Development of Very-Low-Temperature Millimeter-Wave Electron-Spin-Resonance Measurement System

Abstract

We report the development of a millimeter-wave electron-spin-resonance (ESR) measurement system at the University of Fukui using a ³He/⁴He dilution refrigerator to reach temperatures below 1 K. The system operates in the frequency range of 125–130 GHz, with a homodyne detection. A nuclear-magnetic-resonance (NMR) measurement system was also developed in this system as the extension for millimeter-wave ESR/NMR double magnetic-resonance (DoMR) experiments. Several types of Fabry–Pérot-type resonators (FPR) have been developed: A piezo actuator attached to an FPR enables an electric tuning of cavity frequency. A flat mirror of an FPR has been fabricated using a gold thin film aiming for DoMR. ESR signal was measured down to 0.09 K. Results of ESR measurements of an organic radical crystal and phosphorous-doped silicon are presented. The NMR signal from ¹H contained in the resonator is also detected successfully as a test for DoMR.

Appendix A: ESR Lineshape for the Homodyne System

The calculation procedure for the magnetic-resonance lineshape discussed in Sect. 4 is presented below.

A resonant cavity is often expressed in terms of the microwave analog of a series RLC-tuned circuit [2]. Figure 9 shows an equivalent circuit. The resonant angular frequency $\omega_{0}$ and unloaded quality factor $Q_{0}$ are expressed as follows:

$$\omega_{0} = \frac{1}{{\sqrt {LC} }} ,$$

(3)

$$Q_{0} = \frac{1}{R}\sqrt {\frac{L}{C}} ,$$

(4)

where R, L, and C are the resistance, inductance, and capacitance, respectively. The impedance of the circuit is as follows:

Fig. 9

LCR circuit equivalent to the FPR coupled with the oscillator through a waveguide

$$Z = R + i\left( {\omega L - \frac{1}{\omega C}} \right) .$$

(5)

For a reflection cavity, such as our FPR, a waveguide with an impedance Z₀ is coupled to the cavity with a coupling coefficient (or coupling parameter) β:

$$\beta = \frac{{Z_{0} }}{{n^{2} R}} .$$

(6)

The impedance from the waveguide $Z_{\text{L}}$ is:

$$Z_{\text{L}} = n^{2} Z.$$

(7)

We measure the loaded quality factor $Q_{\text{L}}$ from the waveguide:

$$Q_{\text{L}} = \frac{{Q_{0} }}{1 + \beta } .$$

(8)

If the cavity is perfectly coupled or matched to the waveguide, then β = 1 and $Q_{\text{L}} = Q_{0} /2.$.

Furthermore, we consider the case when the cavity contains a magnetic sample with a magnetic susceptibility $\chi = \chi^{\prime} - i\chi^{\prime\prime}$. The effect of the electric permeability is assumed to be negligible. The parameters L, R, Z, and β are expressed as follows:

$$L^{*} = L\left( {1 + q \chi^{\prime}} \right) ,$$

(9)

$$R^{*} = R + q\omega L\chi^{\prime\prime} = R\left( {1 + qQ_{0} \frac{\omega }{{\omega_{0} }}\chi^{\prime\prime}} \right) ,$$

(10)

$$Z^{*} = R^{*} + i\left( {\omega L^{*} - \frac{1}{\omega C}} \right) ,$$

(11)

$$\beta^{*} = \frac{{Z_{0} }}{{n^{2} R^{*} }} ,$$

(12)

where q is the filling factor of the sample. The reflection coefficient without the sample Γ(ω) is:

$$\varGamma \left( \omega \right) = \frac{{Z_{L } - Z_{0} }}{{Z_{L} + Z_{0} }} = \frac{{1 - \beta + iQ_{0} \left( {\frac{\omega }{{\omega_{0} }} - \frac{{\omega_{0} }}{\omega }} \right)}}{{1 + \beta + iQ_{0} \left( {\frac{\omega }{{\omega_{0} }} - \frac{{\omega_{0} }}{\omega }} \right)}} .$$

(13)

The reflection coefficient with the sample Γ^*(ω) is obtained in the same manner:

$$\varGamma^{*} \left( \omega \right) = \frac{{1 - \beta^{*} + iQ_{0}^{*} \left( {\frac{\omega }{{\omega_{0}^{*} }} - \frac{{\omega_{0}^{*} }}{\omega }} \right)}}{{1 + \beta^{*} + iQ_{0}^{*} \left( {\frac{\omega }{{\omega_{0}^{*} }} - \frac{{\omega_{0}^{*} }}{\omega }} \right)}} ,$$

(14)

where,

$$\omega_{0}^{*} = \frac{{\omega_{0} }}{{\sqrt {1 + q\chi^{\prime}} }} ,$$

(15)

$$Q_{0}^{*} = Q_{0} \frac{{\sqrt {1 + q\chi^{\prime} } }}{{1 + qQ_{0} \frac{\omega }{{\omega_{0} }}\chi^{\prime\prime}}} ,$$

(16)

$$\beta^{*} = \frac{\beta }{{1 + qQ_{0} \frac{\omega }{{\omega_{0} }}\chi^{\prime\prime}}} .$$

(17)

We can rewrite Eq. (14) as:

$$\varGamma^{*} \left( \omega \right) = \frac{{1 - \beta + qQ_{0} \frac{\omega }{{\omega_{0} }}\chi^{\prime\prime} + iQ_{0} \left( {\frac{\omega }{{\omega_{0} }}\left( {1 + q\chi^{\prime} } \right) - \frac{{\omega_{0} }}{\omega }} \right)}}{{1 + \beta + qQ_{0} \frac{\omega }{{\omega_{0} }}\chi^{\prime\prime} + iQ_{0} \left( {\frac{\omega }{{\omega_{0} }}\left( {1 + q\chi^{\prime} } \right) - \frac{{\omega_{0} }}{\omega }} \right)}} .$$

(18)

We define the reflection coefficient for an empty cavity at $\omega = \omega_{0}$ as:

$$\varGamma_{0} = \frac{1 - \beta }{1 + \beta } ,$$

(19)

and define γ  = $qQ_{\text{L}} \chi$ ($\gamma^{\prime}$ = $qQ_{\text{L}} \chi^{\prime}$, $\gamma^{\prime\prime}$ = $qQ_{\text{L}} \chi^{\prime\prime}$); we obtain:

$$\varGamma^{*} \left( \omega \right) = \frac{{\varGamma_{0} + \gamma^{\prime\prime}\frac{\omega }{{\omega_{0} }} + iQ_{L} \left( {\frac{\omega }{{\omega_{0} }} - \frac{{\omega_{0} }}{\omega }} \right) + i\left( {\frac{\omega }{{\omega_{0} }}} \right)\gamma^{\prime}}}{{1 + \gamma^{\prime\prime}\frac{\omega }{{\omega_{0} }} + iQ_{L} \left( {\frac{\omega }{{\omega_{0} }} - \frac{{\omega_{0} }}{\omega }} \right) + i\left( {\frac{\omega }{{\omega_{0} }}} \right)\gamma^{\prime}}}.$$

(20)

For the resonance frequency ($\omega_{0}$):

$$\varGamma^{*} = \frac{{\varGamma_{0} + i\gamma }}{1 + i\gamma } .$$

(21)

The measured ESR signal is expressed by:

$$\delta = \frac{{\varGamma^{*} - \varGamma_{0} }}{{\varGamma_{0} }} .$$

(22)

It is well known that $\chi ''\left( \omega \right)$ and $\chi '\left( \omega \right)$ correspond to the absorption and dispersion lines of the magnetic resonance, respectively, related to each other by the Kramers–Kronig relation. Therefore, it is useful to express $\delta$ in terms of γ:

$$\delta = \eta \left[ {\frac{{\gamma '' + \gamma ''^{2} + \gamma '^{2} + i\gamma '}}{{\left( {1 + \gamma ''} \right)^{2} + \gamma '^{2} }}} \right] ,$$

(23)

where:

$$\eta = \frac{2\beta }{1 - \beta } = 1 - \frac{1}{{\varGamma_{0} }} .$$

(24)

The real and imaginary parts of $\delta$ contain both $\chi ''\left( \omega \right)$ and $\chi '\left( \omega \right)$; therefore, the observed ESR signals are not perfectly separated into absorption and dispersion signals. Nevertheless, it is known that the mixing becomes significant when the spin density is approximately larger than 3 × 10¹⁵ cm⁻³ [38]. As the spin density is sufficiently small for our measurements, we can observe almost pure dispersion and absorption signals as the real and imaginary parts of $\delta$, respectively.

In the calculation shown in Sect. 4, we used a Lorentzian-like lineshape:

$$\chi^{\prime}\left( \omega \right) = \frac{{\chi_{0} \gamma_{\text{e}}^{2} H_{0}^{2} \left( {\gamma_{\text{e}}^{2} H_{0}^{2} - \omega^{2} } \right)}}{{\left( {\gamma_{\text{e}}^{2} H_{0}^{2} - \omega^{2} } \right)^{2} + 4\omega^{2} /T_{2}^{2} }} ,$$

(25)

$$\chi^{\prime\prime}\left( \omega \right) = \frac{{2\chi_{0} \gamma_{\text{e}}^{2} H_{0}^{2} \omega /T_{2} }}{{\left( {\gamma_{\text{e}}^{2} H_{0}^{2} - \omega^{2} } \right)^{2} + 4\omega^{2} /T_{2}^{2} }} ,$$

(26)

where $\gamma_{\text{e}}$ is the gyromagnetic ratio of the electron, H₀ is the applied field, and $\chi_{0}$ is the uniform susceptibility. These equations are derived from the Bloch’s equations, without saturation and with a long spin–spin relaxation time T₂ [55]. Slightly different equations are obtained in Refs. [4] and [33], though the obtained lineshape is very close to that corresponding to the above equations. The free parameters for the calculations shown in Fig. 4 are the relaxation rate 1/T₂ revealing the width of the lineshape, Q_L determining the frequency dependence, and scaling factor including q and $\chi_{0}$.