Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field
The nonlinear response to an ac magnetic field with and without parametric pumping was studied experimentally on a rod-like beryllium single crystal at low temperatures in a quantizing magnetic field applied parallel to the hexagonal axis. At low temperatures (≤ 3 K) giant parametric amplificatio...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1203532017-06-12T03:05:50Z Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field Tsindlekht, M.I. Logoboy, N. Egorov, V.S. Kramer, R.B.G. Jansen, A.G.M. Joss, W. К 100-летию со дня рождения Б.Г. Лазарева The nonlinear response to an ac magnetic field with and without parametric pumping was studied experimentally on a rod-like beryllium single crystal at low temperatures in a quantizing magnetic field applied parallel to the hexagonal axis. At low temperatures (≤ 3 K) giant parametric amplification of the nonlinear response is observed in narrow windows near de Haas– van Alphen (dHvA) beating antinodes where the amplitude of the magnetic oscillations is maximal. This effect is of threshold character with respect to the amplitude of the dHvA oscillations and occurs as a result of a diamagnetic phase transition of a strongly correlated electron gas into nonuniform state with formation of Condon domain structure. It is shown that the nonuniform phase appears periodically in a magnetic field. The components of rectified signal field reproduce the envelope of H–T critical curves and can be used for reconstruction of diamagnetic phase diagrams. 2006 Article Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field / M.I. Tsindlekht, N. Logoboy, V.S. Egorov, R.B.G. Kramer, A.G.M. Jansen, W. Joss // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1129–1135. — Бібліогр.: 23 назв. — англ. 0132-6414 PACS: 71.70.Di, 74.25.Nf, 74.60.Ec, 75.60.–d http://dspace.nbuv.gov.ua/handle/123456789/120353 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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К 100-летию со дня рождения Б.Г. Лазарева К 100-летию со дня рождения Б.Г. Лазарева |
| spellingShingle |
К 100-летию со дня рождения Б.Г. Лазарева К 100-летию со дня рождения Б.Г. Лазарева Tsindlekht, M.I. Logoboy, N. Egorov, V.S. Kramer, R.B.G. Jansen, A.G.M. Joss, W. Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field Физика низких температур |
| description |
The nonlinear response to an ac magnetic field with and without parametric pumping was studied
experimentally on a rod-like beryllium single crystal at low temperatures in a quantizing magnetic
field applied parallel to the hexagonal axis. At low temperatures (≤ 3 K) giant parametric
amplification of the nonlinear response is observed in narrow windows near de Haas– van Alphen
(dHvA) beating antinodes where the amplitude of the magnetic oscillations is maximal. This effect
is of threshold character with respect to the amplitude of the dHvA oscillations and occurs as a result
of a diamagnetic phase transition of a strongly correlated electron gas into nonuniform state
with formation of Condon domain structure. It is shown that the nonuniform phase appears periodically
in a magnetic field. The components of rectified signal field reproduce the envelope of H–T
critical curves and can be used for reconstruction of diamagnetic phase diagrams. |
| format |
Article |
| author |
Tsindlekht, M.I. Logoboy, N. Egorov, V.S. Kramer, R.B.G. Jansen, A.G.M. Joss, W. |
| author_facet |
Tsindlekht, M.I. Logoboy, N. Egorov, V.S. Kramer, R.B.G. Jansen, A.G.M. Joss, W. |
| author_sort |
Tsindlekht, M.I. |
| title |
Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field |
| title_short |
Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field |
| title_full |
Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field |
| title_fullStr |
Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field |
| title_full_unstemmed |
Giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field |
| title_sort |
giant parametric amplification of the nonlinear response in a single crystal of beryllium in a quantizing magnetic field |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
К 100-летию со дня рождения Б.Г. Лазарева |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120353 |
| citation_txt |
Giant parametric amplification of the nonlinear response in
a single crystal of beryllium in a quantizing magnetic field / M.I. Tsindlekht, N. Logoboy, V.S. Egorov, R.B.G. Kramer, A.G.M. Jansen, W. Joss // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1129–1135. — Бібліогр.: 23 назв. — англ. |
| series |
Физика низких температур |
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2025-07-08T17:43:01Z |
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2025-07-08T17:43:01Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9, p. 1129–1135
Giant parametric amplification of the nonlinear response in
a single crystal of beryllium in a quantizing magnetic field
M.I. Tsindlekht1,2, N. Logoboy1,2, V.S. Egorov1,3, R.B.G. Kramer1,
A.G.M. Jansen4, and W. Joss1
1Grenoble High Magnetic Field Laboratory, Max-Plank-Institute f�r Festk�rperforschung,
B.P. 166X, Cedex 9, Grenoble F-38042, France
2The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Email: mtsindl@vms.huji.ac.il
3Russian Research Center Kurchatov Institute, Moscow, 123182, Russia
4Service de Physique Statistique, Magnetisme et Supraconductivity, DRFMC,
CEA, 17 rue de Martyrs, Cedex 9, Grenoble, F-38054, France
Received March 3, 2006
The nonlinear response to an ac magnetic field with and without parametric pumping was studi-
ed experimentally on a rod-like beryllium single crystal at low temperatures in a quantizing mag-
netic field applied parallel to the hexagonal axis. At low temperatures (� 3 K) giant parametric
amplification of the nonlinear response is observed in narrow windows near de Haas– van Alphen
(dHvA) beating antinodes where the amplitude of the magnetic oscillations is maximal. This effect
is of threshold character with respect to the amplitude of the dHvA oscillations and occurs as a re-
sult of a diamagnetic phase transition of a strongly correlated electron gas into nonuniform state
with formation of Condon domain structure. It is shown that the nonuniform phase appears period-
ically in a magnetic field. The components of rectified signal field reproduce the envelope of H–T
critical curves and can be used for reconstruction of diamagnetic phase diagrams.
PACS: 71.70.Di, 74.25.Nf, 74.60.Ec, 75.60.–d
Keywords: de Haas—van Alphen effect, Condon domains, paramrtric phenomena.
1. Introduction
Quantum oscillations of resistivity and magnetiza-
tion in bismuth were discovered in 1930 by
Schubnikov and de Haas [1] and de Haas and van
Alphen, respectively [2]. The observation of quantum
oscillations in other than bismuth metals was reported
about ten years later after the first publications. Very
important works concerning the the Schubnikov—de
Haas and de Haas—van Alphen effects in single crys-
tals of zinc, tin, beryllium were published by the
group of B.G. Lazarev [3]. Only in the early fifties
was it understood that quantum oscillations are a
powerful tool for the study of the electronic structure
of metals. Since that time the experiment and theory
of quantum oscillations in metals at low temperatures
were improved and detailed information about the
Fermi surfaces of metals was obtained. However, even
now there are a lot of loose ends and unsolved prob-
lems in this part of low-temperature physics.
Nonlinear phenomena in a quantizing magnetic field
are one of these problems.
Nonlinear properties of normal metals with a long
mean free path of electrons were studied in detail about
35 years ago. It was understood that for the observation
of a noticeable nonlinear response in a quantizing mag-
netic field H, the ac field amplitude h0 has to be com-
parable to the period of the magnetization oscillations
�H H /F� 2 , where F is the de Haas–van Alphen
(dHvA) frequency [4]. This means that at low ampli-
tudes of excitation, when h H0 �� � , the amplitude of
the nonlinear response should be small, and this phe-
nomenon could be well described in the frame of per-
turbation theory [5–7]. Perturbation theory provides a
good explanation for the majority of nonlinear experi-
© M.I. Tsindlekht, N. Logoboy, V.S. Egorov, R.B.G. Kramer, A.G.M. Jansen, and W. Joss, 2006
ments with normal metals in low magnetic fields; see
for example [8] and references therein. The nonlinear
helicon resonance in a quantizing magnetic field was
studied in aluminium [9]. Experiments show that reso-
nance curve shifts when the amplitude of the excitation
in the metal exceeds the dHvA period. A complete the-
ory of nonlinear helicon resonance does not exist up to
now. The recent experimental study of the dHvA oscil-
lations in single crystal beryllium at low temperatures
showed that strong nonlinear response appears even at
very low amplitude of excitation in an antinode area of
dHvA oscillations [10].
All the above mentioned experiments are related to
nonlinear effects, such as self-influence [9], harmonic
generation, and/or rectification [8,10]. Another class
of nonlinear effects is related to parametric phenom-
ena. Parametric phenomena are important in funda-
mental physics and for applications. For example,
Josephson junctions have been used as parametric am-
plifiers of weak microwave signals [11]. Experimental
observations of parametric amplification of the weak
microwave signals were possible because of the non-
linearity of the inductance of the Josephson junction
[11]. Recently, giant parametric amplification of the
nonlinear response was observed in the surface super-
conducting state, for H H Hc c2 3� � , of niobium
single crystals [12,13]. Application of parametric
pumping in the latter case leads to a two or-
der-of-magnitudes enhancement of the nonlinear re-
sponse which cannot be described by perturbation
theory. The main physical reason for the observed phe-
nomenon is a transition between metastable states in a
surface superconducting state upon application of an
ac magnetic field [11,13].
In this paper we present the results of an expe-
rimental study of parametric phenomena in a
nonsuperconducting single crystal of beryllium in a
quantizing magnetic field. It is well known, that the
amplitude of the dHvA effect in beryllium in a
quantizing magnetic field is very high [4]. The condi-
tions of diamagnetic phase transition, where instabili-
ties appear in the system, develop periodically with
variation of the magnetic field. As a result, the line
shape of the dHvA oscillations, M H( ), becomes
sawtooth-like in rod-like samples (Shoenberg effect),
while in plate-like samples, Condon domains form
[4,15]. Such a sawtooth shape of magnetization is
somewhat similar to flux jumps in superconductors
mentioned above. However, parametric phenomena in
normal metals in a quantizing magnetic field, to the
best of our knowledge, have not been studied either
experimentally or theoretically so far. Therefore, it is
interesting to study parametric phenomena in normal
metals too. The best candidate for this role is beryl-
lium which has strong dHvA effect. The relatively
high amplitude of dHvA oscillations in beryllium is
explained by the cylinder-like Fermi surface. It has
well known folders of electrons «cigar» with two
extremal cross-sections of almost equal areas. Conse-
quently, the dHvA oscillations show a beat with a
period 33 times larger than the fundamental frequency
period. Due to this effect, at constant temperature the
amplitude oscillations of magnetization can be
changed drastically by small changes in dc magnetic
field.
2. Experimental setup
We have intended to apply the technique used in
the experiment with Nb [12,13] for the study of
parametric phenomena in beryllium. A rectangular
(8 1 1� � mm) sample was cut by an electric spark
from a high quality single crystal beryllium bar (the
same bar which was used for the sample on which dia-
magnetic domains were observed in [16]). The long
axis of the sample is parallel to the [ ]0001 direction.
Measurements of the rectified signal were carried out
as follows (see [12,13] for details). The sample was
excited by an amplitude modulated high frequency
field h t h t t( ) ( cos ) cos� �0 1 � �� . A small high fre-
quency coil was used for this purpose. Typical values
of the above parameters were h0 12� . Oe, � 0 9. ,
�
/2 1 4� . MHz, and �/2 1609
� Hz. When the ac
excitation is applied to a sample in the nonlinear
state, the magnetic moment of the sample oscillates at
the harmonics of the fundamental frequency �, at fre-
quencies � � �, and at the frequency of modulation �
and its harmonics. Secondary coils convert these
1130 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9
M.I. Tsindlekht, N. Logoboy, V.S. Egorov, R.B.G. Kramer, A.G.M. Jansen, and W. Joss
HFG HPF BPF LCK
Sample
Secondary
Coils
Primary
Coil
dc magnet
Parametric
Pumping
Coil
LFG 2
Fig. 1. Block diagram of the experimental setup. HFC is
high frequency generator, LFG is low frequency generator,
HPF and BPF is high and band pass filters, LCK is
lock-in amplifier.
magnetic moment oscillations into an ac voltage sig-
nal. In addition, a parametric pump field h tp ( ) �
� �h t2 22� �� �cos( ) was applied using a separate
copper driving coil.
We would like to emphasize the difference between
our and standard parametric experiments. Usual para-
metric experiments have to deal with two signals. The
system is excited by weak and strong (pump) signals.
The frequency of the pump signal is twice as large as
the frequency of the weak signal. Due to the
nonlinearity of the system, power from the pump sig-
nal could be transferred to the weak signal [17]. In
our experiment the sample of beryllium with nonlin-
ear M B( ) dependence was excited by an amplitude
modulated ac field with carrier frequency� and modu-
lation frequency �. The spectrum of this ac field does
not contain frequency�. A signal with frequency� in
a system appears when the sample is in a nonlinear
state. Application of parametric pumping with fre-
quency 2� could lead to amplification of the rectified
signal on the frequency �.
In the nonlinear experiments the amplitude A� and
the phase �� of the signal at frequency� as a function
of the dc magnetic field H, ac field amplitude h0, am-
plitude h2� and phase � �2 of the parametric pump h2�
have measured by means of lock-in detection. A block
diagram of the experimental setup is shown in Fig. 1. A
primary high frequency copper coil, driven by a high
frequency generator, produces an amplitude modulated
ac field h t( ). Signals with frequencies � and 2� were
provided by another generator. The modulation fre-
quency � was used as a reference frequency for the
lock-in amplifier. The ac field amplitudes h0 and h2�
were measured by an additional small probe coil wound
under the primary coil. For the sake of clarity the probe
coil is not shown in Fig. 1. The high-pass and band-pass
filters are a crucial part of this set-up (Fig. 1). These
filters should be passive and they have to prevent (i)
the leak of the hindrance signal of frequency� through
the high frequency generator (high-pass filter) and (ii)
penetration of signals of frequency � and 2� to the in-
put of the lock-in amplifier (band-pass filter). One
should note that the ratio between the pumping fre-
quency and the modulation frequency was exactly
equal to 2 in our experimental setup.
3. Experimental results
The linear susceptibility
� dM/dH was measured
by the pick-up coils method [4] at 22 Hz using the same
coil system. The field dependence of the
at 4.2 K is
shown in Fig. 2,a. The observed spectrum is typical for
the dHvA effect in beryllium for the H || [ ]0001 direc-
tion. Analysis shows that this spectrum consists of two
dHvA frequencies, F1 940� T and F2 970� T,
corresponding to two extremal cross-sections of the «ci-
gar»-like Fermi surface [4,15]. Decreasing the tempera-
ture leads to a sharp increase of the amplitude of the
dHvA oscillations (see inset of Fig. 2,a). In the present
work we measure the nonlinear properties of beryllium
at T � 1 4. K.
Measurements of the in-phase component of the recti-
fied signal, ReA� , (inset to Fig. 2,a) show that its shape
looks like the first derivative of the susceptibility
and
Giant parametric amplification of the nonlinear response in a single crystal of beryllium
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1131
T = 4.2 K a
0 2 4 6
20
10
0
T = 1.4 K
ar
b
.u
n
its
,R
e
A
(a
rb
. u
n
its
)
20
10
0
ReA
�
�
5.68 5.70 5.72 5.74
b
0 2 4 6
T = 1,4 K
0
h = 0.6 Oe
2�h = 1.6 Oe
R
e
A
,a
rb
.u
n
its
0
–1
–2
c
0 2 4 6
Im
A
,a
rb
.u
n
its
T = 1,4 K
0
h = 0.6 Oe
2�h = 1.6 Oe
3
2
1
0
Magnetic field, T
Fig. 2. Linear susceptibility � of beryllium single crystal
with [ ]0001 || H versus magnetic field. Inset: Field depend-
ence of � � dM/dH and in-phase signal, ReA�, without
parametric pump (a). In-phase, ReA�, component of recti-
fied signal field dependence (b). Out-of-phase, ImA�,
component of rectified signal field dependence (c).
the amplitude dependence of A h� ( )0 is almost qua-
dratic. The amplitude of the out-of-phase component of
signal, Im A� , is much smaller than Re A� . This means
that perturbation theory can be used for an explanation
of the experimental observations presented above, and
the ratio h / H0 1� �� is a «good» small parameter for
this problem.
Application of parametric pumping introduces an
essential change into the nonlinear response. Fig-
ures 2,b, and 2,c show field dependencies of the Re A�
and Im A� components of the rectified signal in the
presence of parametric pumping.
Parametric pumping affects the rectified signal
only at low temperatures and for magnetic fields near
the antinodes of the magnetization beat and does not
affect nonlinear response near the nodes of the beats.
In magnetic fields H � 6 T the application of paramet-
ric excitation does not produce any changes in recti-
fied signal. For T � 4 2. K parametric pumping has no
influence on the nonlinear response at any values of
magnetic field.
Figure 3 presents the field dependence of Re A�
and Im A� with and without parametric pumping
near an antinode of the beat in more detail. First of
all, as mentioned above, Re A� without parametric
pumping is small and looks like the first derivative of
. With the application of parametric pumping the
rectified signal becomes unusual for magnetic fields
near the antinode, 3 49. T � �H 3 74. T. In this region,
at a given h0 and h2� , the signal grows steeply with
the dc magnetic field, whereas the dHvA amplitude
changes smoothly. This behavior leads us to speculate
about the threshold character of this phenomenon
with respect to the dHvA amplitude.
Figure 4 shows the field dependence of Re A� and
Im A� in the small region near the maximum of the
dHvA at different pump amplitudes, h2� . Parametric
pumping affects the out-of-phase component of the
rectified signal more strongly than the in-phase one.
In Fig. 5 we present the magnitude of Re A� and
Im A� as a function of h2� for a magnetic field near
the beating maximum H � 3 6. T and for H � 5 8. T.
Both Re A� and Im A� increase almost quadratically
at low pump levels h2 1� � Oe. Perturbation theory
predicts a linear dependence at low amplitude of the
parametric pump [17]. With further increase of the
pump h2� the amplitudes don’t change significantly,
showing only a small decrease. Therefore, there seems
to be an optimal parametric pump level of about 1 Oe.
1132 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9
M.I. Tsindlekht, N. Logoboy, V.S. Egorov, R.B.G. Kramer, A.G.M. Jansen, and W. Joss
T = 1.4 K
3.5 3.6 3.7
h = 1.6 Oe2�
�ImA
2�H = 0
ReA�
2�h = 1.6 OeR
e
A
&
Im
A
,a
rb
.u
n
its
Magnetic field, T
Fig. 3. Re A� and Im A� signals versus magnetic field near
antinode of magnetization beat.
R
e
A
, a
rb
. u
n
its
3.57 3.58 3.59
2.0 Oe
1.0 Oe
0.4 Oe
0.2 Oe
2 = 0
0h = 0.35 Oe
T = 1.4 K
0
–1
–2
a
R
eA
m
ax
h = 02
Im
A
,a
rb
.u
n
its
)
b4
3
2
1
0
2.0 Oe
1.0 Oe
0.4 Oe
0.2 Oe
h = 02
T = 1.4 K
2 = 0
0h = 0.35 Oe
Im
A
m
ax
3.57 3.58 3.59
Magnetic field, T
Fig. 4. Re A� versus magnetic field (a). Im ( )A H� for dif-
ferent amplitude h2� and � �2 0� (b).
ReA max
ImA max
maxA
H = 5.801 T
H = 3.584 T
T = 1.4 K
h = 0.35 Oe0
m
ax
m
ax
R
eA
&
Im
A
4
3
2
1
0
2�h , Oe
0 0.5 1.0 1.5 2.0
Fig. 5. Re Amax and Im Amax versus h2� near the beat
maximum (H � 3584. T) and Amax for H � 5801. T.
At H � 5 8. T parametric pumping almost does not af-
fect the rectified signal.
4. Discussion
Diamagnetic phase transition into a nonuniform
(Condon domains) phase is responsible for the ob-
served giant paramagnetic amplification of the nonlin-
ear response. The currently available experimental
data allow us to roughly define the range of magnetic
fields in the dHvA period where a nonuniform phase
exists. Figure 6 presents the field dependence of the
rectified signal without and with parametric pumping.
The line of the rectified signal is nonsymmetric. One
can readily see that parametric pumping begins to
affect strongly the rectified signal for magnetic field
when H H� 1. Using these experimental data one can
measure the value of the dHvA period itself, �H, and
the center of the dHvA period, H0. The relative part
of the dHvA period occupied by nonuniform phase,
�H/ H� can be roughly determined for each dHvA
period. Here �H H H� �2 0 1( ). Figure 7,a shows
�H/ H� as a function of H0. The edge experimental
points, �H/ H� � 0, were determined from the dHvA
oscillation when the changes of the rectified signal
line due to parametric pumping are negligible small.
The technique of parametric pumping amplification
of the nonlinear response in single crystal of beryllium
allows us to define the critical values of the magnetic
field, when the amplitude of the dHvA oscillations be-
comes high enough for realization of a diamagnetic
phase transition. It is interesting to compare the ex-
perimental curve of Fig. 7,a with theoretical calcula-
tions.
The theoretical description of experimental data
should be based on construction of diamagnetic H–T
phase diagrams. Undoubtedly, for correct explanation
of the relatively high amplitude of the dHvA oscilla-
tions, the real shape of the Fermi surface of beryllium
has to be taken into account. The Fermi surface of be-
ryllium was investigated very carefully in the past [4].
This Fermi surface differs significantly from the free
electron model and consists only of the first and sec-
ond zone monster («coronet») and the third zone «ci-
gar». Three maximum cross-sections of the «cigar»
(«waist» and «hips») are characterized by a small cur-
vature, which results in relatively high amplitude of
dHvA oscillations. Thus, the standard approach,
based on Lifshitz–Kosevich formalism [4] giving satis-
factory results for a free electron gas, fails to describe
the electron instability in beryllium. Clearly, the an-
omalously low curvature of the Fermi surface of beryl-
lium near the extremal cross-sections has to be taken
into account. The model developed in [20] is a more
realistic one. It can be considered as an attempt to
take into account the actual 3D Fermi surface geome-
try for beryllium. It is consistent with the previous ex-
periments on observation of the Condon instability at
a value of the magnetic field up to � 3T. However, it
gives overestimated values for higher field and incon-
sistent with our present experiments and recent publi-
Giant parametric amplification of the nonlinear response in a single crystal of beryllium
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1133
1H 0H
2�h = 0
2�h = 1 Oe
4.09 4.10 4.11
Magnetic field, T
0
–0.8
–1.6R
e
A
,a
rb
.u
n
its
T = 1.2 K
�H
h = 0.35 Oe0
Fig. 6. Field dependence of Re A� with and without para-
metric pumping for one dHvA oscillations period.
Magnetic field, T
Magnetic field, T
4 5 6
a
b
T = 1.2 K
T = 1.2 K
T = 2 KD
H
/
H
0.6
0.4
0.2
0
H
/
H
0.6
0.4
0.2
0
T
,K
3
2
1
0
Magnetic field, T
4 5 6
0 1 2 43 5 6 7
Fig. 7. The field dependence of �H/ H� (a). This ratio de-
fines the part of dHvA period occupied by nonuniform
phase. The calculated ratio of the range of existence of the
Condon domains �H to the period of dHvA oscillations
�H (b). The typical envelope of the H–T phase diagram
of beryllium for Dingle temperature TD � 2 K is shown on
the inset.
cation on the observation of Condon domains in beryl-
lium [10]. Our analysis of experimental data is based
on the recent theory of diamagnetic phase transitions
[22], which allows one to reconstruct the phase dia-
grams consistent with present experimental results.
For every period of dHvA oscillations when the re-
duced amplitude of oscillations a dM/dB� � 1, the
instability of an electron gas, e.g. the diamagnetic
phase transition, results in the formation of a magneti-
zation jump for an infinitely long cylinder or of Con-
don domain structure in the plate-like sample with
strong temperature and magnetic field dependencies of
the local magnetic moments [21]. In the domain phase
the metastable states respect to the magnetic field are
determined by � � ��/ M 0 with � � �2 2 0�/ M , identi-
fying the local minima of the thermodynamic poten-
tial� relatively to the magnetization M. In the center
of the period of dHvA oscillations (H H� 0) the two
minima have the same energy corresponding to the
remnant magnetization. The instability points are
points where the two conditions � � ��/ M 0 and
� � �2 2 0�/ M must simultaneously hold, e.g. inflec-
tion points, restrict the range of existence of the non-
uniform phase. According to [21] the ratio of this
range �H to the period of dHvA oscillations �H is de-
fined as follows
�
H/ H a
a
� � � �
1
1
12( )arccos . (1)
The results of calculation of �H/ H� Eq. (1) in the
model of a slightly corrugated cylinder Fermi surface
of beryllium [22] are represented in Fig. 7,b. In the in-
set of Fig. 7,b the typical theoretical phase diagram
for beryllium is shown. It can be emphasized that the
in-phase and out-of phase components of rectified sig-
nals (Fig. 2), being dependent on the phase state of
strongly correlated 3D electron gas, recall the shape
of the phase diagrams, reproducing the envelope of the
diamagnetic phase curves H–T (see, inset of Fig. 7,b).
There is a rather good agreement between experimen-
tal data (Fig. 7,a) and the theoretical calculations
(Fig. 7,b). Thus, the technique of parametric pumping
is a powerful tool for reconstructing the phase dia-
grams of the diamagnetic metals under the conditions
of a strong dHvA effect.
5. Conclusions
Though the dHvA effect itself is a strong nonlinear
phenomenon, perturbation theory can be applied for
an explanation of the rectification without parametric
pumping even at low temperatures. In this sense beryl-
lium is a weak nonlinear system. However, at low
temperatures application of parametric pumping leads
to giant amplification of the nonlinear response in nar-
row windows of the dc magnetic field near the beating
antinodes in a part of dHvA period (see Fig. 4) when
the diamagnetic phase transition takes place [18,19].
The period of the dHvA oscillations grows as H2 and
the diamagnetic phase transition becomes less steep.
At the values of magnetic fields, H � 6 T, parametric
pumping does not affect the rectified signal, which is
consistent with recent theoretical considerations [22].
The non-linear problems of harmonic generation be-
yond perturbation theory, under conditions of a strong
dHvA effect were discussed theoretically long ago [7].
However, direct application of this theory to observed
parametric phenomena is not possible at the present
time. For a detail comparison of the experimental
findings presented in this paper and the detailed the-
ory of the non-linear response with parametric pump-
ing has to be developed.
The giant parametric amplification of nonlinear re-
sponse was detected in the surface superconducting
state of niobium and in normal metal beryllium under
conditions of a strong dHvA effect. Both systems dem-
onstrate: (i) a phase transition between metastable
states in niobium and (ii) diamagnetic phase transi-
tion in beryllium. One can believe that we have to
deal with a general phenomenon inherent to systems
with phase transitions. This phenomenon could be
used for the experimental reconstruction of the H–T
phase diagram of a system with a phase transition.
It is supposed, also, that the parametric amplifica-
tion of the nonlinear response may be observed in any
nonlinear system. In our experiments with niobium in
a sweeping magnetic field [23] and with beryllium in
the present study we show that nonlinearity of the sys-
tem itself is a necessary but not a sufficient condition
for the observation of the parametric phenomena.
We thank Professor I. Vagner and Dr. I. Sheikin
for useful discussions. One of the authors (M.I.T.) is
grateful for the support of the European Community
from the contract RITA-CT-2003-505474.
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