CAPÍTULO VIII. ANÁLISIS Y RESULTADOS
8.2. Análisis grupos de enfoque
Theoretical description
In the case of the IR ZPL transition, the only magnetic response is the Zeeman splitting of orbital states of the 1E level. The 1A
1 level is an orbital singlet and a
spin singlet, so there is no magnetic response. Since both levels are spin singlets, all spin-orbit and spin Zeeman eects are not applicable. The energy levels of the
1E state are Zeeman split by ∆↑↓ =±lµ
BBz, wherel is the orbital g−factor. Since
absorption of the meta-stable singlet states is dicult [67] this spectra was actually taken in emission not absorption. Absorption measurements are dicult because the singlet levels are metastable states that must be continuously pumped by exciting the NV− with a 532 nm laser. This means the absorption measurements have to be
done in a two light source pump-probe fashion, as such MCD in emission was chosen for experimental simplicity. The excitation laser was carefully polarisation scrambled so the left and right circularly polarised emissions have equal probability. However, the laser is only there to pump the visible transmission. The system goes through the inter-system crossing to the 1A state before infra-red emission so no character
Figure 9.4: MCD emission of the NV− IR ZPL, the left and right circularly
polarised emission are given by the red and blue arrows.
The previous arguments of the optical selection rules and analysis of the absorption spectra is exactly the same for emission. The relevant dipole strengths between the excited and ground states are σ↑ and σ↓. Labelling the energy levels as EA, E↓ and
E↑, the emission band is,
ZL+R(E)
E3 =CTOT(σ↓L(E↓−EA−E) +σ↑L(E↑−EA−E)), (9.19)
where CTOT is a constant which contains information about the total emission
strength. For emission the spectra is divided by E3 = (hν)3, unlike absorption
which was divided by E. This is because the emission strength is essentially the
Einstein spontaneous emission rate which is ∝ ν3 [212]. As with absorption this
scaling is required to ensure linearity of the absorption but its eect is fairly min- imal over such a small wavelength range. Similarly, the dierential emission signal will be,
ZL−R(E)
9.5. RESULTS AND ANALYSIS Applying moment analysis the total area of the band is,
I =
Z +∞
−∞
ZL+R(E)
E3 dE =CTOT(σ↓+σ↑). (9.21)
The centroid of the band is then,
¯ E = 1 I Z +∞ −∞ EZL+R(E) E dE = ((E↓−EA) + (E↑−EA))/2, (9.22) the average energy dierence between the excited and ground states. The zeroth spectral moment of the band ishZL+R
E3 i0 =CTOT(σ↓+σ↑). The rst spectral moment
of the MCD signal is hZL−R
E3 i1 = CMCD(σ↓E↓−σ↓E↑). The energy levels of the
ground states are simply E↑↓ = ±glµBBz. The transition strengths |hA|σ↓| ↓i|2 =
|hA|σ↑| ↑i|2 [207], are equal σ↑ =σ↓, giving the ratio of the moments,
hZL−R E3 i1 hZL+R E3 i0 = CMCDσ↑↓2glµBBz 2CTOTσ↑↓ =glµBBz. (9.23) Gaussians as shown in equations9.17and9.18were also tted to the spectra. Using equation (9.15) the separation of the two Gaussians can be compared to the ratio of moments giving d=glµBBz.
Results and analysis
The ratio of moments is shown in gure 9.5(b), the gradient of which is µBlcosθ=
0.251(1) GHz/T using the values from curve tting, µBlcosθ = 0.208(4) GHz/T
using the values from spectral moment and µBlcosθ = 0.230(3) GHz/T from the
average of the two. Since the magnetic eld is aligned along the h100i direction the
eld projection along the NV axis is Bz = Bcosθ with cosθ = 1/
√
3. Using the
average gradient above, gives an orbitalg−factor ofl= 0.0284(4), to our knowledge,
this is the rst time this parameter has been measured.
The orbitalg−factor found here is signicantly reduced from the free atom value
of l = 1 or the experimentally determined NV− excited 3E state values. The only
measurements of the orbital g−factor of the 3E are by Reddy et al who found
l = 0.1by performing MCD [47] and by Hanzawa et al [148] who measured al = 0.2
from the Zeeman shift of the NV− ZPL directly using very large magnetic elds. It
(a)
(b)
Figure 9.5: (a) Example of MCD dierential emission spectra (upper) and total emission spectra (lower) for the IR ZPL of NV− at a temperature of 1.46 K, notice
the sign change due to magnetic eld. Fits are of the form shown in equations
9.11 and 9.10 using Gaussian lineshape functions. (b) Zeeman shift of 1E orbitals
determined the separation of two Gaussian ts (points) and from spectral moments (stars) vs B, where B k[100], line is simple linear t.
9.5. RESULTS AND ANALYSIS
Figure 9.6: Depiction of the linear Jahn-Teller interaction between anE electronic
level and single E and A phonons. The linear Jahn-Teller interaction of an E
electronic state with an E phononE⊗E =E⊕A1⊕A2 vibronic levels. A further
quadratic interaction will break the degeneracy of the A1 and A2 levels. Note that
E ⊗A1 = E, so interactions with symmetric A1 phonons do not introduce new
levels. The linear energy separation of the A1 and A2 levels from the E level is ∆
and the quadratic splitting of theA1 andA2 levels isδ. The application of transverse
strain slightly mixes the A1 and E levels such that the A1 level becomes apparent
in emission.
should be twice that of the3E orbitalg−factor. This is because in the1Elevel, two
holes occupy theeorbitals (e2), whereas in the3E one hole occupies theeand other
hole occupies the a orbital, which has no angular momentum [75, 213]. To explain
the loss of angular momentum, a Ham reduction factor from a dynamic Jahn-Teller interaction is assumed [145]. The Ham reduction is a consequence of the mixing of the electronic states with vibrational levels, creating multiple new vibronic levels. Properties of the electronic state such as spin-orbit and orbital angular momentum are distributed into the new vibronic states which can reduce the apparent magni- tude of these properties. This eect is increased for larger Jahn-Teller coupling. The Ham reduction of the orbital magnetic moment isp= exp[−1.974×(EJ T/~ω)0.761],
where EJT is the Jahn-Teller energy and ω is the vibrational frequency [149]. To
get an estimate of the Ham reduction, an unquenched estimate of the 1E orbital
g−factor is required. Abtew et al [136] theoretically determined the dynamic Jahn-
Teller energy in the3E to beE
J T/~ω= 0.35with a vibrational frequency of~ω = 71
meV. Combining the experimentally measured orbitall3E with the Jahn-Teller and
Ham quenching factor p3E gives the unquenched orbital l03E for the 3E state (the
unquenched parameters are denoted by a prime).
p3E = exp[−1.974×(EJ T,3E/~ω)0.761] (9.24)
The singlet 1E is expected to have an unquenched orbital l0
1E twice that of the 3E
l0
1E (i.e. l
0
1E = 2l
0
3E), which gives an expected Ham reduction of the singlet orbital
l1E of, p1E = l1E 2l0 3E (9.26) = 0.05↔0.025, (l3E = 0.1↔0.2) (9.27)
This reduction factor gives the 1E Jahn-Teller energy ratio of, EJ T ~ω 1E = ln (p1E) −1.974 0.7611 (9.28) = 2.0↔2.3, (l3E = 0.1↔0.2). (9.29)
Abtew et al [136] determined that the triplet 3E level has a Jahn-Teller energy
ratio of EJ T/~ω = 0.35, as such the Jahn-Teller interaction found in the 1E is
approximately 6 times larger. As shown by Rogers et al [66] for the NV− IR ZPL and
by Davies for the NV0 ZPL [65], Jahn-Teller can explain the emergence of new A 1
emission level under uniaxial stress, thisA1 level was 115 cm−1 above the groundE
vibronic level. As shown in gure9.6, these extra levels appear when anE electronic
state interacts with E vibrations producing vibronic states E⊗E =E⊕A1 ⊕A2.
Using a relation from Ham [149], the energy of the A1 and A2 states above the
groundE state can be predicted using linear Jahn-Teller theory and the vibrational
frequency, where the A1 is ∆ = ~2ω
~ω EJ T
above the E level. This linear relation
assumes that the quadratic Jahn-Teller interaction is much smaller than the linear, but enough to break the degeneracy of the A levels (i.e ∆δ). Using the phonon
energy from Abtew et al of~ω= 71 meV, the1E orbitalg−factor valuel1E = 0.025
determined from this MCD measurement and the 3E orbital g−factor l
3E = 0.1
value from Reddy et al gives a feature at ∆ = 167 cm−1 (20.7 meV). Using the
l3E = 0.2 from Hanzawa et al instead of Reddy et al gives a feature at at ∆ = 127
cm−1 (15.7 meV). These values agree fairly well with the result from Rogers (115
cm−1) et al and the characteristic energies by Robledo et al (16.6 meV) [83] and
Acosta et al (15 meV) [64]. It is likely that the predicted values of the A1 feature
are slightly higher because of a non-negligible quadratic Jahn-Teller (δ) which is not
included in the above expression for ∆. As such, it is expected that the measured
A1 level is the lower of the twoA levels. The values from Robledo et al, Acosta et
al, Rogers et al and this work are from very dierent types of measurements that provide observations and quantitative agreement of a similar energy value. This is
9.5. RESULTS AND ANALYSIS further evidence of a strong Jahn-Teller in the 1E and its likely role in the lower
ISC.
Kehayias et tal [67] cite the presence of a dynamic Jahn-Teller eect in the 1E
to explain the disagreement of the absorption and emission bands of the IR ZPL. Using the Jahn-Teller energy measured here the vibronic levels and the strange phonon side-band disagreement of the singlet levels can be further investigated. This and other ab initio studies of the lower ISC can use this Jahn-Teller energy to try and un-pick the mysteries of the lower singlet levels. A strong Jahn-Teller interaction doesn't easily explain why IR ZPL strain parameters are much larger than expected, so there are still remaining unanswered questions surrounding the NV− singlet levels but it is increasingly clear that strong vibronic eects are critical
in the understanding of lower singlet level.