
‘Images’ can be converted to stability diagrams (without gates!)
Coherent interdot tunneling coupling and double-dot T1
Weak coupling tip-qdot (=linear response) measures dynamic response of charge in dots, allowing T1 to be extracted; this is NOT the same physics as in QPC measurements. The oscillating tip-field modulates the barrier between dots. The tip also measures the response to this modulation (back-action), which contains all of the quantum mechanical dynamics.

46. Summary: method
AFM tip serves as an electrostatic gate and a charge sensor
fm-AFM technique – key is low frequency dependent phase noise to cleanly measure dissipation
Qdot - sample tunneling barrier needs to be engineered: ~resonant frequency of cantilever
Interpretation of line shapes: quantum back action of single electron tunneling on sensor
Review:
Nanotechnology 28, (2017)

48. Dr. Yoichi Miyahara, Romain Stomp
Lynda Cockins
Antoine Roy-Gobeil
Prof. Aashish Clerk

49. Take home messages:
Substantial surface potential variations on semiconductor surfaces(>100mV , 50 nm lateral scale). Depends on processing conditions, affects transport properties and mobility.
EFM allows full characterization of energy levels of individual and coupled quantum dots, including dynamics. Physics is back action of quantum system (single electron) on cantilever.
Tuning fork based AFM systems (‘qPlus’) less sensitive, but can still be used as SGM to obtain intriguing results.

50. Can one measure spins?
P. Grutter

51. What information can be extracted quantitatively?
Coulomb blockade energy
Eigen state energy (energy levels)
Shell structure
Excited state energies
Stability diagrams (coupling between dots)
Capacitive coupling (‘weak’ coupling)
Tunneling coupling (‘strong’ coupling)
Power spectrum of electrical noise of environment, spatial localization of 2 level fluctuators.
Coherence time
Reorganization energies