Abrikosov vortex lattice nonlinear dynamics and its stability in the presence of weak defects

12 Pages·0.25 MB·English

by  A. Yu. Galkin

Abrikosov vortex lattice nonlinear dynamics and its stability in the presence of weak
defects.
Alexander Yu.Galkin∗
Institute of Metal Physics, National Academy of Sciences of Ukraine, Vernadskii av. 36, 03142, Kiev, Ukraine
2
0 Boris A.Ivanov
0 Institute of Magnetism, National Academy of Sciences of Ukraine, Vernadskii av. 36 ”B”, 03142, Kiev, Ukraine
2 (Dated: February 1, 2008)

n Abrikosov vortex lattice dynamics in a superconductor with weak defects is studied taking into
a accountgyroscopic(Hall)properties. Itisdemonstratedthatinteraction ofthemovinglatticewith
J
weak defects results in appearance of the additional drag force, F, which is F(V) √V at small
1 velocities,whileathighvelocitiesF(V) 1/√V. Thus,thetotaldragforcecanbea∝nonmonotonic
3 ∝
function of the lattice translational velocity. It leads to the velocity dependence of the Hall angle

andanonlinearcurrent-voltagecharacteristicofthesuperconductor. DuetotheHallcomponentin
n] thelatticemotion,thevalue[dF(V)/dV]−1 doesnotcoincidewiththedifferentialmobilitydV/dFe,
o whereFeistheexternalforceactingonthelattice. TheestimatessuggestthatconditiondV/dFe<0
c canbemetwithdifficultyincontrasttodV/dF <0. Theinstabilityofthelatticemotionregarding
- nonuniform perturbations has been found when the value of (-dF/dV) is greater than a certain
r
p combination of elastic modulii of the vortex lattice, while the more strict requirement dV/dFe <0
u is still not fulfilled.
s
. PACSnumbers: 74.20.De,74.25.Fy,74.60.-w
t
a
m
I. INTRODUCTION modes excited in the moving vortex lattice.
-
d In the present work we investigate vortex lattice dy-
n namics in the presence of weak defects. We are inter-
Vortex dynamics in superconductors is still a contro-
o estedinanadditionaleffectofdissipationinducedbythe
versial matter in physics of superconductivity. As a rule
c
vortex lattice motion and interaction with weak defects,
[ dynamics of Abrikosov vortices was considered in the
overdamped regime,1 when the viscosity is strong and which are not strong enough to pin the vortex lattice,
1 but this motion results in excitation of normal vibra-
it namely determines lattice dynamics. The Hall force
v tional modes and irreversible transfer of the interaction
1 wasgenerallyneglectedinconventionalsuperconductors,
energy to these modes (Landau damping).
9 where viscosity is high. Interest in the Hall force has
Due to the additional dissipation effect nonlinearity
5 quickened in the past few years with fabricationof high-
1 T superclean single crystals with the large Hall angle.2 of the lattice motion has been revealed and it has been
c
0 Moreover, among other things, Hall properties are man- demonstrated that the nonlinear regime may be even
2 dominant one in lattice dynamics being compared with
ifested in special collective modes existing in the vortex
0 the bare linear dissipation. When the lattice is the sub-
system. These modes were observed in resonant experi-
/
t ments with 4He.3 The interestto these modes asapplied ject of the Hall force we predict appearance of a lattice
a
motion instability. The Hall angle is found to be lattice
m tohigh-Tc superconductorshasincreasedconsiderablyin
recent years in connection with experiments in BSCCO4 velocitydependentandthehighervelocitythelargerHall
d- and YBCO5 compounds in which resonant effects were angle as far as it is saturated.
n observed. InYBCO systemresonancesoccur mostlikely The paper is composed of five sections as follows. In
o because of cyclotronic vortex modes, while the resonant Sec.IIwedescribeamicroscopicalmodelofvortexlattice
c dynamicswiththeHallandviscouscontributions. InSec.
effects in BSCCO are associated either with Josephson
v: plasma modes emerging there owing to Josephson cou- III we study the steady motion of the vortex lattice and
i pling ofadjacent CuO layers6 or with the vortex modes discuss the stability of this motion. Discussion of the
X 2
excitations.7 results of the model is presented in Sec.IV. Finnaly, the
r summary is given in Sec. V.
a The vortex modes may manifest themselves not only
inresonancesbutalsoindissipationofthemovingvortex
latticecausedbyexcitationofthesemodes. Thiseffectis
II. MODEL
owing to irreversible transfer of the translational vortex
motion energy into elementary bosonic excitations - the
quanta of vortex modes in the presence of the vortex- Letus considerdynamics ofthe vortexlattice (formed
defects interaction. For the single vortex it has been by Abrikosov vortices parallel to the z axis). Let vor-
−
demonstrated that excitation of vortex modes results in tices inequilibriumto be placedinthe points ofanideal
an anomalous behavior of the friction force, F , which is lattice,~l. Dynamicsofthevortexlatticecanbedescribed
v
divergent at V 0.8 The next step is study of vortex on the basis of an effective equation for the 2D vector
→
2
~u=~u (z,t),lyinginthexy planeanddescribingthedis- translational and steady (~u (z,t)=u (0)+V~t, where V~
l l l
−
placement of the~l - th vortex in the lattice (see Fig.1).1 isthetranslationalvortexlatticevelocity)thedissipative
The following Lagrangian determines nondissipative dy- function yields the friction force per unit length of the
namics of ~u: vortex, F~ = V~(Q/V2)= V~β.
− −
In the case of ”weak” defects the microscopical origin
H ∂u ∂u
L ~u = dz u l,y u l,x of defects is not essential.8 We determine interactionbe-
l,x l,y
{ } 2 ∂t − ∂t −
l Z (cid:20) (cid:18) (cid:19)(cid:21) tween the vortex lattice and defects in the crystal as a
X
W ~ul Uimp ~u . (1) variation of a local critical temperature, Tc, depending
− { }− { } on coordinates. This can be taken into account by in-
Here H is the Hall constant of the single vortex per troducing a coordinate-dependent coefficient [a +F(~r)]
0
unit length, W = W ~u is the energy of the ideal (de- into the Ginzburg-Landau expansion.1 We assume that
l
{ }
fectless)deformedlattice, U ~u defines theenergyof the distribution of the order parameter in the vortex
imp l
{ }
interaction of the vortex with defects (its structure will does not change during the vortex motion and is de-
beconsideredbelow). Todescribedynamicsinthelinear scribedbyaknownfunctionf(r ),wherer2 =x2+y2.9
⊥ ⊥
approximation W ~u should have the following struc- Thus, regarding the displacement of the l-th vortex, we
l
{ }
ture: canwrite the vortexlattice energy Ψ 2f(r ), where
0 l⊥
| |
l
r =[(x l u )2+(y l uP)2]1/2 and Ψ is the
l⊥ x lx y ly 0
2 − − − −
1 ∂~u (z,t) orderparameter. Inthiscase,theenergyU associated
l imp
W ~u = κ + (2)
{ l} 2X~l Z ( (cid:20) ∂z (cid:21) wofitthhecrfyusntcatliionnhofm(ro⊥g)enineittiheescfoarnmbeofwaritfutennctiinonthaleotefrtmhes
vortex lattice displacement:
+ Uik ~l−~l′ δuil,l′ ·δukl,l′)dz,
~Xl′6=~l (cid:16) (cid:17)
U = dx˜dy˜dzf(r )F[x˜+l +u ,y˜+l +u ,z],(5)
where−δ→ul,l′ =~ul(z,t) ~ul′(z,t),Uik(~l)arecomponentsof imp ⊥ x lx y ly
the force tensor which−are expressed through the second X~l Z
derivativesofawell-knownpotentialofthevortex-vortex
where x˜=x l u ,y˜=y l u .
interactionwithrespecttoequilibriumvortexpositions,9 − x− lx − y− ly
Finally, one arrives at the equation for the displace-
κ is the energy per unit length. We neglect in the above
ment of the l th vortex, ~u (z). It is presented in the
expression of the Lagrangian the inertial term, which − l
form:
comprisesthevortexmassbecause,aswasdemonstrated
in8, the vortex mass plays a substantial role in the dissi- ∂~u δW ~u ∂~u
pation of the moving vortex lattice only at high enough H z˜ l = { l} β l +F~l,imp. (6)
× ∂t − δ~u − ∂t
velocities. In the framework of a macroscopic approach, (cid:18) (cid:19) l
i.e. if~u variesinsignificantlyoveradistanceoftheorder
l Herethelefthandsideisthedynamicaltermgoverned
of the lattice constant a , the last expression (2) goes
v bythe Hall(gyroscopic)constant,the terms onthe right
over to the standard expression as with the theory of
hand side are forces acting on the l th vortex because
elasticity, −
duetointeractionwithothervorticesintheideallattice,
friction and the presence of defects.
c ∂u ∂u 2 Using the Eq. (5), the force caused by defects can be
11 x y
W ~u = dxdydz + + (3) written in the form of the Fourier expansion in x and y:
{ } Z " 2 (cid:18) ∂x ∂y (cid:19)
2 2
c ∂u ∂u c ∂~u
+ 66 x y + 44 , F~ = 1 i~q f(q ) (~q)eiq~⊥~l+iqzzeiq~⊥u~l(z) , (7)
2 (cid:18) ∂y − ∂x (cid:19) 2 (cid:18)∂z(cid:19) # l,imp Ω ⊥ ⊥ F
q~
X
where c ,c ,c are the elastic modulii of the lattice,
11 44 66
c44 can be expressed through κ, c44 =κ/a2v. where Ω = LxLyLz is the superconductor volume, F(~q)
Eachvortexmovinginthemediumexperiencestheac- is the Fourier transform of the function F(~r), which de-
tionofthedragforceandtherateofitsenergydissipation fines inhomogeneities in the system, see Eq.(5), ~q⊥ =
is proportional to (∂~ul/∂t)2. For this reason, we choose (qx,qy,0), f(q⊥) is the vortex form-factor.
the dissipative function in the form: In the absence of dissipation and defects the equation
(6) can be simplified and because of translationalinvari-
β ∂~ul 2 ance the force tensor components Uik depend on ~l ~l′
Q= dz, (4) −
2 ∂t only and do not depend on z. Then, the equation of
l Z (cid:18) (cid:19)
X motion can be presented as it is done in the lattice dy-
where β is the dissipation coefficient per unit length of namics theory, i.e. by means of the Fourier transform
the vortex. When the motion of the vortex lattice is of the force tensor components.10 By using the Bloch
3
theorem it is possible to present the equation of mo- the problem more complicated as equations are still lin-
tion in the normal coordinates ~u considering that ear. However,thepresenceofdefectsresultsinnonlinear
q,x(y)
~u = ~u exp i~q ~l+iq z . equations as the interaction energy (5) and force (7) de-
l,x(y) q~,x(y)· ⊥ z pend nonlinearlyon~u. A generalsolutionof this nonlin-
q~⊥,qz
The equPation of motion(cid:16)acquires the(cid:17)following form: ear equation (6) cannot be found. In line with8 we can
carry out a complete analysis assuming that deviations
Hu˙q.x = Cyy(~q) uq,y+Cxy(~q) uq,x; (8) fromtheuniformtranslationalandsteadymotionofvor-
· ·
Hu˙q.y = Cxx(~q) uq,x+Cxy(~q) uq,y, ticesdue to defects aresmall. Putting~u=~exVt+~u˜(~r,t)
− · · and linearizing (7) in respect to~u˜, we obtain the expres-
where Cik = Uik(~l)·exp(i~q~l ) is Fourier representa- sion for Fimp without~u˜l(z):
l
tion of the foPrce tensor Uik, see (2). Here and below
dot denote the time derivative. The Fourier transform F~ = 1 i~q f(q ) (~q)eiq~⊥~l+iqzz−iqxVt . (11)
of ~ul is discrete in the xy plane and continuous along l, imp Ω ⊥ ⊥ F
−
the z axis, the quantity ~q⊥ has the sense of a quasimo- Xq~
−
mentum, and q is a regular linear momentum, (we put
z Thus, the lattice motion through the nonuniform
¯h=1). The quasimomentum ~q there is within the first
⊥ medium results in excitation of small oscillations of vor-
Brillouin zone, ~q 1/a , whereas q is only restricted
| | ≤ v z tices in the lattice (vortex lattice normal modes), which
bythephysicalreason,i.e. q 1/ξ,whereξisthecoher-
z ≤ are described by ~u˜. It was discussed by the authors of
ence length of the superconductor. Though these values
Ref. 12 in the terms of ”the lattice melting”. Never-
~q and q , have somewhat different physical meanings,
⊥ z
theless, it is essential for us that oscillations bring into
below we will use a macroscopical approach and retain
appearence of the additional contribution to dissipation
these symbols.
of the lattice, ∆Q=(β/2) (∂~u˜ /∂t)2dz, and in turn
The variables ~uq~ comprise the time dependent mul- l l
tiplier, ~u exp[ iω(~q)t], where ω(~q) is periodic with into the additional drag force, F~(V),
q~ ∝ ± P R
regard to ~q , ω(~q+~g)=ω(~q), ~g is the vector of the re-
⊥
ciprocal lattice. Then, one can derive the dispersion law
F(V) ∆Q(V)
usingnewparametersC1 =Cxx ∆,C2 =Cyy+∆, 2∆= F~(V)= V~ , F(V)= . (12)
− − V V
(C C )+ (C C )2+4C2 :
xx− yy xx− yy xy
q The general equation for~u˜q (compare with (8)) reads
1
ω2 = C C . (9) 1
H2 1 2 u˜ = [(C +iβq V)f (C iHq V)f ];(13)
q, x yy x x xy x y
D − −
At large ~q 1/a the dispersion law of vortex lat- 1
v
| | ∼ u˜ = [(C +iβq V)f (C +iHq V)f ],
tice oscillationsbecomes periodicalin the ~q⊥ plane, with q, y D yy x y− xy x x
a period determined by the reciprocal lattice constant
Canxdx(ytyh)eamnadxiCmxayl avraelupeeqrxio,ydi=ca1l/fξunscinticoensthoefc~qo⊥effi. cNieenatsr w(HheqrxeVC)2ik, ≡fx,Cyiiks(~qr)e,laDted=to(C(111+)iβqxV)·(C2+iβqxV)−
the center of the first Brillouin zone, ~q 1/a , the
v
| | ≪ i
dispersion law (9) can be presented via elastic moduli, f = q f(q ) (~q)e−iqxVt. (14)
x,y x,y ⊥
C =a2 c q2+c q2 ,C =a2 c q2+c q2 : Ω F
1 v 44 z 66 ⊥ 2 v 44 z 11 ⊥
(cid:0) (cid:1) (cid:0) (cid:1) It is clearly seen that the condition D = 0 at β =
a2 0 defines the dispersion law found in (9), if one would
ω = v (c q2+c q2)(c q2+c q2) , (10)
H 44 z 66 ⊥ 44 z 11 ⊥ replace ω qxV. It is a typical evidence of resonant
→
q character of excitation of normal modes because of the
which has been also obtained in Refs.8,11. This law cor-
lattice uniform motion.
responds to a low frequency gapless mode in the limit of
To analyze specifically a contribution of these small
small q and q .
z ⊥ oscillationstodissipationofthemovingvortexlatticewe
Fromhereonwe willemploy aDebye-likemodel, with
assumethatinhomogeneityiscausedbyasystemofpoint
approximation of the dispersion law by the long wave
defectswhosesizeissmallerthantheradiusofthevortex
equation (10) and with taking into account periodicity
core. In this case, the function F(x,y,z) in (5) can be
in the q ,q directions. The dispersion law for the vor-
x y written as the sum of Dirac delta functions:
tex lattice is schematically drawn in Fig.2. The Debye
frequency for ω(q⊥), ωD, is marked in Fig.2 by a doted F(~r)= αδ(~r ~ra). (15)
line. −
a
We have concerned lattice oscillations without dissi- X
pation and defects and found the dispersion law of nor- Here ~r is the coordinate of the a th defect and α
a
−
malmodes. Thedissipationconsiderationdoesnotmake characterizes the intensity of interaction of the defect.
4
The force acting on the l th vortex with regard to the where star denotes the complex conjugation, and carry-
−
model (15) is written as (11) with substitution ing out averaging over defects with the help of the rela-
tion exp(i~q~r )=N δ , where N is the number
a imp q,0 imp
(~q)=α eiq~~ra. (16) a
F of dePfects, δq,0 is the Kronecker symbol, we can find the
Xa additional dissipation of the moving lattice. It may be
By substituting (13)withthe takinginto account(16) associated with the drag force F(V). In the framework
of the model (15) one can obtain the equation for the
inthedissipativefunction,employingthetransformation
additional drag force F(V) per unit length of the vortex
2
1 dz ∂~u˜l Ω d3q(q V)2~u˜ ~u˜∗,
Ω l Z ∂t ! → (2π)3a2v Z x q q
X
(17)
βVα2 q2 q V2(β2+H2)+C2 +(C +C )(q √∆ q √C C ∆)2
F(V)=C d3q q2f2(q ) ⊥ x 2 1 2 x − y 1− 2− , (18)
imp(2π)2 · x ⊥ [C C (β2+H2)(q V)2]2+β2(q V)2(C +C )2
Z (cid:2) 1 2− (cid:3) x x 1 2
where C = N /Ω is the defect concentration. type δ (q V)2 ω2(~q) , where ω(~q) is the frequency of
imp imp x
−
normal mode (10). Since the equation q V = ω(~q) pos-
Because of the factor βV appearing in (18) in front x
(cid:2) (cid:3)
sess a solution at any finite velocity the value of Q(V)
of the integral one could assume a linear velocity depen-
dence of the drag force, F(V) β˜V, where β˜ is some can be finite even at β 0.
→
∝
constant. Then, it seems that a consideration of vibra- In the case β 0, it is obvious that the additional
→
tions of vortices associated with interaction with defects dissipationbecomesthemainsourceofdissipationinthe
resultsonlyinasmallcorrectiontothebare frictionforce, system. This may be realised if the friction force has a
F~ = βV~. However, the integral in (18), as well as dependence like F Vδ,δ <1, as it was revealed in the
fobrartehe c−ase of the single vortex,8 contains singularities, case of the single vo∝rtex.
therefore, the additional relaxation channel can become There is an alternative explanation of the additional
significantandevenpredominant. The mostvividexam- dissipation appearance. The modes excitation in the
ple of this fact is that the additional dissipation can be limit of small dissipation can be described on the ba-
finite inthe limitofzerobaredissipation,8 F(V)is finite sis of momentum and energy conservation. Let us go
at β 0, as Fbare 0. over to a reference frame moving with vortices. In this
→ →
At first glance, this appears as paradoxical. However, reference frame defects move at a velocity V~ = V~e
x
−
such a colissionless damping (Landau damping) stem- parallelto thex axis and cantransferthe momentum ~q
−
ming from the energy transfer from one mode to an- tothevortexlatticeonlysimultaneouslywiththeenergy
otherappearsinmanybranchesofphysics. Forinstance, ~qV~ = q V. This momentum is redistributed between
x
flux-flowdynamicscausedbyexcitationsoflowfrequency the lattice as a whole and an elementary excitation, and
fermionic modes has been proposed in Ref. 13 and the the energy is transferred to elementary excitations only.
finite friction force independent of the bare friction is In particular, in the case of the single vortex the vortex
predicted there. The friction force finite at V 0 and as a whole acquires momentum perpendicular to its axis
→
causedbyexcitationofbendingoscillationofthedomain (z axis), while the wave propagating along the vortex
−
wall in ferromagnets with microscopic defects was pre- obtains the z axis momentum projection.
−
dicted in Ref. 14. As it will be demonstratedbelow here
Let us analyse the general expression (18). Note that
F(V) Vδ, δ <1 at V 0 and F(V)>F at any β
∝ → bare integration has to be done not only inside the first Bril-
and small enough velocities.
louin zone. The integration is limited only by cut off q˜
⊥
This behavior can be explained as following. over 1/ξ, when the vortex form factor f(q )=0. In the
⊥
The expression (18) contains β in the combination case of the vortex lattice, the lattice constant6 a is usu-
v
β/ β2+G2 ~q,(qxV)2 , where the function G is such ally assumed to be larger than ξ. In order to resolve the
that the condition G ~q,(q V)2 = 0 with the substitu- vortex lattice problem one should go beyond the long-
(cid:8) (cid:2) (cid:3)(cid:9) x
tionq V ω defines the frequenciesofnormalmodesof wavelimit. Generallyspeaking,onehastointegrate(18)
x
→ (cid:2) (cid:3)
vortex vibrations in the lattice (10). For β 0 the ex- taking into accountcomplicated periodical dependencies
→
pression (18) is transformed into the δ-function, and be- of C , C , ∆ of the wave vectors, and a non-periodic
1 2
comes πδ G ~q,(q V)2 . After simple transformations dependence of f(q ), terms with q V and so on. Nev-
x ⊥ x
this expression can be reduced to the δ-function of the ertheless, the situation is simplified in limiting cases, for
(cid:8) (cid:2) (cid:3)(cid:9)
5
example, at small or high velocities. inC ,C or∆shouldbereplacedbycorresponding~gvec-
1 2
Firstwe concernthe caseofsmalldissipation,β H, tors then one cansubstitute in (18) both g =(2π/a )n
x v
≪
for which the conservation law ω(~q) = q V defines an instead of q and g = (2π/a )(n2+m2)1/2 instead of
x x ⊥ v
area that contributes to the integral in (18). To provide q .
⊥
an estimation, we assume that c c c , then The sum over m and n may be replaced by the inte-
11 66 ⊥
ωH = a2(c q2+c q2), and the con∼dition ω∼(~q) = q V gral as we consider large m and n, m,n a /ξ and we
v 44 z ⊥ ⊥ x ≤ v
may be written as integrateover~q˜onlyasmallregionneareachvectorofre-
ciprocallattice, ~q˜ V/V ,andconsideringlargeenough
c
| |≤
valuesof~g till1/ξ. The expressionforthe dragforceper
c c q2+(c q VH/2a )2+c2q2 =(VH/2a )2.
44 ⊥ z ⊥ x− v ⊥ y v single vortex in the limit of small viscosity, H β, and
(19) ≫
small velocities, V V , finally is
c
≪
ThisexpressionindicatesthatatsmallV thesizeofthe
region in the ~q plane, which contributes to the integral
⊥ F =γ V1/2, (21)
in (18) is proportional to V. If one considers that c = H
11
c , then of course the above consideration remains tru6 e 2πα2C J√H 3/√2, c c
66 γ = imp 11 ∼ 66
andonly the expression(19)becomes more complicated. H 3√c11c44c66avξ3/2 ·((c11/c66)1/4, c11 c66,
Thus, at V 0 the region near ~q = 0 gives rise to a ≫
→
very small contribution to Q, which is proportional to where J = ξ3/2dg dg f2(g )g3/2g2 is the constant of
Vδ, δ >1.11 x y ⊥ x ⊥
the order of unity.
However, if periodicity is taken into account, then the R
In the limit of high viscosity, β H, and V 0 the
areas with ω(~q) → 0 emerge near all points in the ~q⊥ similar estimation demonstrates t≫hat again onl→y small
plane, which are equivalent to the origin. If one puts
areas near~g are important and calculations yield
~q =~g+q˜ , whereq˜ isthe wavevectorwithinthe first
⊥ ⊥ ⊥
Brillouin zone and~g is the wave vector of the reciprocal
lattice, then these points are located at q˜⊥ → 0. It is F =γ V1/2,γ = √2πα2Cimp J√β 1 + 1 ,
awroert(hg to+mq˜e)nVtion 2tπhantVv/aalu.esTohfeqcxoVrrensepaorndthinesgeapreoainitns β β √c44 avξ3/2 (cid:18)c66 c11(cid:19)
x x v (22)
≈
the ~q plane is now given by ω(~q)=2πnV/a .
⊥ v
The value of g is limited by the cutoff g 1/ξ or where J is the same constant as in (21).
x x
≤
n (a /2πξ). Hence,theconditionoftheareasmallness So for small velocities, V V , and any relation be-
v c
in≤the ~q˜ plane with substitution of q g 1/a in tween H and β the drag for≪ce is F(V) = γ(H,β)√V,
⊥ x ∼ x ∼ v
q V and with ~q˜ in C ,C without changes takes the wherethecoefficientγ(H,β)inlimiting casesisgivenby
x ⊥ 1 2
form: (21, 22).
When V V one has to take into account not only
c
∼
small regions of the wave vectors near ~g, but perform
a2c (c q2+c ~q˜2)=(VH/ξ)2. (20) integration over all ~q 1/ξ as well. But this is an
v ⊥ 44 z ⊥ ⊥ | | ≤
extremely complicated problem and we are not able to
Thus, these regions can be small being compared to solve it even numerically as the detailed behaviour of
the size of the first Brillouin zone at velocity V ≪ Vc, Cik at large ~q is not known. Far above the critical ve-
wtiohnereofVcc11=,c66ξc¯(/mHor∼e aωccDuξr,atweheexrperec¯ssiisonsofmoreVccomwibllinbae- lloemcit.y,InVt≫hisVcca,soe,neitciasnpsoisgsnibifilecatnotlynesgilmecptlicfyontthreibpurtoiobn-
given below, see Eq. (25)). A rough estimate of the of the terms like (c , c )q2 having the order of the
11 66 ⊥
characteristicvelocityVc canbedonethroughthe Debye value ωD to the denominator in (18), whereas the terms
frequency, Vc ∼ ωD ·ξ. The Debye frequency appears c44qz2, βqxV, HqxV should be taken into consideration.
because of periodicity of the dispersion law and the co- Then, the calculations give at V >V
c
herence length indicates that we take into consideration
all ~q⊥ wave vectors from 1/av to 1/ξ. (Note Vc is much F = η(H,β), η = πα2CimpJ′A(H,β), (23)
smallerthatthecharacteristicphasevelocityofcollective √V √c44avξ1/2
modes, V ω a .)
ph D v
∼ ·
The integral in (18) breaks into the sum over differ- where A(H,β) = 1√2H at H β and
ent ~g. In the framework of a macroscopical approach A(H,β) = 1/(2√β) at H β, the co≫nstant J′ =
≪
theabovederivedformula(18)canbepresentedwithco- ξ1/2dq dq f2(q )q1/2q2 J. Below the difference be-
efficients C1,C2 and ∆ that acquire macroscopic form: tween Jxanyd J′⊥wilxl be⊥d∼isregarded. The dependence
C1/a2v = c11q⊥2 +c44qz2, C2/a2v = c66q⊥2 +c44qz2 as were RF 1/√V correspondstothe singlevortexlimit,8 ifone
a(dcfet1fie1nr−erdce6pa6lb)aqcox2evame2vea.nnTtdh~q∆en,=~tq˜h,(ecis1l1aes−qtubca6rl6at)coqky2ezate2vri,noC,tth1he−antCusm2im−epral∆itfioe=rs, wtthooe∝uωlli(dm~q)ritempVluas≫ctebVce4c4ttaah2vkee→onnilκny.tcoIotnaitcsrcionbuountttiso.unrTpthhriiessiqnizsgwebxaeavcceatvulyescetthoiner
→
integral calculation. All ~q vectors those are not included caseofthesinglevortex,forwhichq andq components
x y
6
ofthe momentum~q areabsorbedbythe vortex,while el- while the linear asymptotic F(V) V appears only at
∝
ementary excitations propagating along the vortex axis V > V V and V = (η/β)2/3. Obviously
trans c trans
acquirethemomentumqz andtheenergyqxV,thatdoes that namely i≫n the latter case the nonmonotonic F(V)
not restrict the integration area over ~q (in contrast to dependence may be realised.
(19) or (20)). The concrete values of this velocity and the relation
Thus, the velocity dependence of the additional drag between V and V will be estimated below.
c trans
force, which results from the vortex-defects interaction
can be described as F = γ(H,β)√V, V < V , and
c
F = η(H,β)/√V, V > V , where the coefficients γ and III. FORCED MOTION OF THE LATTICE
c
η are derived in Eqs. (21-23). To describe the total fric-
tion force the bare contribution F~ = βV~ should be In the previous section we have employed a model of
bare
−
concerned. The total friction force acting on the mov- aratherweakrandomforceinduced bydefects actingon
ing vortex lattice having the different velocity depen- the lattice. Now we proceed to description of the vortex
dence for small and large velocities can be presented as lattice dynamics within a somewhat broader macroscop-
F~ = (V~/V)F(V), with the interpolating equation for ical approach. Having use the macroscopicalform of the
−
F(V) averagefriction force (24) instead of the random vortex-
defects forceconsideredin(6) weareable to reformulate
γ(H,β)√V the vortex lattice dynamics problem on the macroscopi-
F(V)= +βV, (24)
1+V/V cal basis.
c
Consider the displacement vector of vortices in the
where we use Vc = η(H,β)/γ(H,β) as a quantity char- lattice U~ = U~(~r,t) assuming that U~ changes insignif-
acteristic of the problem. The schematic picture of the icantly over the scale of the lattice constant. We use
F(V) is presented in Fig.3. the trivial averaging of the Lagrangian (1) with the dy-
This value does not contradict to estimates provided namical part presented as in (6) and with substitutions
abovein which only the orderofVc has been considered. (...) dxdy/a2(...) and ~u (z,t) U~(~r,t). The en-
From (21-23) one can extract the characteristic velocity l → v l →
ergy of lattice deformation is described by the elasticity
V in limiting cases P R
c theory(3). Weassumethepresenceoftheexternalforce,
F ,whiletheeffectivedragforceisdeterminedbyEq.(24)
e
(√2/H)√c c , H β, c c , with the change V~ U~˙.
Vc = 2√ξ2 ·(3/H) c11c13616616/4, H ≫≫β,c1111≫∼c6666, The equation of m→otion takes the similar form as (6):
In the procb11lecm66/(cid:0)[oβn(ec11m(cid:1)+orce66)c],harβac≫terHis.tic velo(c2it5y) H z˜×U~˙ +U~˙ F(U~|˙U~˙|) + ∂W∂U{~U~} =F~e. (26)
(cid:16) (cid:17) | |
emerges, V , at which the bare dissipation starts
trans Usingtheequation(26)asthe baseweconsidermacro-
to be dominant compared to the additional one, i.e.
scopical dynamics of the vortex lattice, in particular, its
βV γ √V/(1+V/V ) at V > V . We would like
H c trans
≥ stability against small perturbations. We present trans-
to emphasize that V is determined as a velocity be-
low that the additiontraalnsdrag force prevails over the bare lational lattice velocity, U~˙ = V, as a sum of the steady
drag force, whereas Vc is defined only by the additional velocity of the lattice, V~0, and the small correction~u˙, U~˙
drag force behaviour. In virtue of this, Vc and Vtrans =V~ +~u˙(~r,t).
0
depend on the problem parametersin different manners.
First we seek the solution for the lattice steady veloc-
Asitisseenfrom(25),Vc doesnotdependonthelattice- ity V~ . It allows us revealing F (V) dependence, which
0 e
defects interactionintensity, i.e. does not comprise C
imp ismerelycurrent-voltage(I V)characteristicofthesu-
or α. On the contrary, V is directly correlated with −
trans perconductor, see Ref.16. Because of the Hall term in
the intensity α. Because of the nonmonotonic velocity
(26)thesteadyvelocityisconvenienttoexpressviacom-
dependence ofthe additionaldragforce,V exists for
trans ponents V~ = V ~e +V ~e , where ~e = F~ /F~ , ~e =
cahnayraacstesmr oafllthase otonteallidkreasgvafolurcee,ofFC(Vim),pαis2cahnadngoendl.y the (~zˆ ~ek). 0To dekscrkibe t⊥he⊥stationarykflow oef|thee| vo⊥rtex
×
If the lattice-defects interaction intensity is small, lattice with velocity V~ a couple of equations of motion
0
while the bare constant β is large, then V falls on are found,
trans
the increasing part of F(V) and V =(γ/β)2. It cor-
trans
respondstothecaseV V andthevelocityV does
notmanifestitselfatatlrla:nFs(≪V)∝c √V atV ≪Vtrancs and HV⊥+ F(VV0)Vk =Fe, −HVk+ F(VV0)V⊥ =0. (27)
F(V) V at V >V . 0 0
trans
∝
In the opposite case Vtrans Vc, F(V) √V at From these equations one can derive the components
≫ ∝
V < V and F(V) 1/√V at V V V , of stationary velocity V and V via V :
c c trans ⊥ k 0
∝ ≪ ≪
7
the real part of Λ is positive for some values of ~q, cor-
responding small deviations cause the motion instability
F /H FF /H2V
V = e , V = e 0 . (28) with the instability increment ReΛ>0.
⊥ 1+(F/HV )2 k 1+(F/HV )2 For the small correction ~u the couple of equations of
0 0
motion are
For simplicity we use F′ = [dF(V)/dV] , F =
V=V0
F(V ). With the help of (28) the Hall angle could be
given0 as (cid:12)(cid:12) uq,x[ΛF′(V)+Cxx]+uq,y(Cxy ΛH)=0; (32)
−
u (C +ΛH)+u [ΛF(V)/V +C ]=0,
q,x xy q,y yy
V HV
tanα= ⊥ = 0 . (29) where Cik Cik(~q) are the coefficients used in (13),
Vk F(V0) strictly spe≡aking, their presentation in the longwave
limit, V is the translational velocity of the lattice (for
So if the drag force is a linear function of the transla-
simplicityweomithereandfuthertheindex0). Itismore
tional velocity V , F =βV (the usual case), then tanα
0 0 convenient to use in Eq.(32)~e , ~e instead of~e , ~e . It
x y k ⊥
does not depend on the velocity and the Hall angle tan-
is necessary to stress that this equation contains F(V)
gent is merely the ratio of the Hall constant and the andF′(V)=dF(V)/dV,whereF isthetotaldragforce,
dissipationconstantβ. However,by consideringthe spe-
rather than differential mobility dF /dV. As it will be
e
cificdragcoefficientinEq.(24)astrikingfeatureappears,
demonstrated below, this is a very essential point which
namely a velocity dependence of the Hall angle. In the
leads to the fact that the instability condition is not re-
limit V 0 tanα = √V /γ(β,H), and the higher the
0 lated directly to the usual one, dF /dV <0.
→ e
lattice velocity the Hall angle larger,then it is saturated
The characteristic equation for Λ takes the form:
above the characteristic velocity V .
trans
BysolvingtheEq.(28)withthetakingintoaccountthe Λ2 F′(F/V)+H2 +ω2(~q)H2+ (33)
dragforcefromEq.(24)thedependenceofF (V)acquires
e +Λ[F′C +(F/V)C ]=0,
the following form: (cid:2) yy (cid:3) xx
where we used the equation C C C2 = ω2(~q)H2,
γ√V 2 xx yy − xy
V2H2+ 0 +βV =F2 (30) ω(~q) is the dispersion law for free vortex oscillations (9)
0 (cid:20)1+V0/Vc 0(cid:21) e obtainedabove. Theequation(33)possesstwosolutions:
TheEq.(30)hasasimpleformofthevectorsumoftwo Λ= Γ(q) iΩ(q), (34)
forces: theHallforce(thefirsttermonthelefthandside) − ±
and the drag force (the second term). The differential where
mobility related to the external force is
(F/V)C +C F′
xx yy
Γ(q) = , (35)
dV0 Fe H2+(F/V)F′
= . (31)
dFe V0H2+F(V0)F′(V0) H2ω2(q)
Ω2(q) = Γ2(q).
H2+(F/V)F′ −
If one puts H = 0 in (31) then the differential mobility
dV /dF coincides with (dF/dV )−1. But for H = 0,
0 e 0 6 In order to get the instability condition we analyse
especially at H ≫ F(V)/V these two quantities differ Eqs.(34,35). IfF′ >0,thenΓ>0andsmalloscillations
substantially.
decayforallvaluesof~q. Thesearesmalldampedoscilla-
The negative value of the differential mobility is usu-
tions with the frequency ω Ω(~q) at Ω Γ, otherwise,
ally considered as a condition (both necessary and suf- ≈ ≪
for Ω Γ, this is the case of overdamped dynamics of
ficient) of the instability appearance. Nevertheless, it is ~u. Thu≫s,theinequalityF′ <0isthenecessary condition
quite possible that dV /dF < 0, whereas dV /dF > 0,
0 0 e for the lattice instability.
see (31). Therefore, one has to reconsider the instabil- But if F′ < 0, it is not sufficient to induce the lattice
ity criterion for the case of macroscopic dynamics in the
instability. The further analysis of Eqs.(34, 35) reveals
presence of gyroforce. This problem is of interest not
two different and independent sufficient conditions im-
onlyAbrikosovlattice dynamics,butitisalsoimportant
posed by the instability requirement ReΛ>0.
for dynamics of any system having gyroforce and a non- The first condition is H2+(F/V)F′ <0 or
monotonic velocity dependence of the drag force, F(V)
(for instance, vortices in He4, Bloch lines, etc.)
To investigate the stability of the vortex lattice we F′(F/V)>H2. (36)
take U~ = V~ t+~u(~r,t) and find the solution in the form −
0
~u(~r,t) ~u exp(Λt+i~q~r), where Λ defines the charac- This requirement is equivalent to the inequality
q
∝
ter of time evolutionof small deviations from the steady dF /dV < 0, i.e. the rise of the negative differential
e
motion. If the real part of Λ is negative, the value of mobility, in a literal sense. When this condition is sat-
ReΛ>0 serves as a damping coefficient. Otherwise, if isfied the values of Λ in (34) are real and the instability
−
8
appears at any ~q =0. [If one considers the inertial term takes the following form:
6
in the equation of motion (26), then even the uniform
deviations U(t) from U (dynamics with ~q = 0) become β ∆T 2 a3λ3 H p
0 <10−2 C a3 c c1 , (40)
unstable when dFe/dV <0.] H ∼ imp ·(cid:18) Tc (cid:19) ξ6 (cid:18) B (cid:19)
Hence,thefulfillmentoftheconditiondFe/dV <0ap- (cid:0) (cid:1)
parentlyindicates the rise ofthe instability ofthe lattice where H =Φ /4πλ2 is the quantity of the order of the
c1 0
translational motion. But this condition is substantially lower critical field and p depends on (c /c )
11 66
more strict than F′(V) < 0 (see Fig. 3), and as we will
demonstrateinthenextSectionneverwillbesatisfiedin 19/8, c c ,
p= 11 ≫ 66 (41)
ourmodel(this condition,though,couldbe metinother
(11/4, c11 c66.
models with dF/dV <0). ∼
The second condition corresponds to the inequality:
If one would take the numerical values of λ and ξ, as
fortypicalhighT superconductorYBCO:1 λ=10−5cm,
F′Cyy >(F/V)Cxx. (37) ξ = 10−7 cm, a c= 5 10−8 cm, ∆T /T 1,C a3
− c c imp
10−3 then · ≈ ≈
The condition (37) is one more sufficient condition of
the instability independent of (36). Depending on the (H /B)p β . (42)
strength of requirements (36,37) one of them should be c1 ≥ H
fulfilled earlier than the other. We will demonstrate be-
This condition is met at magnetic fields close to the
low that the condition (37) will be met in realsupercon-
lower magnetic field, H , and the microscopical param-
ductors unlike the requirement (36). c1
eters used above for YBCO compound. The maximal
valuesofβ =Φ2/2πξ2c2ρ ,where ρ is the normalcon-
0 n n
ductivity of the superconductor. So the maximal value
IV. DISCUSSION
of the ratio β/H is β/H = (1/ρ )(λ/ξ)2(h¯/2πmc2) and
n
at ρ = 5 10−16 s (the typical value for YBCO) it is
n
Intheprevioussectionwehaveestablishedasetofcon- equalappro·ximatelyto 10−2. Note this is justthe upper
ditions of the vortex lattice instability which comprises limit of β/H and for realsuperconductors this value can
boththenecessaryandsufficientrequirements. Letuses- be even smaller. Thus, for the case of low viscosity the
timate the least strict condition, the necessary condition instability necessary condition may be fulfilled at high
F′(V)<0. By differentiating (24) one obtains enough values of magnetic fields.
In the case of high viscosity, H β, the necessary
≪
conditioncanbesatisfiedwithdifficultyasγ(β)contains
γ (V/V 1)
c− >β. (38) β and the righthand side of the inequality (38) becomes
2√V (1+V/Vc)2 unity (in conrast to the case H β where the small
≫
parameter β/H takes place).
The left hand side of the inequality (38) as a function Now we study the first sufficient condition
of the variable V/Vc has the maximal value 0.04γ/√Vc F′(F/V) > H2. We carry out the calculation in
−
at V/Vc 2. Using Vc =η/γ one can rewrite (38) as the similar manner as for (39) and obtain (neglecting
≃
the bare dissipation as H β) that 0.1 γ3/2/√η >H.
β 0.04 γ3/2/√η. (39) Apparently this conditio≫n cannot be ·satisfied since
≤ ·
though the same inequality as (42) is valid for this
The parameters η and γ can be taken from the limit- condition, but on the right hand side instead the small
ing formulae (21-23) and expressed through microscopic parameter β/H unity is found. Then, we consider the
characteristics of the superconductor (see, e.g., Refs. 1, second sufficient condition F′/(F/V) > C /C
xx yy
16): the elastic modulii c11 = c44 = B2/4π, c66 = which acquires the following f−orm:
BΦ /(8πλ)2, where B is the magnetic induction, Φ
0 0
is the flux quanta, λ is the penetration depth. The γ ((Cyy 2Cxx)V/Vc (Cyy+2Cxx))
− − >β. (43)
vortex lattice constant is determined by magnetic in- 2√V (C +C )(1+V/V )2
yy xx c
duction a2 = 2/√3 (Φ /B). The Hall constant is
v 0
H =Φ en/c,whereeistheelectroncharge,cisthespeed It is seen that the inequality (43) has an obvious re-
0
(cid:0) (cid:1)
of light, n is the density of superconducting electrons, semblance with Eq.(38). The difference is caused by ~q-
n=mc2/4πλ2e2 withthe electronmass,m. Theparam- dependentcoefficientscontainingC andC inEq.(43).
xx yy
eter α can be written as8 α = a3 H2 /8π (∆T /T ), For 2C > C , which is realised, for instance, when
cm c c xx yy
where H is the superconducting thermodynamic criti- c = c , the left hand side of (43) is negative. There-
cm 11 66
(cid:0) (cid:1)
cal field, ∆T /T is a relative suppression of the critical fore,for somevalues of C , C the condition(43)can-
c c xx yy
temperature nearly a defect, a3 is the volume of the de- not be fulfilled and no instability appears. However, by
fect (a is of the order of the interatomic distance). As using the fact that c (c /4)(H /B) c and
66 11 c1 11
≃ ≪
a result, in the case of the small dissipation, the Eq.(39) choosinganappropriate~q itcanbesatisfied. Thesimple
9
analysis shows that when c c the most favorable V. CONCLUSION
11 66
≥
caseisthewavevectorsalongthey-axis,~q ~e . Then,the
y
k
inequality (43) takes the following form:
We have demonstrated that due to the interaction of
weak defects with the Abrikosov vortex lattice the addi-
γ ((c 2c )V/V (c +2c ))
11 66 c 11 66
− − >β. (44) tional channel of dissipation appears, which contributes
2√V (c +c )(1+V/V )2
11 66 c to the total dissipation of the moving lattice. As a re-
sult the additional friction force has a non-linear depen-
Themaximalvalue oflefthandside of(44)is attained
dence of the translational lattice velocity, F √V for
at 3(c11−2c66)(V/Vc) = (3c11+4c66+2c¯), where c¯= both weak bare dissipation, H β, and high∝viscosity
3c2 +6c c +c2 . It is equal to ≫
11 11 66 66 H β. Thischannelofdissipationturnsouttobeanes-
≪
p sentialsourceofthe systemdissipationandevenprevails
γ3/2 (3√3/4)(c 2c )5/2(c¯ c ) over the bare dissipation, βV, below the characteristic
√η (c +c )√3c 1+14−c 6+62c¯(3c− 6c6 +c¯)2. velocity Vtrans.
11 66 11 66 11− 66 The nonlinear character of the additional drag force
(45)
results in a nonlinear current-voltage characteristic of
the superconductor. Besides, the tangentof the Hall an-
Thus, the sufficient condition (44) is more strict than
gle tanα appears to be the translational lattice velocity
necessaryonegivenby(38)owingtothemultiplierswith
V dependent up to the characteristic velocity V ,
c and c . In particular, it can be met only at c > 0 trans,H
11 66 11
above which it is saturated.
2c . However, in the case of interest c 10 c , the
66 11 66
≥
necessary and sufficient conditions are close each other Studyingthestabilityofthevortexlatticeagainstuni-
and the sufficient requirement (44) can be satisfied as form and nonuniform perturbations we have found sev-
well. As the instability is expected to arise when H eral salient features.
≫
β and V 2Vc, one can estimate Vc for this case. In The instability ofthe vortexlattice motionis foundto
≈
accordance with Eq.(25) and with the use of the above occur in the limit of small viscosity, H β. We have
presentedmicroscopicalparametersforYBCO Vc canbe revealedthatthe negativedifferentialmo≫bility,dV0/dFe,
presented as: isnotapriori the latticemotioninstabilityrequirementl.
Due to the Hall component in the lattice motion, the
Vc 10−2 (B[Oe])3/2, (46) value [dF(V)/dV]−1 does not coincide with the differen-
≈ ·
tial mobility dV/dF . The estimates suggest that condi-
andatB =103 Oe,V 300cm/s. Thus,the instability e
c tion dV/dF < 0 can be met with difficulty in contrast
≈ e
mayappearunderthe ratherstrictcondition(42)andat
to dV/dF < 0. The instability of the lattice motion re-
thevelocitiesoftheorderof2V 103sm/s. Hence,both
c garding nonuniform perturbations has been found when
∼
the necessary and sufficient conditions can be fulfilled at
the valueof(-dF/dV)is greaterthanthe combinationof
lowviscosity,β H,relativelysmallmagneticfieldsand
the elastic modulii c and c , while the more strict re-
≪ 11 66
quite high velocities of the vortex lattice motion.
quirement dV/dF <0 is still not fulfilled. It is worthto
e
At small velocities another effect, namely a nonlinear
note that the vortex lattice instability concerned in this
I-V characteristic of the superconductor emerges. This
paper has different origin than instabilities predicted in
effectdoesnotimposelimitsforsuperconductorparame-
Ref. 17.
ters. Itcanbeobservedatβ >H andatstrongmagnetic
fileds, when V V and no instability arises.
c trans
≫
At V V , F(V) dependence is monotonic and
c trans
≫
contains only part with F √V, V < V , F
∝ trans ∝ Acknowledgments
V, V >V . Inthiscase,thequadraticI-V character-
trans
isticisrealisedatV <V . Thiseffectcanbefoundat
trans
very small defect concentrationand rather dense lattice. The authors are indebted to V. G. Baryakhtar and
For instance, if C a3 = 10−6, V 10 cm/s when A. L. Kasatkin for stimulating discussions. This work is
imp trans
≈
H β and V 70 cm/s for H β at B =103Oe. supported by INTAS Foundation grant No 97-31311.
trans
≫ ≈ ≪
∗ Electronic address: [email protected] Rev.Lett. 73, 1711 (1994).
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