# Measurement Techniques of the Magneto-Electric Coupling in Multiferroics

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measurement Issues

## 3. Magneto-Electric Coupling Coefficient

^{E}= $\partial M/\partial E$.

_{ij}, with nine components:

_{11}, α

_{22}, α

_{33}or non-diagonal α

_{31}and α

_{13}is non-zero.

_{r}, µ

_{r}are the dielectric relative permittivity and relative magnetic permeability of the multiferroic, respectively. One method of deriving Equation (9) makes use of the relation linking speed of light to ε

_{0}, µ

_{0}($c=1/\sqrt{{\epsilon}_{0}{\mu}_{0}}$) and refractive index n to ε

_{r}, µ

_{r}($n=\sqrt{{\epsilon}_{r}{\mu}_{r}}$). Equation (9) becomes:

_{SI}. The term n/c in Equation (10) can be interpreted as the inverse velocity of propagation of electromagnetic radiation in a multiferroic medium, $\alpha =\frac{n}{c}=\frac{1}{v}$. From here we apply Einstein’s Special Relativity postulate, forcing v to be v ≤ c, so that $v\cdot n\le c$. Dividing by c on both sides, we get $v\cdot \frac{n}{c}\le 1$, which leads to Equation (9), $\alpha \le \sqrt{\epsilon \mu}$. Therefore, the magneto-electric coupling coefficient can only take positive values in the interval $\alpha \in (0,\sqrt{\epsilon \mu}]$. It is important to specify that the above formalism is valid for single-phase multiferroic materials. There are indeed reports of negative coupling values, but this is the case for composite multiferroics in which the magneto-electric coupling is strain/elastically mediated, and negative coupling values are allowed.

_{V}

^{H}is the magnetically induced voltage magneto-electric coefficient defined as:

^{H}= ε

_{0}⋅ε

_{r}⋅α

_{V}

^{H}[22]. We have shown that, in SI units, α

^{H}and α

^{E}are both expressed in [s/m] units. However, the more practical voltage magneto-electric coefficient, α

_{V}

^{H}(see Equation (12)) is expressed as [V/A] in SI units and [V/cm⋅Oe] in CGS units, which are also utilized in most practical applications and scientific measurements [6,23]. In what follows, we will review some useful measurement techniques of the magneto-electric coupling coefficient.

## 4. Measurement of Magnetically Induced Magneto-Electric Coupling

- (a)
- Bias the multiferroic sample under an optimal DC magnetic field bias;
- (b)
- Apply an AC magnetic field of fixed frequency and amplitude at 0 or π angle, or any non-transverse direction, to the DC magnetic bias field [24];
- (c)
- Measure the voltage response output of the multiferroic at various amplitudes of the applied AC magnetic field, at fixed DC magnetic bias and fixed frequency of the AC field;
- (d)
- Plot the measured voltages as a function of the amplitude of the applied AC magnetic fields;
- (e)
- From the obtained linear graph, as predicted by Equation (13), the magneto-electric coupling coefficient is determined as the slope of the graph divided by the thickness of the dielectric.

_{V}

^{H}is a complex function of material parameters, compliances, DC magnetic bias field and frequency. This complex function has only been derived analytically for some special cases of composite multiferroics [41,42,43]. Experimental studies revealed that the magnetically induced voltage magneto-electric coefficient is highly non-linear with the DC magnetic bias field (see Figure 2b). This non-linear behaviour is related to the fact that the voltage magneto-electric coefficient depends, among other parameters, on the piezo-magnetic coefficient. The piezo-magnetic coefficient is defined as the first derivative of the magnetostriction/magnetic strain with respect to the DC magnetic field. Figure 2a shows the typical magnetic strain as a function of the DC-applied magnetic field. At zero applied magnetic fields, the strain is zero. Increasing the applied magnetic field, the strain increases rapidly and, at a given field called the saturation field, the magnetic strain becomes saturated. Beyond this point, further increases in the magnetic applied field have no effect on the sample’s strain. This saturation magnetic field roughly coincides with the saturation magnetization of the sample on the magnetic hysteresis loop. The non-linearity of the magnetic strain is transferred to the piezo-magnetic coefficient (defined as the derivative of the strain in respect with the applied field) and this in turn is transferred to the magneto-electric coupling coefficient (see Figure 2b). A typical voltage response of a multiferroic composite to applied DC magnetic fields at constant AC magnetic field amplitude and frequency is shown in Figure 2b. The optimum DC bias field corresponds to the point where the magneto-electric voltage response is maximum. This optimum DC bias field corresponds to the maximum piezo-magnetic coefficient, which in turn corresponds to the point of largest gradient in the magnetic strain–field curve (Figure 2a).

## 5. Measurement of Electrically Induced Magneto-Electric Coupling

- (a)
- Place the multiferroic sample in a suitable magnetometer;
- (b)
- Under zero applied magnetic field, excite the sample with an AC electric field/voltage of fixed frequency;
- (c)
- Measure the magnetization of the multiferroic sample at various amplitudes of the applied AC electric field/voltage;
- (d)
- Plot the measured M values as a function of the amplitude of the applied AC electric field/voltage;
- (e)
- From the obtained linear graph, as predicted by Equation (14), the magneto-electric coupling coefficient is determined as the slope of the graph if M = M(E), or the slope of the graph times the thickness of the dielectric if M = M(V) is measured;

_{E}coefficient, and the effect is expected to be largest when the frequency of the AC electric field/voltage matches the electro-mechanical resonance frequency of the sample.

## 6. Measurement of Magneto-Electric Coupling from Piezo-Electric Measurements

^{e}is the piezo-electric coefficient, α

^{eff}is the effective magneto-electric coupling coefficient, H is the applied magnetic field and the above equation has been written using condensed matrix notation with m = 1, 2, 3, 4, 5, 6 and i, j = 1, 2, 3. Imposing short-circuit measurement conditions, so that E = 0, then P = D, where D is the electric displacement, so Equation (15) becomes:

^{e}and we want to determine α

^{eff}. To achieve this, we rearrange Equation (16):

_{ij}

^{eff}in units of (C/m

^{2}⋅Oe) and the differential is the slope of the linear function (Equation (17)):

_{31}

^{eff}is determined here [64]. The DC magnetic bias is generated using a large DC electromagnet, as seen in Figure 4, although a set of permanent magnets can also be used. The AC field is produced by a set of Helmholtz coils. The magnetically induced magneto-electric coupling effect is only observed when the frequency and phase of the applied AC magnetic field match those of the AC mechanical load. In this experiment, this has been achieved by using the same function generator for the two excitations.

^{eff}expressed in units of V/m⋅Oe. The open circuit α

^{eff}coefficient can be determined from piezo-electric measurements by using the relationship between the short circuit piezo-electric coefficient d

^{e}and the open circuit piezo-electric coefficient g

^{e}[65]:

_{0}is the permittivity of the vacuum (ε

_{0}≅ 8.85 × 10

^{−12}C/m⋅V) and ε

_{r}is the relative dielectric constant of the material. Using Equations (18) and (19) we obtain the general expression of the open circuit magneto-electric coupling coefficient (units of V/m⋅Oe) as:

- (a)
- Place the multiferroic sample in a suitable piezo-electric testing instrument;
- (b)
- The instrument must be modified to allow the simultaneous application of AC and DC magnetic fields to the sample;
- (c)
- Measure the piezo-electric coefficient at various amplitudes of the applied AC magnetic field at fixed DC optimum bias field;
- (d)
- Plot the piezo-electric coefficient values as a function of the amplitude of the AC magnetic field;
- (e)
- From the slope of the linear graph, determine the magneto-electric coupling coefficient using either Equation (18) or (20), depending whether the experimental conditions are short-circuit or open-circuit.

## 7. Measurement of Magneto-Electric Coupling via Scanning Probe Microscopy

_{a}is the total atomic repulsive force on the surface and occurs only in contact mode; F

_{mag}is the interaction force between the sample magnetization and the tip’s magnetic moment, defined for the case of a magnetic tip as:

_{e}is the electrostatic force interaction between the charge on the tip and the surface charge distribution. This force occurs in both contact and non-contact mode if a voltage is applied to the tip:

_{tip}= voltage applied on the tip, V

_{surface}is the surface potential, C is the tip–surface capacitance and depends on the tip geometry.

_{piezo}is the piezo-mechanical force that occurs only in contact mode for samples that display piezo-electricity. If x is the piezo-electric strain, Δz the sample displacement and z is the sample thickness, then:

_{H}is the interaction force between the magnetic tip and the applied external magnetic field, defined as:

_{33}measurement is performed. Essentially this is the technique described in Section 5, but applied at the nanoscale using an SPM. This method has already been successfully implemented [69,70], further validating the method proposed in Section 5.

## 8. Measurement of Magneto-Electric Coupling via Frequency Mixing/Conversion

_{mod},

^{2}. The total field experienced by the sample is then:

_{mod}and ω

_{ac}are the angular frequencies of the alternating bias and the small AC signal, respectively. The fundamental effect of the application of an alternating magnetic bias field is to dynamically change the slope of the magnetostriction curve, which is seen by the small AC magnetic excitation field inducing the magneto-electric effect. In turn, the slope describes the signal transfer characteristics of the superposition of B

_{mod}and B

_{ac}into a magnetostrictive elongation at the instantaneous operating point corresponding to the modulation frequency.

_{res}is known, then performing the experiment so that either ${\omega}_{res}={\omega}_{\mathrm{mod}}-{\omega}_{ac}$, or ${\omega}_{res}={\omega}_{\mathrm{mod}}+{\omega}_{ac}$, resonant operation of the device is possible at literally arbitrary excitation frequencies. This is simply achieved by tuning the bias modulation frequency at the correct value to fulfil one of the conditions ${\omega}_{res}={\omega}_{\mathrm{mod}}-{\omega}_{ac}$, or ${\omega}_{res}={\omega}_{\mathrm{mod}}+{\omega}_{ac}$.

## 9. Measurement of Magneto-Electric Coupling from Thermal Measurements

_{0}, ε

_{0}are the magnetic permeability and dielectric permittivity of vacuum, χ

^{m}and χ

^{e}are the magnetic and electric susceptibilities, α

^{E}and α

^{H}are the electrically and magnetically induced magneto-electric coupling coefficients, M is magnetization and P is the electric polarization of the multiferroic system. A full derivation of the multicaloric effect is given in [80]. According to Equations (30) and (31), when both $\partial M/\partial E$ < 0 and $\partial P/\partial T$ < 0, a cooling effect (ΔT

_{E,H}< 0) is achieved for an adiabatic depolarisation/demagnetisation, making this effect very attractive for solid state refrigeration. However, besides solid-state refrigeration applications, the multicaloric caloric effect could also be used to develop metrologies for magneto-electric coupling coefficient estimation from thermal measurements. In order to maximize the multicaloric solid state cooling effect, a multiferroic material must have identical (or similar) ferroic phase transition temperatures to the constituent phases and the device must be operated at exactly (or close) to this transition temperature, where the partial derivatives $\partial M/\partial T$ < 0 and $\partial P/\partial T$ are maximum, i.e., T

_{c}

^{m}≈ T

_{c}

^{e}≈ T. Contrary to this requirement, a magneto-electric coupling measurement based on the multicaloric effect requires multiferroics with very different transition temperatures of their ferroic constituent phases. In this way, the different contributions to ΔT

_{E,H}in Equations (30) and (31) can be easily separated by performing the experiment at a suitable base temperature T. Let us make the following substitutions in Equations (30) and (31), $\partial $M/$\partial $T = γ

^{m}and $\partial $P/$\partial $T = γ

^{e}. Let us also assume that the transition temperature of the magnetic phase is much larger than the transition temperature of the electric phase, T

_{c}

^{m}> T

_{c}

^{e}. If the measurement is performed at an operating temperature T close to one of the transition temperatures, then depending whether T = T

_{c}

^{m}or T = T

_{c}

^{e}, either $\partial $M/$\partial $T = γ

^{m}or $\partial $P/$\partial $T = γ

^{e}at the operating temperature is negligible as the slope is almost zero. Since T

_{c}

^{m}> T

_{c}

^{e}then choosing the operating temperature T = T

_{c}

^{e}results in $\partial $M/$\partial $T = γ

^{m}= 0 and Equations (30), (31) in integral form become:

_{c}

^{m}< T

_{c}

^{e}, then choosing the operating temperature T = T

_{c}

^{m}results in $\partial $P/$\partial $T = γ

^{e}= 0 and applying the above formalism, the electrically induced magneto-electric coupling coefficient can be estimated as:

^{m}or $\partial $P/$\partial $T = γ

^{e}as they are reduced from the equations. One only needs to know the values of the magnetic and dielectric susceptibilities of the material under test, the values of the applied external fields, which are controlled by the experimenter and the values of the temperature change, which are measured experimentally. A possible instrument capable of measuring magneto-electric coupling coefficient from thermal measurements is shown in Figure 6, modified from [83]. According to this method, the magneto-electric coupling coefficient is determined experimentally in the following way:

- (a)
- Place the multiferroic sample in vacuum chamber in adiabatic conditions;
- (b)
- The instrument must be capable to apply magnetic field and electric fields to the sample, as well as to measure accurately the temperature change of the sample;
- (c)
- A temperature reservoir can be set at a desired operating temperature and then brought in contact with the multiferroic sample via a thermal switch;
- (d)
- Apply a large E field to the sample;
- (e)
- While the E field is ON, if T
_{c}^{m}> T_{c}^{e}, choose the operating temperature T = T_{c}^{e}and bring the sample at T = T_{c}^{e}via the thermal switch; - (f)
- Cut the thermal link to the reservoir;
- (g)
- Reduce the E applied field to zero;
- (h)
- Measure the temperature change ΔT
_{E}; - (i)
- Bring the sample back to the operating temperature T = T
_{c}^{e}; - (j)
- Apply a large magnetic field and then bring the sample to adiabatic conditions;
- (k)
- Reduce the applied magnetic field to zero and measure the temperature change ΔT
_{H}; - (l)
- Use Equation (34) to derive the magnetically induced magneto-electric coupling coefficient.
- (m)
- If T
_{c}^{m}< T_{c}^{e}, choose the operating temperature T = T_{c}^{m}and repeat the above procedure; - (n)
- Extract the electrically induced magneto-electric coupling coefficient using Equation (35).

## 10. Measurement of Non-Linear Magneto-Electric Coupling Coefficients

_{0}is the DC applied magnetic field and h(t) is the total AC applied magnetic field, h(t) = h

_{1}(t) + h

_{2}(t), where h

_{1}(t) and h

_{2}(t) are simultaneously applied AC magnetic fields:

_{1}and f

_{2}are the frequencies of the h

_{1}(t) and h

_{2}(t) AC magnetic fields, respectively, then the total field experienced by the sample is H = H

_{0}+ h(t), with h

_{1}, h

_{2}<< H

_{0}. The authors showed that the voltage generated by the multiferroic due to the application of combined magnetic fields is:

^{e}is the piezo-electric coefficient and λ(H) is the magnetostriction. Considering the non-linearity of the magnetostriction, which can be expanded into a Taylor series around H

_{0}, and using Equations (36) and (37), after some algebraic manipulation, the voltage induced due to the application of H = H

_{0}+ h

_{1}(t) + h

_{2}(t) is given by [84]:

^{(0)}; the AC components u

_{1}

^{(1)}and u

_{2}

^{(1)}at the frequencies f

_{1}and f

_{2}, respectively, due to the linear magneto-electric effect; the frequency doubling voltage components u

_{1}

^{(2)}and u

_{2}

^{(2)}at 2f

_{1}and 2f

_{2}frequencies, respectively; the frequency mixing voltage component u

_{mix}with frequencies f

_{1}+ f

_{2}and f

_{1}− f

_{2}, describing the non-linear mixing of magnetic fields. In this method the frequency mixing is due to AC magnetic fields of different frequencies being simultaneously applied to the multiferroic structure in addition to a DC magnetic bias field, while the Kiel frequency mixing method involves an AC magnetic field being applied simultaneously with an AC magnetic bias field, with frequencies carefully selected so that the mixed frequencies match the electro-mechanical resonance frequency of the device: f

_{res}= f

_{1}+ f

_{2}or f

_{res}= f

_{1}− f

_{2}. If t

_{e}is the thickness of the dielectric component, the following magneto-electric coupling coefficients can be extracted from Equation (38):

- (1)
- The standard linear magneto-electric coupling coefficients (units of V/cm⋅Oe):$$\begin{array}{l}{\alpha}_{E,1}{}^{(1)}=\frac{{u}_{1}{}^{(1)}}{{t}_{e}\cdot {h}_{1}}\\ {\alpha}_{E,2}{}^{(1)}=\frac{{u}_{2}{}^{(1)}}{{t}_{e}\cdot {h}_{2}}\end{array}.$$
- (2)
- The non-linear magneto-electric coupling due to the frequency doubling voltage component (units of V/cm⋅Oe
^{2}):$$\begin{array}{l}{\alpha}_{E,1}{}^{(2)}=\frac{{u}_{1}{}^{(2)}}{{t}_{e}\cdot {h}_{1}{}^{2}}\\ {\alpha}_{E,2}{}^{(2)}=\frac{{u}_{2}{}^{(2)}}{{t}_{e}\cdot {h}_{2}{}^{2}}\end{array}.$$ - (3)
- The non-linear magneto-electric coupling due to the frequency mixing voltage component (units of V/cm⋅Oe
^{2}):$${\alpha}_{E}{}^{(mix)}=\frac{{u}_{mix}}{{t}_{e}\cdot {h}_{1}\cdot {h}_{2}}.$$

## 11. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Diagram of the magnetically induced magneto-electric coupling coefficient experiment. The system allows simultaneous application of DC and AC magnetic fields, while electrically induced signals are amplified and detected to determine frequency response via a spectrum analyser or amplitude response via a lock-in amplifier. Refinements such as cryogenic or high temperature measurements and/or simultaneous applied mechanical stress are possible to implement into this generic instrument. Image source: [25].

**Figure 2.**(

**a**) Typical magnetostriction coefficient dependence on the DC-applied magnetic field; (

**b**) typical dependence of magneto-electric induced voltage as a function of DC magnetic bias field, at constant AC magnetic field amplitude and frequency; Figure (

**a**,

**b**) have the same horizontal axis. (

**c**) Typical dependence of magneto-electric induced voltage as a function of AC magnetic field, at constant DC magnetic bias field.

**Figure 3.**(

**a**) Schematic diagram of a MOKE measurement system modified for multiferroic coupling measurements. Note the absence of the DC electromagnet. Parts of the system are: 1. CW or pulsed Laser source; 2. Beam chopper (not required if AC laser source ort if V

_{ac}excitation signal is used as the reference signal for the MOKE lock-in detection; 3. λ/2 wave plate; 4. Polarizer; 5. Sample; 6. λ/4 wave plate; 7. Polarizer/Analyser; 8. Photodetector 1; 9. Photodetector 2; 10. Differential amplifier and signal output. (

**b**) Diagram showing the multiferroic sample under MOKE test while subjected to electrical excitation, i.e., applied electric field.

**Figure 4.**1. Electromagnet’s poles; 2. Lower contact and ac load system; 3. Electromagnet support; 4. BerlinCourt d

_{33}measurement system; 5. Top sample contact; 6. DC coils of the electromagnet; 7. Sample; 8. AC coils generating the AC magnetic field. Image developed by the authors and a CAD engineer at NPL as part of the Multiprobe MET 2.1 Project.

**Figure 5.**Schematic diagram of a scanning probe microscope adapted to perform multiferroic coupling measurements.

**Figure 6.**Schematic of the multicaloric testing system. HS = heat switch; ΔT is measured under adiabatic demagnetization and depolarization. The multiferroic material is kept adiabatically under vacuum. The HS can connect/disconnect the material to/from the temperature reservoir, providing the operating T. The temperature change is measured using non-contact IR thermometry or low heat capacity temperature sensors. Image modified from [83].

**Figure 7.**The multiferroic structure is placed in a uniform bias DC magnetic field. Alternating magnetic fields h

_{1}cos(2πf

_{1}t) and h

_{2}cos(2πf

_{2}t) with amplitudes h

_{1}, h

_{2}and frequencies f

_{1}, f

_{2}are created by two electromagnetic coils K

_{1}and K

_{2}, powered by two independent generators “AC Gen1” and “AC Gen2”. Figure reprinted with permission from [85].

Measurement Mode | Contact SPM | Non-Contact SPM | |
---|---|---|---|

Non-zero applied magnetic field and tip voltage | magnetic tip | F_{a} + F_{e} + F_{piezo} + F_{mag} + F_{H} | F_{e} + F_{mag} + F_{H} |

non-magnetic tip | F_{a} + F_{e} + F_{piezo} | F_{e} | |

Non-zero tip voltage, zero applied magnetic field | magnetic tip | F_{a} + F_{e} + F_{piezo} + F_{mag} | F_{e} + F_{mag} |

non-magnetic tip | F_{a} + F_{e} + F_{piezo} | F_{e} |

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**MDPI and ACS Style**

Vopson, M.M.; Fetisov, Y.K.; Caruntu, G.; Srinivasan, G.
Measurement Techniques of the Magneto-Electric Coupling in Multiferroics. *Materials* **2017**, *10*, 963.
https://doi.org/10.3390/ma10080963

**AMA Style**

Vopson MM, Fetisov YK, Caruntu G, Srinivasan G.
Measurement Techniques of the Magneto-Electric Coupling in Multiferroics. *Materials*. 2017; 10(8):963.
https://doi.org/10.3390/ma10080963

**Chicago/Turabian Style**

Vopson, M. M., Y. K. Fetisov, G. Caruntu, and G. Srinivasan.
2017. "Measurement Techniques of the Magneto-Electric Coupling in Multiferroics" *Materials* 10, no. 8: 963.
https://doi.org/10.3390/ma10080963