# Molecular Field Calculation of Magnetization onNdRh2Ge2 Single Crystal

Authors: A. Himori; K. Hattori; T. Shigeoka
Journal: Research Letters in Physics
 (2008)

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## Abstract

Calculation of magnetization of the ternary single crystal compoundNdRh2Ge2 has been carried out by using the wave-like molecular field model to explain the complex magnetic behavior. The field-induced magnetic structures having the propagation vectors, Q2=(0,0,39/40), Q3=(0,0,35/40), Q4=(0,0,31/40), and Q5=(0,0,0/40) (= the field-induced ferromagnetic phase) were proposed. Calculation on the basis of these structures and the antiferromagnetic phase with Q1=(0,0,1) well reproduces the experimental magnetization processes and H-T magnetic phase diagram.


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## Body

## 1. Introduction

Ternary rare
earth compounds RM2X2 (R = rare earth, M = Rh, Ru, X = Si or
Ge) crystallize in the tetragonal ThCr2Si2-type structure
(I4/mmm) [1, 2].  In the numerous series, the RRh2Ge2 is
worth studying because of the great diversity of its magnetic properties [3]. Recently,
we reported very interesting magnetic behavior of NdRh2Ge2 single crystal [4] as follows.(i)
The compound shows an antiferromagnetic behavior with neel temperature
of 50 K.(ii)
In the temperature dependence of magnetic susceptibility, there
is another anomaly at 37 K which indicates a magnetic transition. A magnetic transition is also suggested at 20 K from
magnetization measurements.(iii)
There is
a strong uniaxial magnetic anisotropy with the easyc-axis which leads to an Ising-like
behavior in the compound.(iv)
At a low
temperature, a four-step metamagnetic process appears along the easyc-axis
(see Figure 1).(v)
TheH-T magnetic phase diagram, where there
are five magnetic phases, was constructed (see Figure 2).Figure 1
Experimental magnetization versus applied field for NdRh2Ge2 at T=4.2 K (bold line) by Himori et al. [4] and calculated magnetization (thin line).Figure 2
Molecular
field coefficientsλ(Qz)
versus propagation vector (0,0,qz).In the present study, in order to explain this complex magnetic
behavior, calculations of
the magnetization and an analysis of moment arrangements at various temperatures and under various
magnetic fields have been carried out with a wave-like molecular
field model [5].


## 2. Wave-Like Molecular Field Model

The s-f
interaction may be an important interaction in metallic rare earth compounds.
Considering this interaction ofith
atom as the effective Hamiltonian in terms of the molecular field Hm(i),(1)H(i)=−gJμBJiHm(i),
where Ji is a good quantum number.The molecular
field acting on an atom on theith
c-plane is given by introducing molecular  field coefficient λ(q) depended on Fourier q component
as
(2)Hm(i)=∑qλ(q)〈Jq〉cos(πqi+φq),
where Jq is the Fourier q component
of Ji. These equations are to be
solved self-consistently. The details have been reported by Iwata [5].


## 3. Results and Discussion

### 3.1. Assumption of the Propagation Vectors at Each Magnetic Phase

For the sake
of calculating  the magnetic processes by the wave-like molecular field model,  knowledge
of magnetic structures is required. Only antiferromagnetic structure has been
reported on the NdRh2Ge2 compound;
it is a simple structure having the propagation vector Q1=(0,0,1)  and magnetic moments along the c-axis [6]. On
the basis of the facts mentioned below, we can, now, propose magnetic
structures for the field-induced magnetic phases.On most of compounds which show an Ising-like
multistep metamagnetic process along thec-axis and have the simple
antiferromagnetic structure with Q1, the field-induced
phases have propagation vectors Qz=(0,0,qz)(z=1,2,…), for
example,  on PrCo2Si2 [7], NdCo2Si2 [8], and so on. Thus, we assume that the field-induced magnetic phases of NdRh2Ge2 compound have Qz.
Wave numbers qz’s are
supposed from the magnetization process as shown in Figure 1. There are five
magnetic phases in the metamagnetic process though it is hardly seen in the
figure. The magnetization in each phase is 0μB, 0.06μB, 0.32μB, 0.58μB, and 2.59μB (= saturation
value Ms), corresponding to (0/40)Ms,(1/40)Ms,(5/40)Ms,(9/40)Ms, and (40/40)Ms, respectively. This large common denominator suggests a
long period structure. Metamagnetic transitions should be responsible for a
spin-flip due to the strong uniaxial magnetic anisotropy. The induced (1/40)Ms-phase requires 40 unit
cells. Then, we can propose that the field-induced phases have the propagation
vectors, Q2=(0,0,39/40),Q3=(0,0,35/40),Q4=(0,0,31/40),Q5=(0,0,0/40) (= the
field-induced ferromagnetic phase). Of course, the antiferromagnetic phase has Q1=(0,0,1).


### 3.2. Determination of the Wave-Dependent Molecular Field Coefficients

Molecular-field
coefficientsλ(Qz) are estimated by
finding the best fit of the calculated values with the experimental data of the
magnetic susceptibility, specific heat, and the magnetization process. The
values of λ(Qz) obtained in this
study are plotted in Figure 2. The similar figure of this characteristic curve
has been seen in PrCo2Si2 and NdCo2Si2.


### 3.3. Moment Arrangements in Nonexternal Magnetic Field

The calculated
Nd magnetic moment arrangements at various temperatures in nonexternal magnetic
field together with the wave-like molecular fields(Hm) are illustrated in Figures 3(a)–3(c). The Q1-structure
(= phase I) is expressed by only one function with qz=1 since the magnetic structure is AFI-type [6], whereas
the Q2-, Q3-structure
(= phase II, phase III) are expressed by a sum of forty one harmonic functions
with qz=0/40,1/40,…,40/40 since the magnetic phases are intermediate phases below the induced
ferromagnetic phase. Obviously,  these moment arrangements have been
corresponded with the wave-like molecular fields. It is very interesting
results that some paramagnetic Nd ions appear in the Q2- and Q3-structure.
It is caused by the Hm (=
0) around the moments and it is seen in TbRu2Ge2 [9].(a) Moment
arrangements and wave-like molecular fields (solid curves) at 4 K, (b) at 33 K,
and (c) at 40 K. The open circles represent paramagnetic Nd.
(a)
T=4.2 K Q1-structure (qz=1)(b)
T=33 K Q2-structure (qz=39/40)(c)
T=40 K Q3-structur (qz=31/40)


### 3.4. Moment Arrangements Under Various Applied Field

Figures4(a)–4(e) illustrate
the calculated Nd magnetic moment arrangements and the behaviors of the total
field (H+Hm) at 4.2 K. It
is seen that the moments flips together with the change of a total field’s sign.
The magnitudes of the moments are smaller   than  gJμB for all
phases because of the CEF effect.  It is noticed that no paramagnetic Nd ions
appear in this case.(a) Moment
arrangements along thec-axis and total Hm+H (solid curves) at 4 K and under H = 0T, (b) H = 5.6T, (c) H = 9.6T, (d) H = 10.6T, and (e) H = 13.6T applied along the c-axis.


### 3.5. Magnetization Process and Magnetic Phase Diagram

The calculated magnetization process at
4.2 K under applied fields is shown in Figure1. The magnetizations are 0μB/f.u., 0.068μB/f.u., 0.32μB/f.u., 0.58μB/f.u., and 2.59μB/f.u. at 0T, 1.4T, 8.9T,
10.1T, and 13.7T, respectively. These values are in good agreement with the
experimental values of 0μB/f.u., 0.06μB/f.u., 0.32μB/f.u., 0.58μB/f.u., and 2.59μB/f.u. It is clear that the magnetization
curve is reproduced by the
calculated values fairly well at 4.2 K.The calculatedH-T diagram
together with the experimental points obtained from the magnetization
measurements are illustrated in Figure 5. It is shown that the calculations reproduce the main feature of the experimental H-T phase diagram.Figure 5
Magnetic
phase diagram along thec-axis. The solid curves and open circles with dash
line represent the calculation critical fields and the experimental ones (by Himori et al. [4]),
respectively.


## 3.1. Assumption of the Propagation Vectors at Each Magnetic Phase

For the sake
of calculating  the magnetic processes by the wave-like molecular field model,  knowledge
of magnetic structures is required. Only antiferromagnetic structure has been
reported on the NdRh2Ge2 compound;
it is a simple structure having the propagation vector Q1=(0,0,1)  and magnetic moments along the c-axis [6]. On
the basis of the facts mentioned below, we can, now, propose magnetic
structures for the field-induced magnetic phases.On most of compounds which show an Ising-like
multistep metamagnetic process along thec-axis and have the simple
antiferromagnetic structure with Q1, the field-induced
phases have propagation vectors Qz=(0,0,qz)(z=1,2,…), for
example,  on PrCo2Si2 [7], NdCo2Si2 [8], and so on. Thus, we assume that the field-induced magnetic phases of NdRh2Ge2 compound have Qz.
Wave numbers qz’s are
supposed from the magnetization process as shown in Figure 1. There are five
magnetic phases in the metamagnetic process though it is hardly seen in the
figure. The magnetization in each phase is 0μB, 0.06μB, 0.32μB, 0.58μB, and 2.59μB (= saturation
value Ms), corresponding to (0/40)Ms,(1/40)Ms,(5/40)Ms,(9/40)Ms, and (40/40)Ms, respectively. This large common denominator suggests a
long period structure. Metamagnetic transitions should be responsible for a
spin-flip due to the strong uniaxial magnetic anisotropy. The induced (1/40)Ms-phase requires 40 unit
cells. Then, we can propose that the field-induced phases have the propagation
vectors, Q2=(0,0,39/40),Q3=(0,0,35/40),Q4=(0,0,31/40),Q5=(0,0,0/40) (= the
field-induced ferromagnetic phase). Of course, the antiferromagnetic phase has Q1=(0,0,1).


## 3.2. Determination of the Wave-Dependent Molecular Field Coefficients

Molecular-field
coefficientsλ(Qz) are estimated by
finding the best fit of the calculated values with the experimental data of the
magnetic susceptibility, specific heat, and the magnetization process. The
values of λ(Qz) obtained in this
study are plotted in Figure 2. The similar figure of this characteristic curve
has been seen in PrCo2Si2 and NdCo2Si2.


## 3.3. Moment Arrangements in Nonexternal Magnetic Field

The calculated
Nd magnetic moment arrangements at various temperatures in nonexternal magnetic
field together with the wave-like molecular fields(Hm) are illustrated in Figures 3(a)–3(c). The Q1-structure
(= phase I) is expressed by only one function with qz=1 since the magnetic structure is AFI-type [6], whereas
the Q2-, Q3-structure
(= phase II, phase III) are expressed by a sum of forty one harmonic functions
with qz=0/40,1/40,…,40/40 since the magnetic phases are intermediate phases below the induced
ferromagnetic phase. Obviously,  these moment arrangements have been
corresponded with the wave-like molecular fields. It is very interesting
results that some paramagnetic Nd ions appear in the Q2- and Q3-structure.
It is caused by the Hm (=
0) around the moments and it is seen in TbRu2Ge2 [9].(a) Moment
arrangements and wave-like molecular fields (solid curves) at 4 K, (b) at 33 K,
and (c) at 40 K. The open circles represent paramagnetic Nd.
(a)
T=4.2 K Q1-structure (qz=1)(b)
T=33 K Q2-structure (qz=39/40)(c)
T=40 K Q3-structur (qz=31/40)


## 3.4. Moment Arrangements Under Various Applied Field

Figures4(a)–4(e) illustrate
the calculated Nd magnetic moment arrangements and the behaviors of the total
field (H+Hm) at 4.2 K. It
is seen that the moments flips together with the change of a total field’s sign.
The magnitudes of the moments are smaller   than  gJμB for all
phases because of the CEF effect.  It is noticed that no paramagnetic Nd ions
appear in this case.(a) Moment
arrangements along thec-axis and total Hm+H (solid curves) at 4 K and under H = 0T, (b) H = 5.6T, (c) H = 9.6T, (d) H = 10.6T, and (e) H = 13.6T applied along the c-axis.


## 3.5. Magnetization Process and Magnetic Phase Diagram

The calculated magnetization process at
4.2 K under applied fields is shown in Figure1. The magnetizations are 0μB/f.u., 0.068μB/f.u., 0.32μB/f.u., 0.58μB/f.u., and 2.59μB/f.u. at 0T, 1.4T, 8.9T,
10.1T, and 13.7T, respectively. These values are in good agreement with the
experimental values of 0μB/f.u., 0.06μB/f.u., 0.32μB/f.u., 0.58μB/f.u., and 2.59μB/f.u. It is clear that the magnetization
curve is reproduced by the
calculated values fairly well at 4.2 K.The calculatedH-T diagram
together with the experimental points obtained from the magnetization
measurements are illustrated in Figure 5. It is shown that the calculations reproduce the main feature of the experimental H-T phase diagram.Figure 5
Magnetic
phase diagram along thec-axis. The solid curves and open circles with dash
line represent the calculation critical fields and the experimental ones (by Himori et al. [4]),
respectively.


## 4. Summary

The
interesting magnetic behavior on the NdRh2Ge2 single crystal had been
reported; successive magnetic phase transitions occur at 20 K, 37 K, and 50 K (=TN)At low temperatures, a four-step metamagnetic process
appears.  We try to explain this complex magnetic behavior by the wave-like
molecular field model. For the sake of calculation, the field-induced magnetic
structures having the propagation vectors, Q2=(0,0,39/40),Q3=(0,0,35/40),Q4=(0,0,31/40), and Q5=(0,0,0/40) (= the field-induced ferromagnetic
phase) were proposed. On the basis of these structures and the antiferromagnetic structure reported, calculation of magnetization
for various temperatures and fields has been performed. The calculation well
reproduces main features of the experimental magnetization processes and H-T magnetic phase diagram. So, we
believe that the magnetic structures proposed for the field-induced phases are
right. To confirm the magnetic structures proposed, neutron diffraction study
is needed.



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*Source: 157070-2008-06-04.xml*