Spin and NMR Techniques
4.2 Nuclear Magnetic Resonance .1 Hyperfine Interaction
We are not interested in the nuclear spin itself, but, due to the hyperfine interaction, NMR can be used to probe the electron spin. The full hyperfine Hamiltonian [6] describing the interaction between an electron and the magnetic moment due to the spin of the nucleus is
( )( ) ( )
,3 8 3
1
4 3 3 2
0
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧ ⎥+ ⋅
⎦
⎢ ⎤
⎣
⎡ ⋅ ⋅ − ⋅
+
⋅
−
= M M M M R
R
R M R M M R
R L m
H q I S I S I S I
e hf
G G G G G G
G π δ
π μ
where q is the nuclear charge, me is the mass of the electron, RG
is the relative separation between the electron and nucleus, LG
is the orbital angular momentum of the electron relative to the nucleus,MGI
is the magnetic moment of the nucleus, G G/=, I g MI = nμn where gn is the nuclear g-factor,μnis the nuclear Bohr magneton, IG
is the nuclear spin, and is the magnetic moment of the electron, where g0 is the free electron g-factor, μB is the electron Bohr magneton, and
G = G g0 S/ MS = μB
SG
is the electron spin. The first term is due to the interaction between the nuclear magnetic moment and the magnetic field created at the nucleus by the orbital angular momentum of the electron. The second and third terms stem from the dipole-dipole interaction between the nuclear and electronic spin, where the last term is the contribution from the singularity that occurs when the electronic wavefunction and nucleus spatially overlap. It turns out that this last term, named the “point contact” term, is the dominant contribution to Hhf for our 2DES samples.
4.2.2 Point Contact Interaction in n-Type GaAs
For electrons in the conduction band at the Γ point in GaAs, the electronic wavefunction is composed mostly of an s-type orbital. This means that the first term in Hhf can be neglected since for L = 0, this first term is zero. Because of the spherical symmetry of the s-type orbital versus the symmetry of the dipole interaction, the second term also vanishes. The third term, however, remains since the s-type orbital wavefunction is nonzero at the origin. Thus,
( )
0 0
8 ,
4 3
hf B n
H μ π μ γg I S δ R
= − π G ⋅ G G
=
where γn =gnμn/= is the nuclear gyromagnetic ratio. The value of the hyperfine correction to the total energy is then
( )
20 0
2 0 ,
hf 3 B n
E = μ g μ γ IG ⋅ SG ψ
=
Where ψ
( )
0 is the value of the electronic wavefunction at the position of the nucleus.Paget et al. have estimated the magnitude of this term for n-type GaAs [7] for each of the three nuclear species present in GaAs: 75As, 69Ga, and 71Ga. The result is given in terms of an effective magnetic field BN due to the nuclear polarization of the host semiconductor:
* , 3
2 0 0
∑
∑
⎟⎟⎠
⎜⎜ ⎞
⎝
= ⎛
=
α α α α α
α α μ γ
I d g x
B g
BN G
=
where Bα is the contribution due to the individual nuclear species, g* is the effective g- factor for electrons in the conduction band at the Γ point in GaAs (g* = -0.44), xα is the fractional concentration of each nuclear species, dα is the electron density at the nucleus, and IGα
is the average value of the nuclear spin. The estimated contribution from each nuclide at T = 0 is B75As = -2.76 T, B69Ga = -1.37 T, and B71Ga = -1.17 T, giving rise to a maximum total contribution of BN ≈ -5.3 T. These effective fields are negative, meaning they will oppose any externally applied magnetic field B0. This leads to a total electronic Zeeman splitting given by
(
N)
B
Z g B B
E = +
Δ *μ 0 .
A decrease in the nuclear polarization will reduce the magnitude of BN, causing an increase in the magnitude of the Zeeman splitting. Also note that BN affects only the Zeeman energy, not the electron’s orbital motion.
4.2.3 Nuclear Polarization
In the case of an externally applied magnetic field B0, it is simple to obtain the equilibrium fractional nuclear polarization of a given nuclear species at temperature T:
, ) exp(
) exp(
0
∑
∑
−
=
−
=
−
−
= I
I m
z I
I
m z z
z z
m I
m m
β β ξ
where .β =γn=B0/kT When β is small we can use the approximation
( )
3 ,
0 1
0 kT
I
n B +
=γ = ξ
which is just the nuclear Curie law.
For reference, Fig. 4.1 shows Bα and BN versus temperature for B0 = 10 T. At conditions roughly similar to those of our experiments, B0 ~ 10 T, T ~ 100 mK, we have BN ~ -0.2 T and a nuclear polarization of ξ0 ~4%.
-5 -4 -3 -2 -1 0
Bα , BN (T)
0.001 2 4 6 80.01 2 4 6 80.1 2 4 6 81 Temperature (K)
total
75As
71Ga
69Ga B0 = 10 T
Fig. 4.1. Effective magnetic field due to thermal equilibrium polarization of host semiconductor nuclides at a static magnetic field of B0 = 10 T. Dotted and dashed lines show the contribution from 75As, 71Ga, and 69Ga separately, and the solid line displays the total contribution due to all three species.
4.2.4 Bloch Equations
The phenomenological equations of Bloch [8] describe the evolution of the nuclear magnetization in the presence of a static magnetic field BG0 =B0zˆ,
and an ac magnetic field BG1 =B1xˆcos(ω1t),
perpendicular to B0. It is assumed that the nuclear magnetization reaches thermal equilibrium with relaxation times T1 and T2, known as the longitudinal and transverse relaxation times, respectively, such that in the absence of the ac magnetic field (B1 = 0)
T2
M dt
dMx x
−
= ,
T2
M dt
dMy y
−
= ,
and
1 0
T M M dt
dMz − z
= ,
where M0 is the thermal equilibrium magnetization. With B1 ≠0, the full Bloch equations are
, ˆ ˆ
ˆ
1 0 2
T z M M T
y M x B M dt M
M
d x y z −
+ −
−
×
= G G G
γ
where γ is the nuclear gyromagnetic ratio and BG BG0 BG1 +
= . It is convenient to transform this equation into a rotating frame of reference that rotates in the x-y plane at the same frequency ω1 as the ac magnetic field. Then
T2
M M dt
M
d x
y
x ′
′ − Δ
′ =
ω ,
z r y x
y M
T M M dt
M
d ω ′ −ω
′ − Δ
−
′ =
2
and
,
1 0
T M M M
dt
dM z
y r
z −
′ +
=ω where MG
′ is the magnetization in the rotating frame, ωr =γB1/2 is the Rabi frequency, and the detuning Δω =
(
ω1−ω0)
/ dM dt′
, where ω0 is the NMR resonance frequency. Under steady state conditions, G =0,
we have for the deviation of the z-component of the nuclear polarization from equilibrium:
( )
2 221 2 2 1 22 0
0 1 T TT
T M T
M M
r z r
ω ω
ω + Δ
= +
− .
This describes a Lorentzian NMR lineshape (NMR-induced change in polarization versus ω) with a half-width at half-max given by
HWHM = 2 1 2
2
1 1
T T +ωr T .