Table Of ContentRipplonic Polaron in the Quasi-One-Dimensional Electron System over Liquid Helium
Sviatoslav S. Sokolov1,2 and Nelson Studart1
1Departamento de F´ısica, Universidade Federal de S˜ao Carlos,13565-905, S˜ao Carlos, S˜ao Paulo, Brazil
2 B. I. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 310164,
Kharkov, Ukraine
(February 1, 2008)
Westudytheenergeticsandtransportpropertiesofpolaronsinthequasi-onedimensionalelectron
system over the liquid helium surface. The localization lengths and the energy of the ground and
excited states are calculated within the hydrodynamic model of the polaron. The polaron bound
energyisfoundandthetemperatureforthepolaronformationisestimatedbelow0.1K.Weexamine
the possibility of observation of the polarons by measuring both the spectroscopic transitions and
the mobility.
9
9
9
1
An electrontogether with its self-induced polarization quencyω0 = eE⊥/mR,whereeandmaretheelectron
n inamediumformsaquasiparticlewhichhasbeennamed charge and mass, respectively, E is the holding electric
a p ⊥
polaron. Besidesitsimportanceasastandardtheoretical field along the z axis, and R is the curvature radius of
J
model of a fermionic particle coupled to a boson scalar the liquid in the channel.
3
1 field, the polaron has been observed in some physical The wave function ψ(x,y) for the electron trapped to
systems. In particular, the search of polaronic states, the dimple satisfies the 2D Schr¨odinger equation
1 predicted theoretically long time ago, [1,2] for surface
2
v electrons levitated over liquid helium, has been the sub- ¯h 2 m 2 2
4 ψ + ω0y ψ+eE⊥ξ(x,y)ψ =εψ, (1)
jectoflargeamountofexperimentalworkduringthelast − 2m ∇ 2
2
decades. [3,4] (cid:0) (cid:1)
1 where the deformation profile of the surface ξ(x,y) de-
1 Theoreticalapproachestoinvestigatethepropertiesof
pends, in turn, on the electron pressure on the surface
0 thesurfacepolaronoverheliumarebasedonthedescrip-
eE ψ(x,y)2. Thetotalenergyofthepolaron,expressed
9 tionofthedimplestate(electronplusthedeformationof ⊥| |
9 asasumoftheelectronenergyandtheenergyduetothe
the helium surface due to the pressing field) within a
/ surface deformation, is
t self-consistent hydrodynamic framework or in terms of
a
m the concept of a Fr¨ohlich polaron (single electron cou- W =ε+ σ (∇ξ)2+k2ξ2 dxdy, (2)
- pled to ripplons). Despite the great difference between 2 c
d the methods, both final results for the structure of the Z Z h i
n ripplonic polaron show fair qualitative and quantitative where k = ρg/σ is the capillary constant and σ and
o c
agreement. [5] ρ are the surface tension coefficient and the density of
c p
: Quite recently, quasi-one-dimensional (Q1D) electron liquid helium respectively. The electron energy can be
v
systemsonthesurfaceofliquidheliumwererealized.[6,7] written as
i
X In such a Q1D system, the electron motion is restricted,
2
r in addition to the quantum well in the direction normal ε= ¯h (∇ψ)2 + mω2y2ψ2
a to the liquid surface, by a lateral confinement, which is {2m 2 0
providedby either geometric or electrostatic means, and Z+eZE ξ(x,hy)ψ2 dxidy, (3)
⊥
}
canbe modelled by a parabolicpotential. In suchcondi-
tions, the formation of an asymmetric ripplonic polaron where we supposed that the characteristic length, which
may become favorable. givesthedecreaseofthewavefunctionalongthey direc-
In this work,we address the possibility of polaronfor- tion,issignificantlysmallerthanthecurvatureradiusR.
mation in the Q1D electron system within the hydrody- The minimization of the total energy W with respect to
namic approach [5,8,9] which allows us to calculate an- ξ leads to the mechanical equilibrium equation
alytically the main properties of the polaron state. Fur-
2 2 2
thermore, this model was employed by Tress et al. to σ(∇ ξ−kcξ)=eE⊥ψ . (4)
interpret the magnetoconductivity data which led them
Equations (1) and (4) form a nonlinear system of equa-
to report the observation of polaronic effect in two-
tions which must be solvedself-consistently,as wasdone
dimensional (2D) electron system over helium films. [4]
numerically in Ref. [9] for the 2D symmetric polaron.
In the Q1D model, described in Refs. [10,11], the sur-
Sincethesecalculationsarecumbersomeintheasymmet-
face electrons are confined by a lateral potential well
U0 = mω02y/2 in the y direction with characteristic fre- ric situation, we prefer to use a variational method and
1
taketrialfunctionslikethe2Dharmonic-oscillatoreigen- energy dissipation, and leads to a finite value of the po-
functionswithlocalizationlengthsl andl asthe varia- laronmobility.[1,5,8,9]Inordertoevaluatethemobility,
x x
tionalparameters.[8]The resultfor the polaronground- weemploytheenergybalanceequationeEkv0 =dρE/dt,
state energy reads as where v0 is the liquid velocity at infinity, and ρE is the
energy density dissipated. The function dρ /dt is ob-
(eE⊥)2 2√2 tained in a straightforward way by finding Ethe normal
W0 ln
≃− 4πσ √γkc(lx+ly) velocity field induced by the polaron from the solution
2 of the Navier-Stokesequation. [1,8] If the driving field is
¯h 1 1 m
2 2
+4m l2 + l2 + 4 ω0ly, (5) applied along the x direction, the mobility is given by
(cid:18) x y(cid:19)
−1
where L2 = (2πσ¯h2)/m(eE )2 and γ is the Euler- e
F ⊥ µ = kk2 ξ 2 , (11)
Mascheroniconstant. Thevariationalparametersareob- x 2ηS x| ~k|
tained from the condition that the polaron energy is a X~k
minimum at these values, which leads to the following
whereη istheheliumviscosity,S istheareaoftheliquid
system of equations:
surface, and ξ is the Fourier transform of the dimple
~k
1 1 profile. The summationinEq.(11)canbe performedan-
l3 − L2(l +l ) =0, (6) alyticallyresultinginacumbersomefunctionofcomplete
x F x y ellipticintegralsdependingonl andl .However,wecan
x y
and simplify significantly the calculation by considering the
limitingcasel l ,whichcorrespondstoholdingfields
1 1 1 =0, (7) of interest 1 x3≫ (y106 V/m). In this case, the polaron
l4 − L2l (l +l ) − L4 − ×
y F y x y 0 mobility can be written as
whereL20 =h¯/mω0.IfLF L0,thesolutionsofEqs. (6) π3/2σ2l
≪ x
and (7) are lx = ly ≃ √2LF. For LF ≫ L0, we obtain µx = √2ηeE2 . (12)
lx LF and ly L0. Note that for realistic values of ⊥
hol≃dingfieldsLF≃ 10−1 µmandL0 10−2 µmandthe If the field is applied along y, the mobility is such
∼ ∼
valuesofthelocalizationlengthsaresignificantlysmaller
that (µ /µ ) = ln(l /l ). In the symmetric case,
x y x y
thanthe curvatureradiusR=1 10µm. Forthelowest
l = l = L , we obtain the well-known result µ =
− x y F
excited states, we find √8πσ2L /ηeE2.
F ⊥
Giventhe recentexperimentalprogress,[4,6,7]wenow
(eE )2 2√2 d (2d +3d )
W10 ≃− 4π⊥σ "ln√γkc(dx+dy) − x4(dxx+dy)2y # fdeieclatceomthpeeellxepdetroimeexnatmalinoebstehrevasitgionnatoufrtehsethpaotlasrhoonuilndtihne-
2 Q1D electron system on the surface of liquid helium.
¯h 3 1 m
2 2
+4m d2 + d2 + 4 ω0dy. (8) First, we calculate the polaronbinding energy,which we
(cid:18) x y(cid:19) define as the ground-state polaron energy W0 minus the
The minimization of the energy with respect to the pa- energy of the free electron in the Q1D channel. At low
rameters d and d leads to temperatures, T ¯hω0, the electron energy in the low-
x y ≪
est subband along the y direction is h¯ω0/2. As we have
3 1 d (d +3d ) seen previously, the values of the variationallocalization
y x y
d3x − L2F(dx+dy)(cid:20)1+ 4(dx+dy)2 (cid:21)=0 (9) lengths are lx ≃ LF and ly ≃ L0 for E⊥ = 1−3×(106
V/m). In this case, the binding energy is given by
and
(eE )2 2√2 ¯h2
1 1 d (d +3d ) 1 E ⊥ ln + , (13)
d4 − L2d (d +d ) 1− 4x(dx+d )y2 − L4 =0. (10) b ≃− 4πσ √γkcLF 4mL2F
y F y x y (cid:20) x y (cid:21) 0
which is slightly smaller than the binding energy of the
In the limit of LF L0, we can disregardthe term L−04 2D symmetric polaron. [8,9] We estimate E 0.03 K
≪ b
in Eq. (10) and the roots of the system aredx 2.12LF and E 0.3 K for holding electric fields o≃f 1−05 V/m
≃ b
and dy 1.73LF. In the opposite limit, L0 LF, we and 3 ≃10−5 V/m, respectively, in the case of a 4He sub-
≃ ≪
obtain dx √3LF and dy L0. strate.×One can conclude that the energetic conditions
≃ ≃
We now investigate the transport properties of the
forthe formationofthepolaronintheQ1Delectronsys-
Q1Dpolaron. Whenadrivingelectric fieldE isapplied
k tem on liquid helium surface are near the same as in the
along the plane, the surface deformation moves together
caseofthe2Dsymmetricpolarononaflatheliumsurface
with the trapped electron inducing a field of hydrody- and the Q1D polaron should be observed at T < E .
b
namic velocities in the liquid, which is accompanied by ∼| |
2
Other possible way to observe the Q1D polaron is polaron formation indicate that the polaron in the Q1D
by measuring the spectroscopic transitions between the channelcanbeobservedataccessibletemperatures. One
ground and excited states of the single electron in the hope that the asymmetric polaron can be observed at
dimple. By determining numerically the electron ener- low temperatures, probably at T < 0.1 K, and mod-
gies in the ground and excited states we obtain that the ernexperimentalmethodsforachievinglowandultralow
frequency ω10 of the spectroscopic transition between temperatures would offer the possibility to observe the
the ground-state and the first excited-state varies from polaronintheQ1Dchannelontheliquidheliumsurface.
9.8 108 Hz, for E = 105 V/m, to 8.0 109 Hz for ThisworkwaspartiallysponsoredbytheFundac¸a˜ode
⊥
E ×= 3 105 V/m. These values were obt×ained for the Amparo `a Pesquisa do Estado de Sa˜o Paulo (FAPESP)
⊥
× 4
Q1D system over He. We remark that, in this range of andtheConselhoNacionaldeDesenvolvimentoCientifico
electricfields,thecharacteristicfrequencyω0 variesfrom e Tecnol´ogico (CNPq), Brazil.
5.9 1010 to 1.0 1011 Hz. As a consequence, the fre-
× ×
quency ω10 is one order of magnitude smaller than the
transitionfrequencybetweensubbandsofthey -confine-
ment. This circumstance can be favorablein experimen-
tal efforts to observe the polaron state by measuring the
frequency of spectroscopic transitions.
[1] V. B. Shikin and Yu.P. Monarkha, Zh. Eksp. Teor. Fiz.
Another possibility is measuring directly the mobility
65, 751 (1973) [Sov. Phys. JETP 38, 373 (1973)]; Yu.
as in Ref. [4]. Our results point out that the mobility of
P. Monarkha and V. B. Shikin, Fiz. Nizk. Temp. 8, 563
the Q1Dpolaronis 40% smaller than the mobility of 2D (1982) [Sov. J. Low Temp. Phys.8, 279, (1982)].
polaronforthesameholdingfield. OurvalueoftheQ1D [2] S. Jackson and P.M. Platzman, Phys. Rev. B 24, 499
polaron mobility, for T =0.1 K and E⊥ =3 105 V/m, (1981); O. Hip´olito, G. A. Farias, and N. Studart, Surf.
×
is threeordersofmagnitudesmallerthanthe mobilityof Sci. 113, 394 (1982); F. M. Peeters and S. A. Jackson,
2D electrons over a flat helium surface and of electrons Phys. Rev.B 31, 7098 (1985).
moving freely along a channel, 102 m2/Vs, [11],which [3] E.Andrei,Phys.Rev.Lett.52,1449(1984);F.F.Mende,
∼
may allow to distinguish them easily. Yu.Z.Kovdrya,andV.A.Nikolaenko,Fiz.Nizk.Temp.
While the above estimates show that our proposed 11,646(1985)[Sov.J.LowTemp.Phys.11,355,(1985)];
X.L.Hu,Y,Carmi,andA.J.Dahm,J.LowTemp.Phys.
mechanismforpolaronformationinQ1Dsystemsonhe-
89, 625 (1992).
lium should be compatible with accessible experiments,
[4] O. Tress, Yu.P.Monarkha, F. C. Penning, H.Bluyssen,
other comments are needed to corroborate or refute our
and P. Wyder, Phys. Rev. Lett. 77, 2511 (1996); O.
predictions. Ofparticularinterestisthepossibilitytouse
Tress, F. C. Penning, G. Mistura, E. Teske, and P.
liquid 3He instead 4He as the substrate for the confined Wyder, Physica B 249-251, 644 (1998).
Q1Delectronsystem.[12]Recallthatfor3He,whosesur- [5] For a review, see N. Studart and S. S. Sokolov, in Two-
face tension coefficient is more than twice smaller than dimensional electron systems in helium and other sub-
thatof4He, the binding energy Eb canbe substantially strates, editedbyE.Y.Andrei(Kluwer,Dordrecht1997).
| |
larger. Also the spectroscopic transition frequencies are [6] Yu. Z. Kovdrya, V. N. Nikolaenko, H. Yayama, A.
twice larger for 3He in comparison with 4He. Further- Tomokiyo, O. I. Kirichek, and I. B. Berkutov, J. Low
more,the descriptionofthe polaroninterms ofa hydro- Temp.Phys.110,191(1998);Yu.Z.Kovdrya,V.A.Niko-
dynamicviscousmodelmaybe doubtfulforpure4Hefor laenko, S. P. Gladchenko, and S. S. Sokolov, Fiz. Nizk.
Temp. 24, 1113 (1998) [Low Temp. Phys. 24, (1998)].
temperatures below 1 K, whereas the hydrodynamic ap-
[7] R.J.F.vanHaren,G.Acres,P.Fozooni, A.Kristensen,
proach of viscous phenomena is well satisfactory in the
M. J. Lea, P. J. Richardson, A. M. C. Valkering, and R.
case of 3He for temperatures down to ∼ 0.1 K. Other W. van der Heijden, Physica B 249-251, 656 (1998);
attractive substrate should be also the low-temperature [8] Yu. P. Monarkha, Fiz. Nizk. Temp. 1, 526 (1975) [Sov.
mixture of 3He and 4He. [13] On the other hand, unfor- J. Low Temp. Phys. 1, 258 (1975)].
tunately, one cannot devise now a mechanism for using, [9] G. E. Marques and N. Studart, Phys. Rev. B 39, 4133
in the Q1Dapparatus,a solid substrate below the liquid (1989).
heliumtoprovidealargereffectiveholdingfieldasinthe [10] Yu. Z. Kovdryaand Yu. P. Monarkha, Fiz. Nizk. Temp.
caseofthe 2Dpolaronona liquidheliumfilm[4],andas 12, 1011 (1986) [Sov. J. Low Temp. Phys. 12, 571
aconsequencegreatlyenlargingtheabsolutevalueofE . (1986)].
b
[11] S. S. Sokolov, G.-Q. Hai, and N. Studart, Phys. Rev. B
In conclusion we predict the possibility for the for-
51, 5977 (1995).
mation of the asymmetric polaron in the Q1D channel
[12] K. Shirarama, S. Itoh, H. Suto, and K. Kono, J. Low
over the liquid helium surface. The localization lengths
Temp. Phys. 101, 439 (1995); S. S. Sokolov, J. P. Rino,
of the asymmetric polaron, and the ground and excited and N.Studart,Phys. Rev.B 55, 14473 (1997).
state energies are determined as functions of the holding
[13] SeeR.B.Hallock, PhysicsToday,p.30(June1998) and
electric field. The numerical estimates of the frequency references therein.
of spectroscopic transitions and the temperature for the
3
