Nuclear Magnetic Resonance

1
Optically Pumped NMR in GaAs
2
OPNMR in GaAs quantum wells

• Outline
• NMR and the Zeeman Hamiltonian.
• The Fermi-Contact interaction.
• Knight Shift.
• Dynamic Nuclear Polarization.
• OPNMR in GaAs.

3
Nuclear Magnetic Resonance

• The interaction energy of a nucleus in a magnetic
  filed is given by
• Where is the gyromagnetic ratio of the
  nucleus.
• The eigenvalues of this Hamiltonian are simply
• Where
• We need an interaction that can cause transition
  from one state to another,
• only for
• The allowed transition are between adjacent
  levels, giving

4

• At equilibrium the nuclei are polarized in the
  field direction, and no signal is detected in the
  coil.
• -(c) After a pulse the magnetization
  rotates in the xy plane, and signal can be
  detected in the coil.

5
(No Transcript)
6
(No Transcript)
7
The hyperfine Hamiltonian

• The Hamiltonian of the electron in the magnetic
  field of the proton
• We treat a single electron atom with a spin ½
  nucleus.
• The hyperfine Hamiltonian is given by

8
Quenching of the Orbital motion.

• The first term in the Hamiltonian corresponds to
  the interaction between the nucleus and the
  orbital moment of the electron.
• The magnetic field produced by a moving charge
• We get a field of about 60Tesla!
• The fact that NMR experiments are done in a much
  smaller field is due to the fact that the orbital
  motion of the electrons is quenched in solids.

9
Coupling with the electron spin

• To tackle the problem with the dipolar
  interaction at the origin the proton is assumed
  to have a finite size.
• Outside of the nucleus the magnetic field is
  exactly dipolar.
• Inside it is assumed to be constant.
• The dipolar magnetic field is
• Where M is the magnetic moment of the nucleus.

10

• The internal field is constant and parallel to
  the moment.
• At the magnetic field is zero.
• The total flux trough the xy plane is also zero.

11
The flux trough a disk centered at the nucleus
with radius a The flux trough the rest of the
xy plane We get that the field in the nucleus
is
12

• Outside of the nucleus the interaction with the
  electron spin is the regular dipole-dipole
  interaction.
• The contribution of the internal field is known
  as the contact term.
• We insert the field we found into the last term
  of the hyperfine Hamiltonian, and calculate its
  expectation value
• Now we let approach zero

13

• The contact term is therefore (in order to get
  the expectation values we found)
• The contact term is non-zero only for the S
  states of the atom because for all the other
  states .
• The interaction can be written in terms of the
  nucleus and electron spins as

14
Knight shift

• Using the contact term we can write a spin
  Hamiltonian
• Where is the probability to find an
  electron at the nucleus site. The electronic wave
  function is in general a complicated many body
  wave function.
• Let us assume that the electrons are
  noninteracting. The wave function of the single
  electron will then be the Bloch functions
• The effective interaction is found by summing
  over all the occupied electronic states
• Where P is the occupation factor.

15

• The term in the brackets is the contribution of
  state k to the electron magnetization.
• Where is the spin susceptibility and
  is the external field.
• We get that the effective interaction depends on
  the average spin susceptibility

16

• When added to the Zeeman term, the Fermi
  contact term shifts the resonance frequency!
• The shift is the result of an additional field
  that is created because of the polarized
  electrons.

17
Dynamic Nuclear Polarization

• Let us assume that the only channel the nuclei
  have for relaxation is the mutual flip of a
  nuclear and an electronic spin via the hyperfine
  interaction.
• The balance equation for the steady-state
  populations has the form
• Where is the projection of the nuclear spin
  on the direction of the magnetic field,
  are the population of the nuclear spins and
  are the population of the
  electronic states with spin up and down.

18

• The transition probabilities and
  are connected by the thermodynamic
  relation
• The Zeeman energy of the nucleus is very small
  compared to the Zeeman energy of the electron.
• The temperature T is the temperature of the
  reservoir from which the energy needed for the
  transition is taken.
• Then we get
• Where is the
  equilibrium value of the average electron spin.

19

• We get for the average nuclear spin
• Any deviation from equilibrium in the electron
  spin system results in nuclear polarization!
• If zero spin polarization is maintained
  then one get the usual Overhauser effect
• The nucleus is polarized as it had the much
  larger magnetic moment of the electron!

20

• In GaAs irradiation with circularly polarized
  light results in a polarization of the electronic
  spins.
• The maximum polarization in GaAs is 0.5.

• Taking into account relaxation processes the
  polarization will be
• Where is the lifetime of the electron and
  is the relaxation time of the electron spin in
  the conduction band.

21

• For the nucleus we define two relaxation times
  
• is the relaxation due to interaction with
  the polarized electrons.
• is the relaxation due to all other
  mechanisms.
• Taking this into account the nuclear polarization
  is
• To get polarization transferred to the nuclear
  spin system it is necessary that

22
Spectrum of bulk GaAs
Characteristic numbers for the relaxation times
in GaAs
We have 3 spin 3/2 nuclei in GaAs

• It was shown that the main contribution to the
  nuclear polarization in bulk GaAs comes from
  electrons trapped in shallow donor states and not
  from free electrons.
• The numerical estimate of the hyperfine field in
  the nucleus site due to free electrons is less
  than 1G, compared to the donors filed of

23
Basic pulse sequence
A train of 20 pulses that destroys any
polarization of the nuclear system.
is the period during which the laser source
is on.
Detection of the NMR signal.
is the period during which the laser source
is off.
24
Results of OPNMR in bulk GaAs

• Changing the helicty of the light results in
  inversion of the NMR signal.
• The NMR signal is observed in the range
  1.47-1.52eV, although the band gap is 1.52eV.

25
OPNMR in quantum wells

• The wells are delta-doped GaAs/AlGaAs.
• We can think of two kind of electrons
• 1) Doped electrons (
  ).
• 2) Photoexcited electrons.
• The electrons are confined in the well so their
  hyperfine interaction with the nuclei can be very
  large, although they are free .
• of the nuclei in the barrier is similar to
  the one in bulk GaAs, but the in the well
  is much shorter ( 30s ).

26
In this figure the Ga spectrum of a single well
is shown.

• The dynamical nuclear polarization occurs in the
  wells via electron-nucleus coupling. Then it
  spread to the barriers by nuclear spin diffusion.
  The polarization diffuses at about 10A/sec.
• The Ga NMR spectra of GaAs wells and AlGaAs
  barriers can be extracted from the short and long
  OPNMR spectra.
• The signal of the nuclei in the wells is shifted
  compared to the signal from the barriers, this is
  due to the 2D electron gas susceptibility.

27

• The solid line is a fit to the function
• Which represents the equilibrium polarization of
  the doped electrons.
• The effective electron g value is

28
Quantum Hall regime.

• The Knight shift increases as function of T.

29

• We can see that at and the
  electron polarization saturates at final
  temperature. At these filling factors the system
  is ferromagnetic!

30
(No Transcript)
31
Summery

• NMR is a powerful method.
• Optical pumping allows the use of NMR in dilute
  or small systems.
• OPNMR allows direct observation of 2D electron
  gas systems.
• In the future the method can be applied to
  systems not in the quantum Hall regime. For
  example in the metal-insulator problem.