1. The Free Electron Fermi Gas
12/31/2018
Early History of Metal Theory
(Drude, Lorentz, Fermi, Dirac, Pauli, Sommerfeld, Bloch, …)
The Basic Hamiltonian
Approximations & Assumptions
The Ground State (T = 0)
Wave-functions, allowed states, Fermi sphere, density of states
Thermal Properties
Expectation values, energy, specific heat
Electrical Transport Properties
DC and AC conductivities
Magnetic Properties
Classical Hall effect, Pauli paramagnetism, Landau quantization, the A-B phase, cyclotron resonance, the quantum Hall effect
This is the outline of this talk. Since this seems to be the only talk on nanotubes today, I will give a very brief introduction to carbon nanotubes, describing their basic crystal structure & band structure, and the condition of metallicity, which is most relevant to what I will be talking about later. The second part will describe more details of the theoretical predictions I just summarized, together with our numerical estimates for the magnitude of the effects we are looking for. In the third part, I will tell you a little bit about the samples we are using. They are individually-suspended in aqueous solutions and show a number of chirality-dependent peaks in optical spectra, which I will show you. Then in the last section I will show you our magneto-optical data, which we believe is convincing evidence of the influence of the Aharonov-Bohm effect on the electronic structure of the nanotubes. Our initial results are already published in this paper, but we have a lot more data since then.

2. Magnetic Properties of a Free Electron Fermi Gas
12/31/2018
The Classical Hall Effect
Pauli Paramagnetism
Landau Quantization
The Aharonov-Bohm Phase
Cyclotron Resonance
The Quantum Hall Effect
This is the outline of this talk. Since this seems to be the only talk on nanotubes today, I will give a very brief introduction to carbon nanotubes, describing their basic crystal structure & band structure, and the condition of metallicity, which is most relevant to what I will be talking about later. The second part will describe more details of the theoretical predictions I just summarized, together with our numerical estimates for the magnitude of the effects we are looking for. In the third part, I will tell you a little bit about the samples we are using. They are individually-suspended in aqueous solutions and show a number of chirality-dependent peaks in optical spectra, which I will show you. Then in the last section I will show you our magneto-optical data, which we believe is convincing evidence of the influence of the Aharonov-Bohm effect on the electronic structure of the nanotubes. Our initial results are already published in this paper, but we have a lot more data since then.

3. The Hall Effect z x y Lorentz force: Balance equation:
12/31/2018 z x y
Lorentz force:
Balance equation:
RH is independent of t and m  An excellent method for determining n

4. The Hall Effect More formal derivation magneto-resistivity tensor
12/31/2018
More formal derivation
magneto-resistivity tensor
magneto-conductivity tensor
Jy = 0

5. Density within the Drude Model
12/31/2018
rm [kg/m3]: mass density
A [kg]: atomic mass (mass of one mole)
rm/A moles atoms per m3
NArm/A atoms per m3, NA = 6.02 × 1023
n = NArmZ/A electrons per m3, Z: # of valence electrons
For Li, rm = × 103, A = × 10-3, Z = 1
n = 4.70 × 1028 m-3

6. Comparison with Experiments
12/31/2018
For Li, rm = × 103, A = × 10-3, Z = 1
n = 4.70 × 1028 m-3
RH = 1.33 × m3/C
good
RH(exp) = 1.7 × m3/C
For Zn, rm = 7.13 × 103, A = × 10-3, Z = 2
n = 1.31 × 1029 m-3
RH = 4.77 × m3/C
bad
RH(exp) = +3 × m3/C
Positive Hall coefficient!

7. Cyclotron Frequency and
the Hall Angle
12/31/2018
Newtonian equation of motion in E and B:
steady
state

8. Deviation from the Classical Hall Effect
12/31/2018

9. How Difficult is wct > 1 ?
12/31/2018
me = cm2/Vs  B > 1 Tesla
me = 1000 cm2/Vs  B > 10 Tesla
me = 100 cm2/Vs  B > 100 Tesla

10. Magnetic Properties of a Free Electron Fermi Gas
12/31/2018
The Classical Hall Effect
Pauli Paramagnetism
Landau quantization
The Aharonov-Bohm Phase
Cyclotron Resonance
The Quantum Hall Effect
This is the outline of this talk. Since this seems to be the only talk on nanotubes today, I will give a very brief introduction to carbon nanotubes, describing their basic crystal structure & band structure, and the condition of metallicity, which is most relevant to what I will be talking about later. The second part will describe more details of the theoretical predictions I just summarized, together with our numerical estimates for the magnitude of the effects we are looking for. In the third part, I will tell you a little bit about the samples we are using. They are individually-suspended in aqueous solutions and show a number of chirality-dependent peaks in optical spectra, which I will show you. Then in the last section I will show you our magneto-optical data, which we believe is convincing evidence of the influence of the Aharonov-Bohm effect on the electronic structure of the nanotubes. Our initial results are already published in this paper, but we have a lot more data since then.

11. Pauli’s Spin Matrices
12/31/2018
Let’s concentrate on electronic spins

12. Zeeman Effect B shifts the energy of each state by U
12/31/2018
quantization of angular momentum
B = 0
B  0
B shifts the energy of each state by U
ml: magnetic quantum number

13. Electron Spin Anomalous Zeeman splitting
12/31/2018
B = 0
B  0
Anomalous Zeeman splitting
Stern-Gerlach experiment (1922)  splitting into an even number of components (should be 2l +1)
Goudsmidt and Uhlenbeck (1925): spinning on its axis
Dirac’s theory (1928): existence of spin angular momentum

14. Spin Angular Momentum ms: spin quantum number
12/31/2018
ms: spin quantum number
ms = +1/2: “spin up” and ms = -1/2 : “spin down”

15. Gyromagnetic Ratio and the Electron g-factor
12/31/2018
: gyromagnetic ratio
g-factor
Quantum Electrodymanics (QED)

16. Spin is Purely Quantum Mechanical
12/31/2018
Orbital angular momentum:
As h  0, we can keep L non-zero by increasing the size of l to infinity
Spin angular momentum:
As h  0, S  0

17. Magnetic Susceptibility
12/31/2018
Total field [T] or [Wb/m2]
Applied field [A/m]
4p × 10-7 [T-m/A]
Induced field [A/m]
: magnetization curve
c > 0: paramagnetic
c < 0: diamagnetic

18. Calculate Spin c “Classically”
12/31/2018
spin “down”
spin “up”
Consider N electrons in volume V at temperature T in a magnetic field H, and calculate the total magnetic moment M
Too large, and temperature dependent

19. Pauli’s Spin Susceptibility
12/31/2018
g↓(e)
2mBm0H e
spin imbalance
magnetic moment per electron
g↑(e)
Net magnetic moment per m3: