De Haas-van alphen effect of FeP in double helical magnetic state

Article abstract of DOI:10.1143/JPSJ.70.192

The de Haas-van Alphen (dHvA) effect has been investigated on high-quality single crystals of FeP in a double helical magnetic state with a 5c period in the magnetic fields up to 14 T and at temperatures down to 0.42 K. Single crystals were grown by the chemical transport technique using iodine as the transport agent. Nineteen branches were observed in three crystallographic planes (100), (010) and (001), and analyzed with simple analytic equations for the Fermi surfaces. Many of the dHvA branches have a characteristic angular dependence and are found to correspond to the orbits on surfaces beyond the Brillouin zone in the magnetic state which is reduced to 1/5 of the non-magnetic one with the MnP-type along the [001] direction. These branches originate from the orbits caused by a magnetic breakdown phenomenon under high fields. Cyclotron mass ratios were obtained to be up to 5.9, whereas the electronic specific heat coefficient is rather small, 2.86 mJK^-2mol^-1.

Full text of DOI:10.1143/JPSJ.70.192

Journal of the Physical Society of Japan
Vol. 70, No. 1, January, 2001, pp. 192-198
de Haas-van Alphen Effect of FeP in Double Helical Magnetic State
Tatsuhiro Nozue∗, Hisao Kobayashi, Noriaki Kimura1,
Haruyoshi Aoki1 and Takashi Kamimura
Department of Physics, Tohoku University, Sendai 980-8578
1Center for Low Temperature Science, Tohoku University, Sendai 980-8578
(Received September 14, 2000)
The de Haas-van Alphen (dHvA) effect has been investigated on high-quality single crystals
of FeP in a double helical magnetic state with a 5c period in the magnetic fields up to 14 T and
at temperatures down to 0.42 K. Single crystals were grown by the chemical transport technique
using iodine as the transport agent. Nineteen branches were observed in three crystallographic
planes (100), (010) and (001), and analyzed with simple analytic equations for the Fermi sur-
faces. Many of the dHvA branches have a characteristic angular dependence and are found to
correspond to the orbits on surfaces beyond the Brillouin zone in the magnetic state which is
reduced to 1/5 of the non-magnetic one with the MnP-type along the [001] direction. These
branches originate from the orbits caused by a magnetic breakdown phenomenon under high
fields. Cyclotron mass ratios were obtained to be up to 5.9, whereas the electronic specific heat
coefficient is rather small, 2.86 mJK−2mol−1
.
KEYWORDS: FeP, dHvA effect, Fermi surface, magnetic breakdown, cyclotron mass, specific heat
§1. Introduction
For transition metal compounds such as pnictides and
chalcogenides, the hybridized electronic states between
d electrons of a metal and p electrons of a ligand atom
dominate their properties of a crystalline structure, elec-
trical resistivity, magnetism and so on. Among these
compounds, the monopnictides have a hexagonal NiAs-
type (P63/mmc) or an orthorhombic MnP-type (Pnma)
structure in which the atoms are displaced slightly from
the atomic position of the NiAs-type. Only Cr, Mn and
Fe compounds have a magnetic order.1) CrSb is an anti-
ferromagnet with a Cr moment of 3.0μB, and Mn1+xSb
and MnAs ferromagnet with 3.2 and 3.3μB, respectively,
at 4.2 K. These compounds have the NiAs-type structure.
On the other hand, MnP, FeP, CrAs and FeAs with the
MnP-type have double helical spin arrangements at the
ground state. The incommensurate propagation vector
is along the [001] direction. Felcher et al. showed by
neutron diffraction experiments that FeP has the double
helices below 125 K with Fe moments of 0.46 and 0.37μB
which depend on the each helix and a nearly commensu-
rate propagation with a vector, q = 0.20 × c∗ at 4.2 K,
as shown in Fig. 1.2)
It is necessary to reveal the electronic structure of
these compounds experimentally on high quality sam-
ples to investigate comprehensively their wide variety of
magnetic and electric properties. So far, we have mea-
sured the de Haas-van Alphen (dHvA) effect on NiAs3, 4)
with a small-distorted NiAs-type5) and on CrP6, 7) with
the MnP-type structure and discussed the results on the
basis of calculated band structures. They are Pauli-
paramagnetic with weakly temperature dependent sus-
Fig. 1. (a) MnP-type crystal structure as projected on the (100)
plane. (b) Double helical spin structure of FeP. The angles be-
tween neighbor atoms are α12 = α34 = +176◦, α23 = α41
−140◦.
=
magnetic breakdown phenomenon were observed.7) The
electron breaks through the crystalline gap in k-space un-
der high fields and makes a large closed orbit which is be-
yond the band structure calculation.8) Such a branch was
observed in some substances such as Cr9) and Ni3Al.10)
In chromium, an incommensurate longitudinal spin den-
sity wave (SDW) occurs below 123 K.11) The results of
the dHvA measurement were interpreted in terms of
the chain model.12) In this model, the Fermi surfaces
consist of the chains along the propagation vector of
SDW of nearly ellipsoid surfaces obtained from the cal-
culated band structure in the paramagnetic state. Some
branches were caused by the orbits due to magnetic
breakdown at the crossing points of the chains. In the
dHvA measurement of Ni3Al which exhibits weakly fer-
romagnetism below 41 K,13) two groups of several close
branches were observed. These groups were interpreted
ceptibilities. In CrP, some branches originating from a in terms of magnetic breakdown between two Fermi sur-
faces attributed to the bands split by the exchange and
∗
E-mail: noz@mail.cc.tohoku.ac.jp
the spin-orbit interactions. Among the magnetic transi-
192

2001)
de Haas-van Alphen Effect of FeP in Double Helical Magnetic State
193
tion metal monopnictides, the dHvA effect on MnP was
measured along the symmetric directions.14) MnP has
the complicated magnetization process dependent on the
magnitude and direction of magnetic field.15) Therefore
the topology of Fermi surface was not obtained.
In this paper, we report the results of dHvA effect
measurements of FeP crystals and discuss the topology
of Fermi surfaces with simple analytic equations. It will
be shown that most of the branches correspond to the
orbits originating from magnetic breakdown, which is at-
tributed to the superstructure due to the double helical
spin order. The cyclotron mass ratios are compared with
results obtained presently for some transition metal com-
pounds.
§2. Experimental
Fig. 2. Temperature dependence of electrical resistivity for a cur-
rent along the [010] direction in FeP. The curve bends at 119 K
where the differential resistivity has a peak as shown in the inset.
Polycrystalline samples were prepared by the following
procedure. Appropriate amounts of starting materials Fe
rods (99.99% purity) and P flakes (99.9999%) were sealed
under vacuum in a silica tube which were progressively
heated to 850◦C for 3 days, kept at this temperature for
3 days and then cooled slowly to room temperature. The
reacted compound was carefully ground and reheated at
850◦C for several days in an evacuated silica tube. The
lattice constants a, b and c were estimated to be 5.196(1),
˚
3.098(1) and 5.792(1) A, respectively, at room tempera-
ture and agree well with those of the previous works.2, 16)
Single crystals were grown by a chemical vapor transport
technique using iodine as a transport agent. The pow-
dered sample of about 1 g was sealed in a silica tube with
2 mg cm−3 amount of iodine under vacuum at about 77 K
to suppress a sublimation of iodine during the evacuation
procedure. The silica tube was placed in a furnace in
which the charge zone was kept at 815◦C and the growth
zone at 675◦C. After 3 weeks, the tube was cooled slowly
to room temperature. The FeP single crystals grew at
the growth zone, where the largest one among these sin-
gle crystals is a size of about 2×2×3 mm3. The obtained
single crystals have an extremely high quality because
the residual resistivity is 0.3 μΩ cm and the residual re-
Fig. 3. Temperature dependence of specific heat in FeP. The peak
at 119 K originates from the magnetic order. The inset shows the
plot of C/T versus T2.
sistivity ratio, ρ(300 K)/ρ(4.2 K), is over 1000, as shown range from 6 to 10 T, an AC modulation field, h0, was
in Fig. 2.
6.9 × 10−3 T at a frequency 64 Hz and the signals were
The specific heat measurements were carried out using detected at the fundamental frequency. On the other
an adiabatic method with a mechanical heat switch in hand, the signals were detected at the second harmonic
the temperature range from 1.5 to 30 K and by means of with h0 = 8.0 × 10−3 T at 90 Hz in the higher field range
an AC-calorimeter with a heat source of chopped light from 10 to 14 T. However, the dHvA oscillation was mea-
from 20 K to room temperature. The specific heat data sured from 1 to 14 T along three symmetric directions.
obtained by the AC-calorimeter were normalized to the The temperature dependence of dHvA oscillation was
absolute value determined by the adiabatic method in measured in the ranges from 0.63 to 1.65 K in the lower
the temperature range form 20 to 30 K.
field and from 0.42 to 1.56 K in the higher field to obtain
The dHvA effect was measured on a cubic single crys- the cyclotron mass ratio.
talline sample, 1 × 1 × 1 mm3, by a field modulation
method in applied magnetic fields up to 14 T at tem-
§3. Results
peratures down to 0.42 K. The field direction was ro-
tated in the three principal crystallographic planes of
orthorhombic symmetry, that is, the (100), (010) and
(001) planes. The dHvA oscillations in which lower and
higher frequencies than about 1000 T were extracted by
the Fourier transform analysis were obtained in the field
ranges from 6 to 10 T at a rate of 0.2 T/min and from
10 to 14 T at 0.1 T/min, respectively. In the lower field
Figure 3 shows the temperature dependence of the spe-
cific heat, C. The small but distinct peak at 119 K cor-
responds to the antiferromagnetic order.2) The entropy
change around 119 K is estimated to be 0.2 JK−1mol−1
and this small entropy change is attributed to the small
ordered moment. As seen in Fig. 2, we have observed
an anomaly in the electrical resistivity curve at the
same temperature. The electronic specific heat coeffi-

194
Tatsuhiro Nozue et al.
(Vol. 70,
cient is obtained to be 2.86 mJK−2mol−1 from the plot
as C/T versus T2 as shown in the inset of Fig. 3, which
is comparable with that of Pauli paramagnetic NiAs,
3.01 mJK−2mol−1 3)
.
The magnetizations along the symmetric directions,
[100], [010] and [001], at 4.2 K were obtained to be pro-
portional to the field up to 5 T, using a SQUID mag-
netometer. The value at 5 T is as small as 0.5% of the
Fe moment obtained by the previous neutron diffraction
measurement.2) The susceptibilities along the three di-
rections were estimated to be 1.8×10−5 emu cm−3. Since
the dHvA signals show the smooth baseline within the
range up to 14 T as described below, the magnetization
seems to be small even at 14 T. Then we regard the ap-
plied field, H, as the magnetic induction, B.
Fig. 5. Fourier spectra of the oscillations in four divided regions
along the [010] direction in FeP. Each spectrum consists of fun-
damental frequencies indicated by the arrows, their higher har-
monics and those mixed fundamental ones.
Figure 4 shows a typical dHvA oscillation observed in
the field along the [010] direction at 0.45 K. The smooth
baseline of the oscillation strongly suggests that no mag-
netic phase transition takes place up to 14 T. Since the
direction of magnetic moments in the double helical mag-
netic state can be changed by the magnetic field, we have
analyzed the dHvA oscillations in four divided regions;
from 2.32 to 2.52, from 3.96 to 4.56, from 7.44 to 9.90,
and from 9.52 to 13.94 T, which correspond to an equal
range in 1/H. The fundamental dHvA frequencies in
the Fourier spectra obtained at the four field ranges are
indicated by the arrows, as shown in Fig. 5, and the oth-
ers were assigned to higher harmonics of the fundamen-
tal frequencies and those mixed fundamental ones. As
seen in Fig. 5, each fundamental frequency in these four
Fourier spectra is considered to be the same value within
the experimental accuracy. As the results, the changes
of the direction of magnetic moments and the propaga-
tion vector induced by the magnetic field are too small
to affect the results of dHvA measurements up to 14 T.
The independence of the fundamental dHvA frequencies
from the magnetic field is different from the results of
MnP.14) This difference is attributed to the small Fe mo-
ment and to nearly antiferromagnetic coupling between
the nearest-neighbor ferromagnetic-sheets parallel to the
(001) plane.2) Thus the magnetic structure of FeP under
magnetic field is regarded as the same structure without
magnetic field.
A lot of fundamental dHvA frequencies were extracted
from the observed dHvA oscillations in the (100), (010)
and (001) planes. The angular dependence of these fre-
quencies is shown in Fig. 6 and below 9800 T there are
nineteen dHvA branches called by the Greek characters
as seen in Fig. 6. The β, ζ, λ, ν, π, ρ, σ, τ and υ
branches have minima at the [010] direction. Among
Fig. 6. Angular dependence of the dHvA frequencies in FeP.
these branches, β, λ and π branches were observed in all
directions. These branches have the similar angular de-
pendence around the [010] direction. In the (010) plane,
they seem to split into several close branches. The ζ,
ν and ρ branches were observed between the angles at
which the frequencies in these branches are almost equal
to those in the β, λ and π branches, respectively, and
also have similar angular dependences. Moreover, the σ
and τ branches are resemble to π and ρ branches, re-
spectively, though they were observed in limited angular
regions. Thus these eight branches are regarded as four
sets of the pair branches, that is, (β, ζ), (λ, ν), (π, ρ) and
Fig. 4. Typical dHvA oscillation in field along the [010] direction
in FeP.

2001)
de Haas-van Alphen Effect of FeP in Double Helical Magnetic State
195
tions, respectively. The parameter rx is calculated with
the frequencies at three symmetric axes, Fx, Fy and Fz,
by the relation rx = ((2e/¯hc)(FyFz/Fx))1/2. The param-
eters of ry and rz are expressed by the same relations. As
seen in Fig. 6, the value of Fy is much smaller than those
of Fx and Fz, each of ellipsoids is found to be elongated
along the [010] direction. Around the [010] direction,
however, the observed frequencies increase more rapidly
than the angular dependence of the branches obtained
from the above ellipsoids. Thus the surface around the
[010] direction, at least within about 40◦ from the [010]
axis, is considered to be hyperboloidal rather than ellip-
soidal. Such the hyperboloid is written as
(σ, τ).
The α, δ, ι, κ, μ and ξ branches which were observed
in the limited angular regions have minima at the [100]
direction and have similar angular dependences. The γ
branch consists of one frequency in the (010) plane and
two frequencies in the (001) plane. The lower branch has
the minimum at 24◦ from the [100] direction in (001).
The η and ε were observed only in the (010) and (001)
plane, respectively. The θ branch is observed at about
660 T in the angular region from [001] to 42◦ in (100)
and to 3◦ in (010).
Cyclotron mass ratios, m∗/m0, were obtained at the
three symmetric directions, [100], [010] and [001]. The
linear relation between ln(A/T(1 − exp(−2XT))) and T
was satisfied within the temperature ranges of this exper-
iment where A is the amplitude of the dHvA oscillation
and X is proportional to m∗/m0. The obtained m∗/m0
are summarized in Table I with their dHvA frequencies.
The minimum value of m∗/m0 is 0.60 of the β branch at
the [010] direction and the maximum value 5.9 of the λ
branch at the [010] direction.
ky2
ry2
kx2
rx2
kz2
rz2
−
+
= 1.
(4.2)
As a result, the whole surface for the β, λ and π branches
is a “cocoon”-like one as shown later. These branches
were fitted to the angular dependences of the hyper-
boloids within about 40◦ from the [010] direction. Con-
sequently, the radii rz for the β, λ and π branches
were obtained to be 0.072, 0.184 and 0.276 A−1, respec-
˚
§4. Discussion
tively. The Brillouin zone of magnetic FeP is reduced
−1
Let us consider now Fermi surfaces for the many com-
plicated and characteristic branches as shown in §3. We
must take into account the superstructure with a period
of 5c as discussed below. The band structure of magnetic
FeP has not yet been calculated owing to the complicated
spin structure. Thus we discussed the observed branches
with simple Fermi surfaces drawn by conventional ana-
lytic equations.
At first, we take up the β, λ and π branches which
were observed in all directions. Thus the Fermi surfaces
drawn by these branches are thought to be closed ones
at high magnetic field. The closed surface is tentatively
considered as an ellipsoid expressed by the equation,
˚
to 1/5 × (2π/c) = 0.217 A along the [001] direction,
where (2π/c) is for the non-magnetic state, that is, for
the MnP-type structure. The elliptic orbits for the λ
and π branches in the (010) plane are, therefore, beyond
the zone boundary perpendicular to the [001] direction.
This suggests that the λ and π branches correspond to
the large orbits attributed to magnetic breakdown at the
zone boundary perpendicular to the [001] direction.
Then we assume a simple elliptic orbit for the π branch
in the first Brillouin zone for the non-magnetic state in
the (010) plane as shown in Fig. 7(a). In the magnetic
state, energy gaps open at which the energy ε(k) satis-
fies the relation, ε(k) = ε(k + G nq). In this equation,
G is a reciprocal lattice vector and n an integer corre-
sponding to the order of the gap resulting from the in-
teraction with the wave vector q of the helical magnetic
ky2
ry2
kx2
rx2
kz2
rz2
+
+
= 1,
(4.1)
where x, y and z denote the [100], [010] and [001] direc-
Table I. dHvA frequencies, F, and cyclotron mass ratios,
m∗/m0, at the symmetric directions in FeP.
F (T) (m∗/m0)
F (T) (m∗/m0)
[100]
α
γ
ι
115
317
810
1.1
1.5
2.3
3.0
2.9
2.6
3.5
3.8
3.1
4.2
4.5
5.0
5.5
5.2
5.4
5.7
5.8
[010]
β
ζ
270
670
0.60
1.2
0.89
1.7
1.2
2.3
1.8
5.1
λ
ν
π
ρ
σ
τ
1216
1710
2150
2800
3070
4070
κ
μ
β
1180
1460
1840
1990
2170
4060
4160
4280
4430
4580
4770
6530
6640
6790
ξ
λ
Fig. 7. (a) Orbit X0 is displaced to the broken ellipses by q of
the double helical spin structure in the Brillouin zone indicated
as an outer rectangle. The Brillouin zone is reduced by q as
indicated by the bold rectangle. The ellipse X0 has the radii rx
and rz obtained from fitting the π branch around the [010] to the
hyperboloidal Fermi surface. (b) Closed orbit X1 and two open
tubular orbits X2 and X3. The possible orbits with magnetic
breakdown at the z1 an z2 points are indicated as bold ellipses
B1 and B2.
[001]
θ
660
3790
4000
4230
4520
4730
4880
5200
3.1
3.7
4.0
5.0
5.2
5.1
4.0
5.9
β
π
λ
π

196
Tatsuhiro Nozue et al.
(Vol. 70,
deviates from the data points in the high field region.
Therefore, the reduction of amplitude due to magnetic
breakdown in high field must be taken into account.
If two neighbor orbits are separated by a small gap
at a point in k-space, the electron tunnel through the
gap with a probability, P, given by exp (−H0/H) and
is reflected at the point with a probability, Q, given by
(1 − exp (−H0/H)), where the breakdown field, H0, de-
pends on the local geometry of both the orbits near the
point of tunneling or reflection.18) The oscillation am-
plitude is multiplied by the reduction factor written as
follow,
order. In Fig. 7(a), the ellipses displaced by q and 2q
are also shown. As shown in Fig. 7(b), the splits of or-
bits in the magnetic state arise at the points z1 and z2,
and there appear one closed orbit, X1, and two tubular
open orbits, X2 and X3. Since a gap energy like this
origin is generally thought to be very small, magnetic
breakdown easily takes place between these orbits. In
this case, the possible closed orbit with magnetic break-
down is an orbit B1; X1X2X1X2X1, which has four times
of tunneling at the point z1. The other is an orbit B2;
X1X2X3X2X1X2X3X2X1, which has four times of tun-
neling at both the points z1 and z2. The larger orbits
than the B2 orbit are also possible to be observed owing
to further many times of magnetic breakdown. Further-
more, the areas enclosed with the orbits, X1, B1 and
B2, are almost equal to those of the β, λ and π frequen-
cies at the [010] direction, respectively. Therefore, these
branches come from the close related Fermi surfaces due
to magnetic breakdown.
RMB = Pn /2Qn /2
,
(4.4)
1
2
where n1 is the number of the point at which the tun-
neling takes place between the relevant orbits and n2 at
which the reflection occurs. Since the β branch origi-
nates from the X1 orbit as shown in Fig. 7(b) and the
electron on the orbit was with no tunneling and with
reflecting at the two points z2, then n1 and n2 are 0
and 2, respectively. Thus RMB decreases with decreas-
ing the probability of the reflection at the point z2, that
is, with increasing the probability of tunneling to the X2
orbit, in the high field region. The resultant formula of
eq. (4.3) multiplied by RMB is fitted to the data obtained
with the fixed TD determined above. The fitted curve is
shown by the solid one in Fig. 8 and agrees well with the
experimental data. Then H0 is obtained to be 4.8 T.
As pointed out in §3, the π and ρ branches are con-
sidered to be paired ones, that is, they are considered to
originate from the different orbits on one “cocoon”-like
Fermi surface as shown in Fig. 9. The central thin orbit
on the surface is assumed to correspond to the π branch
and to be hyperboloidal described as eq. (4.2) and two
thick orbits at the both ends of surface to the ρ branch
and to be ellipsoidal as eq. (4.1). The (β, ζ) and (λ, ν)
pair branches on the similar “cocoon”-like surfaces have
the same relation to the (π, ρ) pair as shown in Fig. 9.
To verify the above model, using the fitted parameter
rx, ry and rz for the π branch as a hyperboloid and the
ρ branch as an ellipsoid, angular dependent frequencies
of (β, ζ) and (λ, ν) branches were calculated within the
In order to confirm magnetic breakdown mentioned
above in this sample, the field dependent amplitude of
the dHvA oscillation was analyzed for the β branch for
the [010] direction. According to the Lifshitz-Kosevich
formula, the field dependent amplitude, ALK, is de-
scribed as follow,17)
exp(−2π2kBTD/βH)
sinh(2π2kBT/βH)
ALK ∼ H−1/2
,
(4.3)
where β = e¯h/m∗c and TD is the Dingle temperature.
In the measurement with the field modulation method,
the amplitude of the dHvA signal is obtained to be
proportional to AAL multiplied by a Bessel function,
ꢀ
ꢁ
Jk 2πFh0H−2 . Figure 8 shows the field dependence
of the amplitude of β oscillation at the [010] direction.
The fitted result for the amplitude to eq. (4.3) below
2 T is shown by the broken curve in Fig. 8, where mag-
netic breakdown is ignored. Then TD is estimated to be
0.47 K using F = 270 T, m∗/m0 = 0.6, T = 0.65 K and
h0 = 9.8 × 10−3 T. As seen in Fig. 8, the broken curve
Fig. 9. Model of a “cocoon”-like Fermi surface corresponding to
(π, ρ) paired branches as projected on the (100) plane. The
central thin part corresponds to the π branch and the thick part
at the both ends to the ρ branch. The broken rectangle indicates
the Brillouin zone in magnetic state. The (β, ζ) and (λ, ν) paired
branches are obtained from the similar surfaces with the different
dimensions as shown in Fig. 7(b).
Fig. 8. Field dependence of the amplitude of β oscillation. The
broken curve is calculated using eq. (4.3) multiplied by the first
ꢀ
ꢁ
order Bessel function, J1 2πFh0H−2 . The solid curve shows
the result with the reduction factor RMB due to magnetic break-
down.

2001)
de Haas-van Alphen Effect of FeP in Double Helical Magnetic State
197
angular region of about 40◦ from the [010] direction in
the (001) and (100) planes. The dimensions rnx and rnz
in the (010) plane are determined by the relation as fol-
lows,
nq
rnz = rz −
2
ꢂ
ꢅ
1/2
ꢃ
ꢄ
2
nq
rnx = rx 1 −
,
(4.5)
2rz
where n=1 and 2 correspond to the (β, ζ) and (λ, ν) pair
branches, respectively. As shown in Fig. 10, the calcu-
lated results were drawn with solid and broken curves
which agree with the experimental data qualitatively.
Furthermore, as shown in Fig. 10, the σ and τ branches
are partly agree with the calculated results using the re-
verse process. However, the υ branch is not explained
with this model.
Subsequently, we discuss the α, δ, ι, κ, μ and ξ
branches observed around the [100] direction. They are
also similar to each other and, therefore, can be ana-
lyzed using the same model with magnetic breakdown
mentioned above. In this case, the parameters rz of the
ellipsoids fitted to these branches are also beyond the
Brillouin zone in magnetic state, except for the α and δ
branches. The angular dependence of the ellipsoids with
rny and rnz calculated from rx, ry and rz of the μ and ξ
branches with eq. (4.5) are shown by the solid and bro-
ken curves, respectively, in Fig. 11. The broken curves
calculated from the ξ branch for n=1 and 2 agree well
with the κ and α branches, respectively. The solid curve
Fig. 11. dHvA branches around the [100] direction in FeP. The
bold solid and broken curves show the results of fitting the μ
and ξ branches to the angular dependence of the ellipsoidal cross
section, respectively. The thin solid and bold curves indicate the
angular dependence with the radii of rnx and rnz calculated for
n = 1 and 2 from the values of rx, ry and rz of the μ and ξ
branch with eq. (4.5), respectively.
obtained from the μ branch for n=1 agrees qualitatively
with the ι branch. The solid curve for n=2 disagrees with
the observed branches, therefore, the δ branch is thought
to be caused by the different origin. Moreover, the γ, ε,
η and θ branches are remained to be explained. So far,
the Fermi surfaces are considered to be centered in the
Brillouin zone. The information about the positions of
surfaces was not obtained in this experiment.
As seen in Fig. 6, there observed the β, λ and π
branches split into 3∼7 branches for the all directions
in the (010) plane. The ζ and ν branches were shown as
one branch in Fig. 6 because of the low resolution of the
Fourier analysis in the narrow field ranges. However, the
ζ and ν oscillations for the [010] direction observed in the
wide field range of 1 to 14 T split into three frequencies.
The similar branches with a small splitting were observed
in chromium9) which has an incommensurate longitudi-
nal SDW below 123 K.11) Reifenberger et al.12) has pro-
posed the chain model, that is, the split branches were
attributed to the segment between the intersections of
ellipsoidal Fermi surfaces displaced by the incommensu-
rate propagation vector multiplied by an integer. These
results suggest that FeP has an incommensurate helical
spin structure with the slightly deviated period from 5c.
Figure 12 presents the plot of the cyclotron mass ra-
tios versus dHvA frequencies for FeP and the other tran-
sition metal compounds.4, 6, 10, 19-22, 24, 25) The cyclotron
masses of MnSi are strongly enhanced by the low fre-
quency fluctuation of the spin density.26) In AuSn, nearly
free electron model is adopted for the s and p electrons,
and thus the cyclotron mass ratio is rather small.25) As
seen in this figure, FeP has relatively heavy cyclotron
masses among the transition metal compounds. How-
ever, the electronic specific heat coefficient is rather
Fig. 10. dHvA branches around the [010] direction in FeP. The
bold solid and broken curves show the results of fitting the π
branch to the hyperboloidal surface and the ρ branch to the
ellipsoid, respectively. The solid curves indicate the angular de-
pendence with the parameter rnx and rnz calculated for n = 1
and 2 from the values of rx, ry and rz of the π branch with
eq. (4.5). The broken curves indicate the angular dependence
calculated from the parameters of the ρ branches in the same
way.

198
Tatsuhiro Nozue et al.
(Vol. 70,
deviated period from 5c. The cyclotron mass ratios of
FeP are relatively heavy among the transition metal com-
pounds.
Acknowledgments
We are grateful to Profs. T. Suzuki and K. Motizuki for
their valuable discussion and encouragement. We wish to
thank Dr. M. Matsuura for his kind support to utilize the
AC-calorimeter. The present work is supported, in part,
by the REIMEI Research Resources of Japan Atomic
Energy Research Institute.
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Fig. 12. Cyclotron mass ratio versus dHvA frequency for FeP and
the other transition metal compounds. The upper and lower
broken lines are to guide the reader’s eye.
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,
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High quality single crystals of FeP were prepared by
the chemical transport reactions, using iodine as the
transport agent. On these samples, the dHvA effect was
measured in the field up to 14 T at temperature down to
0.42 K. Nineteen branches are assigned in the three prin-
cipal planes. Four pairs of branches observed around the
[010] direction have a similar angular dependence to each
other. It was found that the pair branches of the lowest
frequency among them correspond to the orbits on sur-
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spin structure and the other originates from the larger
orbits with several times of magnetic breakdown. The
branches around the [100] direction are thought to be in
the same situation. The occurrence of magnetic break-
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dependent amplitude of the β oscillation at the [010] di-
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into several branches in the (010) plane, however, sug-
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