1. Transport in nanostructures
M. P. Anantram
( )
Center for Nanotechnology
NASA Ames Research Center, Moffett Field, California
Modeling of transport in ultra small MOSFETs using techniques other than traditional DD methods has been a topic of recent active research in the Computational Electronics community. In this talk, I will discuss our work on modeling of transport in ultrasmall MOSFET from a purely quantum mechanical approach. The work to be presented is not phenomenological theory, which is central to explaining experimental results. Rather, it can be regarded to be an experiment performed on a computer under idealized conditions that is often used in the modeling and simulation world. So it helps understand certain aspects of ballistic MOSFETs both qualitatively and quantitatively.
11/20/2018

2. Outline Transport: What is physically different? Applications
- Resonant Tunneling Diodes (RTD)
- Carbon Nanotubes
- DNA

3. Trends in Device Miniaturization
Down scaling of semiconductor technology
Molecular devices
Smaller and Faster Devices
Conventional
Hybrid
New
- Ultra Small MOSFET
- Interference based devices
- Resonant tunneling diodes
- Single electron transistors
- Carbon nanotube
- Molecular diodes
- DNA?

4. Conventional Methods of Device Modeling
Electrons are waves. de Broglie wavelength of an electron is,
h/p,
where p is the momentum
Device dimensions are much larger than the electron wave length
Transit time through the device is much larger than the scattering time
Diffusion equation for semiconductors
Ballistic Phase-coherent
Diffusive

5. 1 Electrons behave as waves rather than particles
T(E)
R(E)
T(E)+R(E)=1
Electrons behave as waves rather than particles
Schrodinger’s wave equation
Poisson equation still important
Landauer-Buttiker Scattering theory
In this theory,
Current,
T(E) – Transmission probability for an electron to traverse
the device at energy E
fLEFT(E) – occupancy factor / probability for an electron to be
incident from the left contact (Fermi- dirac factor)

6. Transport in molecular structures – Interplay between chemistry and physics
Quantum chemistry tools (perform energy minimization) or Molecular Dynamics (MD) are used to find chemically and mechanically stable / preferable structures
Schrodinger equation describes electron flow through the device
Poisson’s equation gives the self-consistent potential profile
Chemically & Mechanically Stable Structures
Energy Minimization, quantum chemistry (hundred atoms)
Molecular Dynamics simulations (millions of atoms)
Number of atoms  accuracy
Current, Electron Density
Schrodinger’s equation / non equilibrium Green’s function
Potential (Voltage) Profile
Poisson’s equation
New Devices

7. Outline Transport: What is physically different? Applications
- Resonant Tunneling Diodes (RTD)
- Carbon Nanotubes
- DNA

8. Resonant Tunneling Diode
GaAs
AlGaAs +
Typical thickness: tens of Angstrom 1
Transmission
Probability
T(E)
Black – semi-classical
Red - quantum
What quantum feature
does the peak represent?
200meV
Energy (E)

9. Example: Resonant Tunneling Diode
Current
Voltage
Negative differential resistance
Peak to valley ratio should be large

10. Outline Transport: What is physically different? Applications
- Resonant Tunneling Diodes (RTD)
- Carbon Nanotubes
- DNA

11. Bonard et al, Appl. Phys. A 69 (1999)
Nanotube Images
Single-wall nanotube Multi-wall nanotube
Bonard et al, Appl. Phys. A 69 (1999)
Thess et. al, Science (1996)

12. Quantum conductance experiment
Near perfect quantum wire
Crossed nanotube junction: Inter-tube metal-
semiconduction junction, rectifier
Frank et. al, Science 280 (1998)
Fuhrer et al, Science (2000)
and not

13. Work horse of conventional computing
IBM
Wind et. al, Appl. Phys. Lett., May 20, 2002
Logic gates, oscillators, … using many nanotubes
on a single wafer has been demonstrated.

14. Bent Nanotube or Intra-molecular junction? Intra-molecular diode?
Cees Dekker, Delft University

15. Electromechanical Switch?
Tombler et al, Nature 405, 769 (2000)
Mechanism for conductance decrease?

16. Graphene Blue box – unit cell of graphene a1 & a2 – lattice vectors
r1, r2 & r3 - bond vectors
Two atoms per unit cell a2 a1
Applying of Bloch’s theorem:
For graphene, symmetry dictates that t1=t2=t3

17. a2 a1

18. a1 a1 a2 a2

19. Graphene to Nanotube eikf=eik(f+2p)
Y = eikxx+ikyy (u v)
Example, (6,0) zigzag tube,

20. Nanotube wavefunction
p - integer

21. Summary of main electronic properties
Metallic nanotubes:
n-m = 3*integer
Semiconducting tubes:
Bandgap a 1/Diameter
Armchair tubes are truly metallic
Other metallic tubes have a tiny curvature induced bandgap
zigzag (n,0)

22. Summary of main electronic properties
zigzag (n,0)
Armchair tubes do not develop a band gap

23. Summary of main electronic properties
Metallic nanotubes:
n-m = 3*integer
Semiconducting tubes:
Bandgap a 1/Diameter
Armchair tubes are truly metallic
Other metallic tubes have a tiny curvature induced bandgap
zigzag (n,0)

24. Shapes in nature
Nanohorns
Torus

25. Armchair nanotube: Bands
Fermi
energy
Figure below is a pictorial representation of the MIT 25nm MOSFET
Close to E=0, only two sub-bands, (6.5 kW)
At higher energies, (< 1kW)
Low bias record (multi-wall nanotube) (500W)
Can subbands at the higher energies be accessed to drive large currents through these molecular wires?

26. Quantum conductance experiment
Frank et. al, Science 280 (1998)

27. VAPPLIED > 200mV, slow increase
Frank et. al, Science 280 (1998)
VAPPLIED < 200mV, G~2e2/h
VAPPLIED > 200mV, slow increase
E ~ ±120meV, non-crossing bands open
At E~2eV, electrons are injected into about 80 subbands
Yet the conductance is ~ 5 e2/h

28. Semiclassical Picture
DENC
Transmission in crossing subband
Bragg refelection
Zener tunneling (non crossing subbands)
(r,r’) are indices for the matrix equation
These equations are solved only inside the green box.
Yet Sch Eqn or the above equations are not being solved for a closed system. The open boundaries in the contacts are included via a self-energy.
The strength of the two processes are determined by:
Tunneling distance, Barrier height (DENC), Scattering
DENC a 1/Diameter. So the importance of Zener tunneling increases with increase in nanotube diameter.

29. Yao et al, Phys. Rev. Lett (2000)
dI/dV 4e2/h for Va < 2DENC ( DENC 1.9eV )
DENC changes with diameter
Effect of diameter on current?
155 mA
25 mA
Yao et al, Phys. Rev. Lett (2000)
The first feature is directly a consequence of how the CB bends in poly silicon.
The differential conductance is not comparable to the increase in the number of subbands. (20,20) nanotube – 35 subbands at 3.5eV
Two classes of experiments with order of magnitude current that differs by a factor of 5!
Our ballistic calculations agree with increase in conductance

30. Summary (Current carrying capacity of nanotubes)
Nanotubes are the best nanowires, at present!
However, Bragg reflection limits the current carrying capacity of nanotubes
Large diameter nanotubes exhibit Zener tunneling
Conductance much larger than 4e2/h is difficult
non crossing  non conducting
crossing  conducting
First term of n(x) is the electron density in the case of 3D bulk and is equal to the doping density.
First term of V(x) says that the potential at x=0 increases with kf / doping
These expressions are valid only at ZERO temp and are a poor approximation to what happens at ROOM TEMP and these doping densities.
Nevertheless they are useful.
Phys. Rev. B 62, 4837 (2000)

31. Tombler et. al, Nature 405, 769 (2000) sp2 to sp3 Upon deformation
Stretching of bonds
Opens bandgap in most nanotubes
[Phys. Rev. B, vol. 60 (1999)]
What is the conductance decrease due to?

32. Approach 1) AFM Deformation 2) Bending Structure Relaxation
Central 150 atoms were relaxed using DFT and the remaining atoms were relaxed using a universal force field
Density of states and conductance were computed using four orbital tight-binding method with various parametrizations

33. Bond Length Distribution & Conductance
(12,0) Zigzag
BENDING
AFM DEFORMATION

34. AFM Deformed versus Stretched

35. What happens to other chiralities?
Metallic zigzag nanotubes develop largest bandgap with tensile strain.
All other chiralities develop bandgap that varies with chirality (n,m).
Experiments on a sample of metallic tubes will show varying decrease in conductance.
Some semiconducting tubes will show an increase in conductance upon crushing with an AFM tip.

36. Summary (electromechanical switch)
Metallic nanotubes develop a bandgap upon strain. Detalied simulations show that this is a plausible explanation for the recent experiment on electromechanical properties by Tombler et al, Nature (2000)
In contrast, we expect nanotube lying on a table to behave differently. A drastic decrease in conductance is expected to occur only after sp3 type hybridization occurs between the top and bottom of the nanotube.
Suspended in air
Table Experiment
AIR
SiO2

37. Outline Transport: What is physically different? Applications
- Resonant Tunneling Diodes (RTD)
- Carbon Nanotubes
- DNA

38. DNA Conductance Double helix – a backbone & base pairs
Building blocks are the base pairs:
A, T, C & G
Example: 10 base pairs per turn, distance of 3.4 Angstroms between base pairs.
Arbitrary sequences possible
A challenge for nanotechnology is controlled / reproducible growth. DNA is an example with some success. However, there are many copies in a solution!
2D and 3D structures with DNA base pairs as a building block have been demonstrated
Lithography? Not yet.

39. basepair

40. Experiments Conductivity in DNA has been controversial
Electron transfer experiments (biochemistry) / possible link to cancer
Transport experiments (physics)

41. Semiconducting / Insulating
Voltage (V)
Current
~ 1nA
Semiconducting / Insulating
Porath et. al, Nature (2000)
Current
~ 10nA
Voltage
Metallic, No gap
20mV
Fink et. al, Science (1999)

42. Counter-ions
Is conduction through the base pair or backbone? - Basepair
When DNA is dried, where are the counter ions?
Crystalline / non crystalline?
Counter ions significantly modify the energy levels of the base pairs
Counter-ion species is also important
Resistance increases with the length of the DNA sample (exponential within the context of simple models)
Counter-ions