Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium
Nonlinear dynamics and transport properties of a 2D Wigner solid (WS) on the free surface of superfluid helium are theoretically studied. The analysis is nonperturbative in the amplitude of the WS velocity. An anomalous nonlinear response of the liquid helium surface to the oscillating motion of the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1759842021-02-04T01:31:12Z Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium Monarkha, Yu.P. Квантовые жидкости и квантовые кpисталлы Nonlinear dynamics and transport properties of a 2D Wigner solid (WS) on the free surface of superfluid helium are theoretically studied. The analysis is nonperturbative in the amplitude of the WS velocity. An anomalous nonlinear response of the liquid helium surface to the oscillating motion of the WS is shown to appear when the driving frequency is close to subharmonics of the frequency of a capillary wave (ripplon) whose wave vector coincides with a reciprocal-lattice vector. As a result, the effective mass of surface dimples formed under electrons and the kinetic friction acquire sharp anomalies in the low-frequency range, which affects the mobility and magnetoconductivity of the WS. The results obtained here explain a variety of experimental observations report-ed previously. 2018 Article Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium / Yu.P. Monarkha // Физика низких температур. — 2018. — Т. 44, № 4. — С. 379-388. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 73.20.Qt, 73.40.–c, 67.90.+z, 71.45.Lr http://dspace.nbuv.gov.ua/handle/123456789/175984 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Monarkha, Yu.P. Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium Физика низких температур |
| description |
Nonlinear dynamics and transport properties of a 2D Wigner solid (WS) on the free surface of superfluid helium are theoretically studied. The analysis is nonperturbative in the amplitude of the WS velocity. An anomalous nonlinear response of the liquid helium surface to the oscillating motion of the WS is shown to appear when the driving frequency is close to subharmonics of the frequency of a capillary wave (ripplon) whose wave vector coincides with a reciprocal-lattice vector. As a result, the effective mass of surface dimples formed under electrons and the kinetic friction acquire sharp anomalies in the low-frequency range, which affects the mobility and magnetoconductivity of the WS. The results obtained here explain a variety of experimental observations report-ed previously. |
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Article |
| author |
Monarkha, Yu.P. |
| author_facet |
Monarkha, Yu.P. |
| author_sort |
Monarkha, Yu.P. |
| title |
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium |
| title_short |
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium |
| title_full |
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium |
| title_fullStr |
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium |
| title_full_unstemmed |
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium |
| title_sort |
nonlinear dynamics of a two-dimensional wigner solid on superfluid helium |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
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http://dspace.nbuv.gov.ua/handle/123456789/175984 |
| citation_txt |
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium / Yu.P. Monarkha // Физика низких температур. — 2018. — Т. 44, № 4. — С. 379-388. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT monarkhayup nonlineardynamicsofatwodimensionalwignersolidonsuperfluidhelium |
| first_indexed |
2025-07-15T13:36:20Z |
| last_indexed |
2025-07-15T13:36:20Z |
| _version_ |
1837720227131949056 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4, pp. 379–388
Nonlinear dynamics of a two-dimensional Wigner solid
on superfluid helium
Yu.P. Monarkha
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: monarkha@ilt.kharkov.ua
Received August 16, 2017, published online February 26, 2018
Nonlinear dynamics and transport properties of a 2D Wigner solid (WS) on the free surface of superfluid he-
lium are theoretically studied. The analysis is nonperturbative in the amplitude of the WS velocity. An anoma-
lous nonlinear response of the liquid helium surface to the oscillating motion of the WS is shown to appear when
the driving frequency is close to subharmonics of the frequency of a capillary wave (ripplon) whose wave vector
coincides with a reciprocal-lattice vector. As a result, the effective mass of surface dimples formed under elec-
trons and the kinetic friction acquire sharp anomalies in the low-frequency range, which affects the mobility and
magnetoconductivity of the WS. The results obtained here explain a variety of experimental observations report-
ed previously.
PACS: 73.20.Qt Electron solids;
73.40.–c Electronic transport in interface structures;
67.90.+z Other topics in quantum fluids and solids;
71.45.Lr Charge-density-wave systems.
Keywords: Wigner solid, 2D electron systems, nonlinear transport, superfluid helium.
1. Introduction
A two-dimensional (2D) electron gas bound to the free
surface of liquid helium is known to exhibit a phase transition
into the Wigner solid (WS) state [1]. In this state, surface elec-
trons are localized in sites of a triangular lattice. The WS of
surface electrons is hovering above the liquid surface at the
average height of about 100 Å. The surface of liquid helium
has no pining centers, therefore, the WS can move along
the interface in an ac driving electric field interacting with
surface excitations of liquid helium. Electrons forming the
Wigner crystal put periodic pressure on the surface of liquid
helium, and, therefore, a lattice of surface dimples is formed
under electrons [2]. This periodic pressure and the dimple
lattice are essential for understanding the linear dynamics of
the WS in an ac driving field [3] and for the description of the
experiment [1]. An important consequence of the theory is
that an oscillating motion of the WS along the surface reso-
nantly excites capillary waves (ripplons) if the driving field
frequency ω is close to the frequency of ripplons
3/2
, = /r q qω α ρ with the wave vector q equal to the elec-
tron reciprocal lattice vector g (here α and ρ are, respective-
ly, the surface tension and mass density of liquid helium).
Since the discovery of Grimes and Adams [1], a number
of remarkable nonlinear effects was observed when studying
the WS transport along the surface of superfluid 4 He. A
nonequilibrium melting of the 2D WS indicated by a sharp
change in the magnetoresistance of the electron system
was reported in Ref. 4. A puzzling nonlinear dependence
of the WS magnetoconductivity xxσ ending by a dynamic
transition was observed [5,6]. This transition was interpret-
ed as the WS sliding over the sublattice of surface dimples.
In the region of small values of the input voltage inV , the
inverse magnetoconductivity 1
xx
−σ rises rapidly with inV up to
its maximum value. Then, in the region of intermediate
values of the input voltage, 1
xx
−σ decreases approximately
as in1/V . A similar region of xxσ proportional to inV ob-
served [7] was explained by an assumption that the Hall
velocity of electrons /cE B is limited by the phase velocity
of ripplons with the wave vector equal to the electron re-
ciprocal lattice vector g . It was suggested that in this re-
gion the electric field E is independent of inV , while
inxx Vσ ∝ . It should be noted also that in that work, the
region of low excitation voltage where 1
xx
−σ rises ( xxσ de-
creases) as well as the linear conductivity regime were not
detected.
© Yu.P. Monarkha, 2018
Yu.P. Monarkha
In the absence of a magnetic field, the nonlinear WS
mobility was studied in Ref. 8. In this experiment, an ini-
tial mobility of the WS is rather high, and it strongly de-
creases with the amplitude of the driving electric field.
This is contrary to the magnetoconductivity data of
Refs. 5,6 which indicate that at a smallest inV mobility of
the WS is low (though there are no pinning centers above
superfluid 4 He) and it initially increases with the input
voltage. This seeming contradiction should be explained in
a strict theory.
Interesting results were obtained in experimental stud-
ies [8] of coupled phonon-ripplon modes of the WS on
liquid helium in high driving electric fields. At a low exci-
tation voltage the response amplitude as a function of the
frequency of the excitation signal shows two maxima whose
positions agree with the frequencies of a coupled phonon-
ripplon mode [3] calculated for two smallest wave vectors 1k
and 2k defined by the geometry of the experimental cell. It
seems strange that with an increase of the amplitude of the
excitation signal positions of these two resonances were
observed to shift in opposite (!) directions. Moreover, new
low-frequency electron-ripplon resonances were observed
away from frequencies of conventional phonon-ripplon cou-
pled modes [3]. These experimental data also require a the-
oretical explanation (a brief report explaining these effects
is given in Ref. 9).
It should be emphasized that contrary to the case of su-
perfluid 4 He, the linear regime of the WS transport over the
free surface of normal and superfluid 3 He was detected [10],
and anomalies of WS mobility data were well described by
the theory [11]. Moreover, there is a good understanding [12]
of the nonlinear WS mobility on the surface of liquid 3He. In
this case, a good theoretical description is possible owing
to strong damping effects in the Fermi liquid which limit
the mobility of surface dimples. We conclude that remarka-
ble nonlinear phenomena reported for the WS transport over
superfluid 4 He are induced by the extremely small damping
of ripplons. According to the quantum hydrodynamical
model [13], the damping of capillary waves
42
1
=
60 vq
T q
π
γ ρ
(1)
decreases fast with lowering temperature. Here 1v is the
first sound velocity, and q is a wave vector. For typical
experimental conditions (electron density 8 2= 6 10 cmen −⋅
and 0.4 K 0.1 KT≥ ≥ ), the ratio ,g r gγ ω varies from 410−
to 710− even for the smallest reciprocal-lattice vector 1= .g g
In this work, we report a theoretical investigation of
nonlinear dynamics and nonlinear transport properties of
the 2D WS over superfluid helium caused by extremely
small damping of capillary waves. Considering the regime
of a given current, we found an exact expression for the me-
dium response force acting on the dimple sublattice which
consists of two parts representing the kinetic friction and
dimple inertia. It is remarkable that the effective mass of
surface dimples and the effective collision frequency of the
kinetic friction change sharply in the vicinity of
subharmonic frequencies of ripplons , /r g mω (here
= 2, 3,...m ), which affects the nonlinear transport of the
WS and phonon-ripplon coupling. In the presence of a
magnetic field directed normally to the surface, the nonlin-
ear magnetoconductivity expression obtained is not re-
duced to a simple Drude form which explains differences
between mobility and magnetoconductivity data reported
previously. At sufficiently strong driving fields, the non-
linear magnetoconductivity xxσ obtained here becomes
negative, which causes instability and WS melting. This
effect can be considered as an alternative explanation of
dynamic transitions observed in the experiments [4–6].
2. Model description
The localization of an electron in a lattice site is ac-
companied by a displacement of the liquid helium surface
( )ξ r from the equilibrium flat shape caused by the electron
pressure. This deformation of the gas-liquid interface called
the dimple lattice is the main origin of nonlinear effects in
the WS transport. Under usual conditions, the dimple lat-
tice does not change the melting temperature of the WS,
but it strongly affects its dynamics, especially in the low
frequency range. In terms of electron displacements ls from
a lattice site lR , the WS coupling with capillary waves is
described by the interaction Hamiltonian
[ ]int = exp ( ) ,q l lH U iξ +∑ ∑q
q l
q R s (2)
where ξq is the Fourier transforms of the surface displace-
ment, qU is the coupling function for the electron-ripplon
interaction [14] (in the limit of strong holding fields E⊥ , it
equals ).eE⊥ Displacements ls consist of a high-frequency
part f ,ls , caused by thermal vibrations, and a low-frequency
part s,ls due to slow WS motion in a driving electric field. In
this work, the low-frequency part is assumed to be uniform
s, ( )l t≡s s .
Taking into account an expression for the Hamiltonian
of free ripplons [14] and Eq. (2), the equation for ξq can be
found in the following form
2 ( )
,2 = e ,g e i t
g r g
U n g − ⋅ξ + γ ξ +ω ξ −
ρ
g s
g g g
(3)
where ( )2 2
f= exp / 4q qU U q s− , a Debye-Waller factor
appears due to averaging over fast (high-frequency) modes,
2
fs is the contribution to the mean-square displacement
from fast modes, and gγ describes ripplon damping. The
right side of Eq. (3) represents the periodic electron pres-
sure acting on the free surface of liquid helium. If ( ) = 0ts ,
the first two terms of Eq. (3) are zero and the rest terms
yield the shape of a motionless dimple lattice.
380 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium
In the conventional theory of the WS coupling with
ripplons [3], the exponential function entering the right side
of Eq. (3) is expanded up to a linear (in ⋅g s ) term. This
term, representing the first harmonic of ω in the electron
pressure, is responsible for a resonant increase of the ripplon
field ξg when ,r gω→ω . In a nonlinear theory, the electron
pressure contains superharmonics of ω due to higher ex-
pansion terms. Thus, the right side of Eq. (3) contains terms
proportional to exp( )im t− ω [here we assume that
( ) exp( )t i t∝ − ωs and = 2, 3,...m ]. These superharmonic
terms of the electron pressure lead to a resonant increase of
ξg if ,r gmω→ω . In other words, we expect a resonant in-
crease of the medium response (the dimple effective mass
and kinetic friction) if the WS is driven at a frequency which
is close to a subharmonic of typical ripplon frequencies:
, /r g mω→ω . In the following, we will not expand the right
side of Eq. (3) assuming that the parameter ⋅g s can be large.
As noted in the Introduction, at low temperatures the
ripplon damping of pure 4 He is extremely small according
to Eq. (1). In the presence of impurity 3He atoms (dilute
3 4He He− solutions), the ripplon damping is substantially
increased. For the viscous regime, one can find
( ) ( )
1/2
2 2 22= / , = 2 1 1 ,q q q x x
x
η γ φ ωρ η φ − + −
ρ
(4)
where η is the viscosity of the solution. In this case, our
model should be improved by an additional frequency de-
pendent term 2 ( )g g−δ ω ξ in the left side of Eq. (3), where
2 2 2( ) = ( / )g gδ ω ω ζ ωρ η and
( )
1/2
2
2
4 1= 1 1 1 .
2
x x
x
ζ + + −
(5)
In the limit 2/ 1gωρ η , we have 2 2( ) / 2gδ ω →ω
which just increases inertia of the dimple lattice and
changes parameters of Eq. (3) by the numerical factor 2 / 3.
At low temperatures, impurity quasiparticles enter the
long mean-free-path regime, and the ripplon damping is
caused by their reflection from an uneven surface. Accord-
ing to Ref. 15, for specular reflection, we have
2 (qp)( ) ( ) /= , ( ) = (0)2 ,
2
s
q
F
T T Tq T F e
T
µ κ γ κ κ ρ
(6)
where
4
2 3
0
(0) = , ( ) = ,
4
F
x
p xF z z dx
e z
∞
κ
π +∫
(7)
FT and Fp are, respectively, the Fermi temperature and the
Fermi momentum of impurity atoms. The ripplon damping
defined by Eq. (6) has the same dependence on q as that ob-
tained for pure 4 He, still it has different dependence on tem-
perature: constqγ → if 0T → . Assuming that a fraction of
incoming quasiparticles ar can be reflected diffusively
from the the surface layer of 3He atoms which is not in-
volved in the horizontal motion of the dimple lattice [15],
we have ( )= (1 3 / 4) s
q a qrγ − γ . For impurity concentration
3 = 6.1%c and electron density 8= 1.4 10en ⋅ cm–2, the ratio
,/g r gγ ω can be increased up to about 0.4. Thus, any reason-
ably small ratio ,/g r gγ ω can be realized in an experiment
with the WS on the surface of 3He–4He mixtures.
A solution ( )g tξ of the model equation (3) can be found
trivially in an integral form [17], and the force acting on
the WS by surface dimples, defined as
int= /D ee f
H− ∂ ∂∑F r (here ... f denotes averaging
over fast modes), can be represented as
2
2= ( ),
ˆ
e gD
e g
n gU
t
N
−
ρω
∑ g
g
F g
(8)
where
[ ]{ }
0
ˆ ˆ( ) = sin( ) sin ( ) ( ) ,g
g gt e t t d
∞
−γ τ
ω ω τ ⋅ − − τ τ∫g g s s
(9)
and 2 2
,ˆ =g r g gω ω − γ . Equations (8) and (9) determine the
nonlinear response of the liquid helium surface to an oscil-
lating motion of the WS.
The equation of motion for an electron displacement
( )ts including the medium response force ( )D tF represents
a complicated nonlinear equation which is very difficult to
solve for a given driving electric field ( )tE . Remarkably,
an inverse problem of finding ( )tE for a given current can
be solved exactly. It should be noted that, in experiments
on WS transport, the driving electric field is often adjusted
to the current owing to electron redistribution which
screens external potential variations. Therefore, the regime
of a given current is realized at least partly. This conclu-
sion is supported by experimental observations [12,16] of
regions with / < 0dE du , where u is the WS velocity.
In the absence of a magnetic field, we shall consider a
simple periodic dependence 0( ) = sin( )t tωs s and assume
that the x-axis is directed along 0s . Then, the periodic
function ( ) ( )x
DF t can be represented as a Fourier series
( )
2
0 0
=1
= cos( ) sin( ) ,
x
D
e k k
e k
F
m s k t s k t
N
∞
− ν ω ω −χ ω ω ∑
(10)
where kν and kχ are two kinds of Fourier coefficients de-
fined by the well known rule. The factors 0em sω and
2
0em sω (here em is the free electron mass) are introduced
in order to make proper dimensions for kν and kχ . We
shall see that 1ν is an effective collision frequency due to
kinetic friction of surface dimples, while 1χ is the ratio of
the dimple effective mass Dm to the free electron mass
1( = / ).D em mχ
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4 381
Yu.P. Monarkha
According to Eq. (10), the force acting on the WS con-
tains the first and higher harmonics of ω. Since the other
terms of the equation of motion for an electron displace-
ment ( )ts are linear, only the first harmonics of ( ) ( )x
DF t are
important for obtaining the WS conductivity and the secu-
lar equation for coupled phonon-ripplon modes. Therefore,
we can ignore the higher harmonics of ω in Eq. (10)
(though there is no problem with calculation of kν and kχ
for an arbitrary ),k and omit the subscript 1 assuming
1 Dν ≡ ν and 1 =χ χ. Direct calculations yield
2
2
03= , , ,
ˆ ˆˆ
e g g
D x x
g ge g
n gU
g g s
m
γω
ν ω ωρω
∑
g
(11)
2
2
04= , , ,
ˆ ˆˆ
e g g
x x
g ge g
n gU
g g s
m
γω
χ ω ωρω
∑
g
(12)
where we introduced the following functions
1( , , ) = ( , , ),a Q a
a ν′ ′ ′ ′ω γ ω γ
′ω
(13)
1
0
( , , ) = 2 sin( ) cos 2 sin ,
2 2
x x xQ a x e J a dx
∞
′−γ
ν
′ ′ω ω ′ ′ω γ
∫
(14)
2
1( , , ) = ( , , ),
( )
Ma Q a
a
′ ′ ′ ′ω γ − ω γ
′ω
(15)
1
0
( , , ) = 2 sin( ) sin 2 sin ,
2 2
x
M
x xQ a x e J a dx
∞
′−γ ′ ′ω ω ′ ′ω γ
∫
(16)
and 1( )J z is the Bessel function. Properties of the function
( , , )Q aν ′ ′ω γ were partly investigated in the low-frequency
limit [17] where a broadening of the Bragg-Cherenkov
threshold occurred. Still, the nonlinear WS mobility and
conductivity were not investigated because the dimple
mass-function ( , , )a ′ ′ω γ was not studied, and super-
harmonic resonances of the ripplon field at ,r gmω→ω
were not disclosed.
It should be noted that even first harmonics of ( ) ( )x
DF t
in Eq. (10) defined by Dν and χ (or equivalently, by Qν
and MQ ) contain contributions from higher harmonics of
the electron pressure entering the right side of Eq. (3). This
can be seen from dependencies of ( , , )a ′ ′ω γ and
( , , )a ′ ′ω γ on the dimensionless frequency ˆ= / g′ω ω ω
calculated for different values of the nonlinear parameter
0= xa g s . Fig. 1 indicates that ( , , )a ′ ′ω γ has sharp max-
ima at = 1/ m′ω ( ˆ= /g mω ω ) whose intensities depend on
a in a non-monotonic way. As the parameter a increases,
the distribution of maxima shifts strongly into the low-
frequency range.
The function ( , , )a ′ ′ω γ shown in Fig. 2 sharply
changes its sign near points = 1/' mω . In the limiting case
ˆ= / 0g g′γ γ ω → , the frequency dependence of ( , ,0)a ′ω
can be fitted by a simple function [9]
2 2
=1
( , ,0) = ,
1/ ( )
m
m
S
a
m
∞
′ω
′− ω
∑ (17)
where the weight of a singularity mS is a non-monotonic
function of a. In the linear regime, only the first term in
the sum is important: 1 = 1,S and = 0mS if > 1m . With an
increase of the nonlinear parameter a, the distribution of
mS is shifted in the range of large m. It should be noted
that a strong increase of a reduces 1S and changes its sign.
Thus, in the nonlinear regime, the both functions
( , , )a ′ ′ω γ and ( , , )a ′ ′ω γ are affected strongly by
Fig. 1. (Color online) The dimensionless collision frequency func-
tion ( , , )a ′ ′ω γ vs. ,ˆ= / r g′ω ω ω calculated for four values of the
parameter 0= xa g s .
Fig. 2. (Color online) The dimensionless dimple-mass function
( , , )a ′ ′ω γ vs. ,ˆ= / r g′ω ω ω calculated for four values of the
parameter 0= .xa g s
382 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium
superharmonic resonances of the ripplon field. The effec-
tive collision frequency Dν and the dimple effective mass
=D em mχ change sharply when the WS is driven by a
frequency ω which is close to subharmonics of the typical
ripplon frequencies ,= /r g mω ω .
The amplitude of the WS velocity 0 0=u sω . Therefore,
the nonlinear parameter 0= xa g s can be represented as
= /a u′ ′ω , where 0 ,= / g xu u u′ and , ˆ= /g x g xu gω . This
allows us to calculate and as functions of the di-
mensionless velocity u′ near the singular points = 1/ m′ω .
For example, consider = 1/ 3′ω and a close frequency
= 0.331′ω . Under these conditions, the function
( / , , )u′ ′ ′ ′ω ω γ is shown in Fig. 3 for three values of ′γ . It
is remarkable that is large far below the Bragg-
Cherenkov threshold condition ( = 1)u′ where a steady mo-
tion of the WS starts emitting ripplons [18]. Moreover,
curves calculated for = 1/ 3′ω and for the close frequency
have different evolutions of their maxima with lowering
the damping parameter. The maximum of calculated
for = 1/ 3′ω monotonously increases with decreasing 'γ
(red dashed and dotted curves), while the maximum calcu-
lated for a very close frequency = 0.331′ω increases only
in a short range of ′γ (blue dashed curve) and decreases at
sufficiently low ′γ (blue dotted curve). A similar behavior
of is disclosed for a frequency slightly larger than 1/ 3
and in the vicinity of other singular points 1/ m.
The dimensionless mass function ( / , , )u′ ′ ′ ′ω ω γ is
shown in Fig. 4 for = 1/ 3′ω and two close frequencies.
Calculations were performed using three values of the
damping parameter ′γ . For the singular point = 1/ 3′ω
(red), the as a function of u′ is practically independent
of the damping parameter. It is remarkable that already
very small changes of the driving frequency = 0.32′ω
(blue) and = 0.34′ω (green) lead to strong (even qualita-
tive) changes in the dependence of on the WS velocity
u′, and near extrema the results of calculations become
dependent on the damping parameter. A similar behavior
of is found also for 1/ 2′ω → and near lower singular
points 1/ m.
The results shown in Figs. 3 and 4 lead to the following
important conclusions. In the low frequency range 1′ω ,
a frequency ′ω can be rather close to a singular point 1/ .m
Then, a small variation in the electron density sn affects
ˆ= / g′ω ω ω and can cause large (even giant) changes in the
nonlinear conductivity of the WS, which may lead to a
mistaken conclusion that data are not reproducible. In an
experiment employing the transmission line model, there
are long wave-length density variations along the electron
pool. Therefore, according to Fig. 3, the area of the pool
which approaches the condition ˆ/ 1/g mω ω → has huge
friction causing a dynamic "pining" of the WS to the liquid
substrate. Of course, this pinning is limited by the wave-
length of the density variation. Moreover, as we shall see,
in the nonlinear regime, data obtained employing different
methods can bring different mobility results.
3. Nonlinear mobility
In the absence of a magnetic field, the WS mobility and
conductivity can be found from the force balance equation
( ) ( )
= ( )
x
D
e e e
e
F t
m u eE t m u
N
− + − ν (18)
where 0= cos( ),u s tω ω and eν is the electron collision
frequency due to scattering with thermally excited
ripplons. For the given current 0= cosx sj en s t− ω ω , this
equation determines the electric field ( )E t whose first
harmonic (usually measured in an experiment) can be gen-
erally written as 0( ) = sin( )E t E tω +β . The expression for
( ) ( )x
DF t obtained above yields
Fig. 3. (Color online) The ( / , , )u′ ′ ′ ′ω ω γ vs. dimensionless
velocity 0 ,= / g xu u u′ calculated for = 1 / 3′ω (red curves) and
= 0.331′ω (blue curves), and for three values of ′γ : 0.01 (solid),
0.0075 (dashed), and 0.003 (dotted).
Fig. 4. (Color online) The ( / , , )u′ ′ ′ ′ω ω γ vs. dimensionless
velocity 0 ,= / g xu u u′ calculated for three values of ′ω and for
three values of ′γ : 0.01 (solid), 0.0075 (dashed), and 0.003 (dotted).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4 383
Yu.P. Monarkha
2 2 2
sin = ν
β −
ν + ω
(19)
and the relationship between amplitudes
2 2 4 2
0 0 0( ) = ,
e
es s E
m
ν ω + ω (20)
where 0 0( , ) = 1 ( , )u uω + χ ω represents dimensionless
effective mass of an electron bound to a surface dimple,
0 0=u sω and = D eν ν + ν .
To obtain the conductivity expression we shall use the
change of the time variable =
2
t t π′ω +β ω + which trans-
forms the ac electric field into the conventional form
0( ) = cos( )E t E t′ ′ω . In this case, using Eqs. (19) and (20)
the WS current can be transformed into the usual two-
component form
0 0= cos( )Re sin( )Im ,j E t E t′ ′ω σ + ω σ (21)
where the first component, oscillating in phase with ( ),E t′
usually determines the real part of conductivity, while the
out-of-phase component determines the quantity which is
called the imaginary part of conductivity:
2
2 2 2Re = ,e
e
e n
m
ν
σ
ν + ω
2
2 2 2Im = .e
e
e n
m
ω
σ
ν + ω
(22)
The WS mobility is defined as = Re / eenµ σ .
The Eq. (20) allows obtaining the velocity-field charac-
teristic for the amplitudes 0E and 0u . Contrary to the case
of pure 3He, here this characteristic depends also on the
effective mass function 0( , )uω and eν . In order to calcu-
late eν , we use a high temperature approximation for the
dynamic structure factor of the 2D WS where the average
kinetic energy of an electron in the WS state replaces the
temperature [19]. For the conditions of the experiment [8],
typical 0 0u E− characteristics are shown in Fig. 5, where
, 11
= /R r gu gω . Assuming that our results can be applied (at
least qualitatively) to the regime of a given field, we con-
clude that at a certain threshold value of 0E the balance of
forces is broken and there should be a bistability jump (tran-
sition) to a high velocity branch. In the regime of a given
current, the region with 0 0/ < 0u E∂ ∂ can be observed.
Using 0 0u E− characteristics of Fig. 5, the Eq. (22) and
the relationship = Re / senµ σ , the nonlinear mobility of the
WS 0( )Eµ is calculated and shown in Fig. 6 for two values
of ω. Before the bistability jump shown by an arrow, the
calculated dependence 0( )Eµ is in a qualitative agreement
with experimental observation [8]. It should be noted that
even a small region with negative 0/ E∂µ ∂ is noticeable in
the experimental data set [8]. The figure indicates that the
threshold field thE and mobility values of small 0E depend
strongly on ω. After the jump, the mobility curve was cal-
culated assuming that the WS state survives the transition
to a high velocity branch and eν is independent of 0E . Ex-
perimental data indicate that after the jump which occurs at
0 10 mV/cmE ≈ , the electron mobility steadily increases
with 0E similar to the mobility of a 2D electron gas, and
most probably the WS is broken.
The experimental method [20] allows measuring the
quantity 2/e em n eω which is proportional to the effective
mass function . Under the experimental conditions [20],
theoretical calculations using the expression for ob-
tained here are shown in Fig. 7 for three typical values of
ω. As expected, at a fixed electron density ( =en
8 212.6 10 cm )−= ⋅ the peculiarity of the function 0( )E
Fig. 5. (Color online) Velocity-field characteristics calculated for
two typical frequencies = / 2 = 2 MHzf ω π (solid) and 3 MHz
(dashed). The damping parameter = 0.005′γ . The other condi-
tions are the same as in the experiment [8].
Fig. 6. (Color online) Mobility of the WS vs the amplitude of the
driving electric field. Conditions are the same as in Fig. 5.
384 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium
strongly depends on the driving frequency ω. For example,
the bistability range, which is prominent at = / 2 =f ω π
5 MHz= (dotted) and noticeable at = 3 MHzf (solid), dis-
appears already at = 2 MHzf (dashed). The observed [20]
shape of 2/e em n eω is in agreement with the solid curve of
Fig. 7. The position of the maxima is about 2.3 times high-
er than the value of the linear regime which also agrees
with experimental observations.
4. Excitation of coupled phonon-ripplon modes
If longitudinal phonons of the WS with a wavevector k
defined by the geometry of the cell can be exited in an ex-
periment, the equation of motion for electron displace-
ments acquires an additional restoring force. Considering
only a long wave-length excitation 1( ),k g the conduc-
tivity of the WS can be represented as
2
22 2 2 2
Re ( ) = ,
( ) /
e
e l
e n
k
m k
ν
σ
ν + −Ω ω ω
(23)
where ( )l kΩ is the spectrum of longitudinal phonons of
the WS. Here the nonlinear effect is included in definitions
of Dν ( = )D eν ν + ν and . In the nonlinear regime, the
effective mass function has new singular points near
,= /r g mω ω , and, therefore, one can expect new resonanc-
es of Re ( )kσ when
2 2( ) / = 0.l k−Ω ω (24)
The Eq. (24) is a secular equation for coupled phonon-
ripplon modes. Solutions of this equation were analyzed in
Ref. 9. Here we investigate the frequency dependence of
Reσ proportional to the energy absorbtion in a nonlinear
regime.
The spectrum of Re ( , )kσ ω is shown in Fig. 8 for dif-
ferent values of the nonlinear parameter 1 1 0=a g s and for
two nearest wavevectors 1k and 2k corresponding to the
conditions of the experiment [8]. At a low excitation level
1( = 0.1)a we have two pronounced maxima corresponding
to conventional phonon-ripplon coupled modes. At a high-
er excitation level 1( = 0.5)a these maxima are shifted in
opposite (!) directions because they were initially placed at
the opposite sides of the singular point 1/ = 1/ 2ω ω
1 , 1
( = )r gω ω where ( )′ω rapidly changes as an odd func-
tion (see Fig. 2). For even higher excitation levels 1( = 1a
and 1 = 2),a new low-frequency resonances appear due to
phonon-ripplon coupling near subharmonics of ,r gω .
It should be noted that in a real experiment the inhomoge-
neous broadening could affect the new resonances espe-
cially for 2=k k . Anyway, the theoretical results presented
in Fig. 8 explain why the positions of conventional pho-
non-ripplon resonances corresponding to 1k and 2k shift in
opposite directions with an increase of the excitation pow-
er. The appearance of new low-frequency resonances is
also in agreement with experimental observations [8].
5. Nonlinear magnetoconductivity
Magnetoconductivity of the WS usually is measured
under the condition that the Hall velocity is much higher
than the drift velocity along the direction of the electric
field (here the x-axis is fixed to be parallel to the electric
field). In the presence of a strong magnetic field B directed
perpendicular to the surface, assuming ,= sin( )y y ss s tω ,
we have to consider , ,= sin( ) cos( )x x s x cs s t s tω + ω to sat-
isfy equations of motion for electron displacements.
Fig. 7. (Color online) The quantity 2/e em n eω vs. the amplitude
of the driving electric field calculated for three frequencies of
( )E t . Other conditions are the same as in the experiment [20].
Fig. 8. (Color online) Frequency dependence of Re ( , )kσ ω calcu-
lated for two lowest wavevectors 1k (red curves) and 2k (blue
curves). For the sake of clarity, each curve of a higher excitation
level is shifted up by 5 17 10− −⋅ Ω with respect to the zero level of
the previous curve. The damping parameter = 0.01′γ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4 385
Yu.P. Monarkha
The amplitude of the Hall motion ,y ss is much larger than
,x ss and ,x cs . Contrary to the mobility calculations given
above, now we should describe two components of the
force acting on the WS by surface dimples: ( )x
DF and ( )y
DF .
According to Eqs. (8) and (9), ( )x
DF and ( )y
DF contain fac-
tors xg and yg , respectively, and, therefore, the function
( )tg should be treated differently when expanding it in
small x xg s . We shall use superscripts ( )x and ( )y to dis-
tinguish these two cases. Considering the y component of
the force, we can neglect xs as compared to ys and use the
following approximation ( )( ) ( )yt tg g with
{ }( )
0
ˆ ˆ= sin( ) sin ( ) ( ) ,gy
g g y y ye g s t s t d
∞
−γ τ ω ω τ − − τ τ ∫g
(25)
because the integral is an odd function of yg .
The Eq. (25) can be calculated in the way similar to that
described above. For major terms of the Fourier series, we
have
( )
2D
s, s,= cos( ) sin( ) ,
y
e y y y y
e
F
m s t s t
N
− ν ω ω −χ ω ω (26)
where
2
2
s,3
ˆ ˆ= ( , / , / ),
ˆ
e g
p p y y g g g
e g
n gU
g g s
m
ν ω ω γ ω
ρω
∑
g
(27)
2
2
s,4
ˆ ˆ= ( , / , / ),
ˆ
e g
p p y y g g g
e g
n gU
g g s
m
χ ω ω γ ω
ρω
∑
g
(28)
and = ,p x y is a subscript. As compared to the mobility
treatment, here we have a different parameter s,= y ya g s
describing an excitation level.
Considering the ( )x
DF component, one cannot use the
approximation of Eq. (25) because the integral is an even
function of xg , and this approximation yields ( ) = 0x
DF .
Therefore, in this case, ( )tg should be expanded up to
linear terms of x xg s :
[ ]( )
0
ˆ ˆsin( ) ( ) ( )gx
g g x x xe g s t s t
∞
−γ τω ω τ − − τ ×∫g
{ }cos ( ) ( ) .y y yg s t s t d × − − τ τ (29)
Here the term independent of x xg s [similar to that of
Eq. (25)] is neglected because it gives zero result after
summation over all g . Formally, we can represent
( ) ( )( ) ( ) ( )y xt t t+g g g , where the first term is im-
portant only for ( )y
DF while the second term is important
only for ( )x
DF .
Inserting , ,= sin( ) cos( )x s x c xs s t s tω + ω and =ys
, sin( )y ss t= ω into Eq. (29) gives two different terms. The
term proportional ,s xs can be represented as a derivative
with respect to a, while the term proportional to ,c xs can
be represented as a time derivative. For the chosen de-
pendence ( )xs t , the WS velocity component along the x-
axis is a sum ( ) ( )x xu t u t′+ , where ,( ) = sin( )x x cu t s t−ω ω
and ,( ) = cos( )x x su t s t′ ω ω (here and below the stroke '
means that a quantity originates from the sin-term of ).xs
An analysis of the Fourier coefficients indicates that now
( )x
DF contains additional friction and inertia components
[ ]
( )
=
x
D
e x x x x x x x x
e
F
m u u u u
N
′ ′ ′ ′− ν + ν + χ + χ (30)
where xν and xχ are described by Eqs.(27) and (28)),
2
2
,3= , , ,
ˆ ˆˆ
e g g
x x d y y s
g ge g
n gU
g g s
m
γω′ν ω ωρω
∑
g
(31)
2
2
,4= , , ,
ˆ ˆˆ
e g g
x x d y y s
g ge g
n gU
g g s
m
γω′χ ω ωρω
∑
g
(32)
and
1( , , ) = ( , , )d a Q a
a ν
∂
ω γ ω γ
ω ∂
, (33)
2
1( , , ) = ( , , ).d Ma Q a
a
∂
ω γ − ω γ
∂ω
(34)
It should be noted that ( , , )d a ω γ contains a derivative
( , , ) /Q a aν∂ ω γ ∂ which at a certain condition can become
negative causing instability and melting of the WS.
In order to proceed with conductivity calculations, we
have to introduce new dimensionless inertia functions
= 1p p+ χ and = 1x x′ ′+ χ . Then, the balance of ampli-
tudes is described by
( ) ( )22 2 2 0
,
= .c
x y x y x y c x y
e s y
e E
m s
ω
′ ′ω ν + ν ω + ω −ω −ν ν
ω
(35)
In the limit of low ω and ν, this equation transforms into
the usual Hall relationship , 0= /s ys cE Bω . The important
point is that new functions x′ and x′ν are much larger than
x and xν respectively because d and d contain the
derivative / a∂ ∂ instead of 1/ a entering and , while
the parameter ,= y y sa g s is very large due to a high Hall
velocity (the first maximum of occurs at 50).a ≈
A procedure similar to that used for obtaining Eq. (22)
yields the following equation for the real part of
magnetoconductivity
2
= e
xx
e
e n
m
σ ×
2 2 2 2
2 2 2 2 2
( ) ( )
.
( ) ( )
y c x y x y y y x x
c x y x y x y y x
′ ′ν ω + ν ν + ν ω + ν ω −
×
′ ′ω + ν ν −ω +ω ν + ν
(36)
386 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 4
Nonlinear dynamics of a two-dimensional Wigner solid on superfluid helium
To include the contribution from direct scattering of elec-
trons by thermally excited ripplons one should add eν to
each pν and x′ν . This equation differs substantially from the
conventional Drude form. The usual magnetoconductivity
equation follows from Eq. (36) if we consider the low fre-
quency limit and assume that = =y x x′ν ν ν . As noted
above, this simple approximation cannot be used here be-
cause x x′ and x x′ν ν if , 1y y sg s .
In the conventional magnetoconductivity treatment based
on the Drude equation, there is a strict relationship between
xxσ and = / ee mµ ν , and one can expect a certain accord-
ance of magnetoconductivity data with mobility data. The
appearance of new quantities x′ and x′ν which differ
greatly from and Dν entering Eq. (22) means that such
an accordance is impossible for the nonlinear WS transport
over superfluid 4He.
Typical dependencies of 1
xx
−σ on the WS velocity
,= y su sω normalized to a ripplon velocity , 11
= /R r gu gω
are shown in Fig. 9 for two electron densities. The condi-
tions of the system were chosen to be very close to those of
the experiment [5,6]. It is important that 1
xx
−σ rapidly falls
down before the Bragg-Cherenkov threshold, approaching
a minimum which is below the experimental data. After
the minimum the dependence 1( )xx u−σ is similar to the de-
pendence 1
in( )xx V−σ observed in the experiment (assuming
that yu is approximately proportional to in )V : 1( )xx u−σ in-
creases, attains a maximum and starts falling down again.
The new fall ends by sharp changes of 1( )xx u−σ leading to
instability because xxσ becomes negative when u exceeds a
certain threshold value thu which is larger than the Bragg-
Cherenkov threshold. Before attaining negative values,
1( )xx u−σ demonstrates a vertical jump, which also agrees with
experimental observations. The condition < 0xxσ means
that any density fluctuation grows [21] and, therefore, the
long-range order is expected to be destroyed. The threshold
value thu changes in a non-monotonous way even for very
small variations of en , and in the interval
8 2 8 21 10 cm < < 1.3 10 cmen− −⋅ ⋅ the ratio th 1/u u can sub-
stantially exceed the values indicated in Fig. 9.
6. Conclusions
In this work, we theoretically investigated the nonlinear
responce of the liquid helium surface to an oscillating mo-
tion of the 2D Wigner solid on the surface of superfluid
helium. The response force applied to the WS consists of
two different terms representing the effective mass of sur-
face dimples and the kinetic friction. The electron pressure
acting on the free surface is a nonlinear function of the
electron current which induces superharmonic resonances
of the ripplon field and the response force. As a result, the
dimple mass and the kinetic friction change sharply when
the WS is driven with a frequency which is close to sub-
harmonics of the frequency of a ripplon whose wave vector
coincides with a reciprocal-lattice vector. In the limiting
case of zero ripplon damping, the dimple mass and the
effective collision frequency as functions of frequency
have an infinite number of singular points. Therefore, any
low driving frequency is close to a singular point, which
means that WS transport over superfluid 4He is singular and
a small variation in the electron density can cause large
changes (even qualitative) in nonlinear transport properties
of the WS.
We found that our calculations of the nonlinear WS mo-
bility and the dimensionless mass function are in good quali-
tative accordance with experimental observations [8,20]
which previously had no theoretical explanations. The anal-
ysis of nonlinear phonon-ripplon coupling given in this work
explain observation of new low-frequency resonances [8]
and the strange behavior of conventional phonon-ripplon
coupled modes with an increase of the excitation signal.
Considering magnetotransport of the 2D WS in an oscil-
lating electric field, we found that the nonlinear magneto-
conductivity tensor cannot be reduced to the conventional
Drude form, and, therefore, an accordance between nonline-
ar mobility and magnetoconductivity data is impossible. In
the direction of the driving electric field, the dimple mass
and effective collision frequency acquire huge values ex-
ceeding greatly those of the mobility treatment. Moreover,
at a sufficiently high electron velocity in the perpendicular
direction, the magnetoconductivity becomes negative which
causes instability and fluctuational melting of the WS. The
dependence of nonlinear magnetoconductivity on the veloci-
ty amplitude obtained in this work can be used also as an
alternative explanation of the remarkable dynamic transition
observed long ago [5,6].
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1. Introduction
2. Model description
3. Nonlinear mobility
4. Excitation of coupled phonon-ripplon modes
5. Nonlinear magnetoconductivity
6. Conclusions
|