CuPc-Enhanced NV Relaxometry: Method & Analysis

• The paper outlines a method using shallow NV centers for T1 relaxometry to extract CuPc spin dynamics and spectral properties.
• CuPc-enhanced NV relaxometry leverages Lorentzian spectral models to quantify spin fluctuations and dipolar interactions in molecular films.
• The technique precisely determines NV–CuPc standoff distances (~1 nm) and reveals detailed hyperfine interactions within the CuPc spin bath.

CuPc-enhanced NV relaxometry refers to the use of $T_1$ relaxometry of shallow diamond nitrogen-vacancy (NV) centers to probe the electron spin ensemble of a polycrystalline copper phthalocyanine (CuPc) thin film. This approach enables the extraction of the CuPc spin ensemble's spectral, dynamical, and orientational properties at room temperature, which are inaccessible to bulk electron resonance techniques. The methodology leverages the sensitivity of NV centers to the fluctuating magnetic fields produced by CuPc electronic spins, providing quantitative access to correlation timescales, hyperfine structure, and local environments, and even yields the NV center’s depth with sub-nanometer accuracy.

1. Microscopic Origin of NV $T_1$ Relaxation in a Fluctuating CuPc Spin Bath

A shallow NV center (electronic spin $S_{NV}=1$) positioned at depth $z$ below a CuPc film is subject to dipolar interactions with the unpaired electron ($S=1/2$) of each CuPc molecule. The relevant NV spin levels $|0\rangle$ and $|-1\rangle$ are split by a zero-field term $D\approx 2.87~{\rm GHz}$ and a Zeeman shift $\omega_{NV} = \gamma_e B_0$.

The temporal fluctuations from the CuPc spin bath create a transverse magnetic field at the NV site: $$\delta B_{\perp,i}(t) = \frac{\mu_0}{4\pi} g_e \mu_B \frac{1-3\cos^2\theta_i}{r_i^3} S_i^\perp(t)$$ where $\theta_i$ is the angle between the NV quantization axis and the vector connecting NV and bath spin $i$, and $S_i^\perp$ is the transverse CuPc electron spin component.

The Hamiltonian in the rotating frame is

$$H_{int}(t) = \gamma_e S_{NV}^x \sum_i \delta B_{x,i}(t) + \gamma_e S_{NV}^y \sum_i \delta B_{y,i}(t).$$

Applying Fermi’s Golden Rule, the NV $|0\rangle \to |-1\rangle$ depolarization rate (relaxation rate $\Gamma_1$) is

$$\Gamma_1 \equiv 1/T_1 = \gamma_e^2 \sum_i \langle \delta B_{\perp,i}(t) \delta B_{\perp,i}(0)\rangle_{\omega_{NV}}$$

where $\langle...\rangle_\omega$ is the Fourier transform at $\omega_{NV}$. Modeling the spin bath as a continuous density $\rho$ and integrating over all CuPc spins in a semi-infinite film yields

$$\Gamma_1 = \rho \left( \frac{\mu_0 g_e \mu_B}{4\pi} \right)^2 S(S+1) \frac{2\tau_c}{1+(\omega_{NV}\tau_c)^2}\frac{1}{z^3}.$$

A common explicit form is

$$(1)\;\; \Gamma_1 = \frac{4\pi}{9} \rho \left( \frac{\mu_0 g_e^2 \mu_B^2}{4\pi \hbar} \right)^2 S(S+1) \frac{\tau_c}{1+(\omega_{NV} \tau_c)^2} \frac{1}{z^3}.$$

Here, $S=1/2$, $\rho\approx1.4~{\rm nm}^{-3}$, and all symbols have their standard values as in the data.

The $1/z^3$ scaling of $\Gamma_1$ provides strong distance dependence, enabling nanoscale spatial resolution.

2. Lorentzian Spectral Density of CuPc Spin Fluctuations

Each CuPc electron spin is modeled as a two-level fluctuator with Markovian dynamics and correlation time $\tau_c$, yielding a transverse autocorrelation

$$\langle S^\perp(t) S^\perp(0) \rangle = S(S+1) e^{-|t|/\tau_c}.$$

The corresponding spectral density is Lorentzian: $$(2)\;\; J(\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t} e^{-|t|/\tau_c} = \frac{2\tau_c}{1+\omega^2\tau_c^2}.$$ The full spin spectral density is $S_S(\omega) = S(S+1) J(\omega)$. The key assumptions are: single-exponential decay, and a homogeneous spin bath with correlation time $\tau_c$ independent of $\omega$. This spectral profile determines the frequency dependence of the NV relaxation rate.

3. NV $T_1$ Relaxometry Protocol

The NV $T_1$ relaxometry experiment uses a three-block pulse sequence:

• Laser Pump ($\sim$5 μs at 532 nm): Polarizes the NV into $|0\rangle$.
• Dark interval ($t$): Spin evolves freely for time $t$ (typically 0 to several ms), allowing relaxation under the influence of the CuPc bath.
• Laser Readout: Measures NV photoluminescence, partitioned into a “signal” and a delayed “reference” window for normalization.

By recording the normalized fluorescence as a function of dark time $t$, and fitting with an exponential or stretched-exponential, one extracts the $1/e$ time constant $T_1$.

Increasing relaxation (shorter $T_1$) in the presence of CuPc reflects enhanced magnetic noise at $\omega_{NV}$ compared to a bare NV under otherwise identical conditions.

4. Resolving the CuPc Electron Hyperfine Spectrum

CuPc molecules exhibit both axial $g$-tensor anisotropy $(g_\perp, g_\parallel)$ and hyperfine coupling $A$ between $S=1/2$ electron and $I=3/2$ Cu nucleus. In an external magnetic field $B_0$, the electron resonance frequencies are

$$\omega_{m_I,\alpha} = \frac{\mu_B g_\alpha B_0}{\hbar} \pm \frac{A_\alpha m_I}{\hbar}$$

with $\alpha \in \{\perp,\parallel\}$ and $m_I = +3/2,...,-3/2$.

As $B_0$ is swept, the NV transition frequency $\omega_{NV}(B_0)$ comes into resonance with these CuPc transitions, leading to cross-relaxation and a series of Lorentzian dips in $T_1$ at matching conditions. By fitting $T_1$ versus $B_0$, hyperfine parameters $(g_\perp, g_\parallel, A_\perp, A_\parallel)$ and the local molecular axis orientation are retrieved.

Experimentally, the field-dependent relaxation has the form: $$\Gamma_1(B_0) = \Gamma_{1,{\rm off}} + \Delta\Gamma_1\, \frac{\tau_c}{1 + (\omega_{NV}(B_0) - \omega_{m_I,\alpha})^2 \tau_c^2}$$ where $\Delta\Gamma_1$ is the cross-relaxation amplitude and $\Gamma_{1,{\rm off}}$ is the baseline NV relaxation rate absent resonant CuPc transitions.

5. Room-Temperature Correlation Time and Dominance of Electron-Electron Interactions

The CuPc spin correlation time $\tau_c$ is governed by both spin-lattice and intra-bath electron-electron interactions. The relaxation processes are:

• (a) Spin-lattice (phononic) relaxation: $T_{1,s-p}$, typically $B_0$ and $T$-dependent.
• (b) Spin-spin (dipolar flip-flop) processes between CuPc electrons.

The effective correlation time is obtained from

$$(3)\;\; \frac{1}{\tau_c} = \frac{1}{T_{1,s-p}} + \langle \Omega^2 \rangle \tau_c$$

where

$$\langle \Omega^2 \rangle = \frac{8\pi}{9} \left( \frac{\mu_0 g_e \mu_B}{\hbar} \right)^2 \rho.$$

This leads to

$$\tau_c = \frac{ \sqrt{1 + 4 \langle \Omega^2 \rangle T_{1,s-p}^2} - 1}{2 \langle \Omega^2 \rangle T_{1,s-p}}.$$

For typical experimental conditions ($T=300$ K, $T_{1,s-p}\approx38$ ns, $\rho\approx1.4\,{\rm nm}^{-3}$), $\langle \Omega^2 \rangle T_{1,s-p} \approx 1$, yielding $\tau_c \approx 70$ ns. Experimental observation shows $\tau_c$ is essentially independent of $B_0$ between 200–800 G, signifying dominance of the field-independent dipolar (spin–spin) term.

6. Extracting NV–CuPc Standoff with Nanometer Precision

With $\tau_c$, $\rho$, and $S$ fixed from spectroscopy, the NV’s $T_1$ is measured with ($T_{1,\rm on}$) and without ($T_{1,\rm off}$) the CuPc film. The additional relaxation rate

$$\Delta\Gamma_1 = \frac{1}{T_{1,\rm on}} - \frac{1}{T_{1,\rm off}}$$

is attributed solely to CuPc spin noise. Inverting Eq. (1), the NV–CuPc standoff is

$$z = \left( \frac{4\pi}{9} \rho \left( \frac{\mu_0 g_e^2 \mu_B^2}{4\pi \hbar} \right)^2 S(S+1) \frac{\tau_c}{1 + (\omega_{NV}\tau_c)^2} \right)^{1/3}\, /\, \Delta\Gamma_1^{1/3}.$$

This procedure yields depth $z$ with approximately 1 nm precision, enabled by the sharp $1/z^3$ dependence of the relaxation rate on distance.

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CuPc-enhanced NV relaxometry provides a versatile experimental platform for probing molecular spin systems, enabling the extraction of dynamical properties, hyperfine structure, and nanoscale spatial information inaccessible to traditional bulk techniques. This methodology yields insights into spin bath engineering and hybrid quantum material characterization, supporting developments in molecular-scale quantum processors and spin-based quantum networks.