TOPOLOGICAL COUPLING

OF DISLOCATIONS AND

VORTICES

IN SPIN DENSITY WAVES

N. Kirova, S. Brazovskii

Laboratoire de Physique Théorique et Modèles Statistiques,

CNRS & Université Paris-Sud, Orsay, FRANCE.

Introduction

Sliding Charge/Spin Density Waves generate the Narrow Band Noise (NBN) a coherent periodic unharmonic signal with the fundamental frequency W being proportional to the mean dc sliding current j with the universal ratio W /j [1,2].

Ø In CDW

W /j= p two electrons per CDW wave length l.

Ø In SDW

W /j = 2p if only electron density is

involved

W /j = p if spins are relevant.

Competing models:

The Wash-Board Frequency (WBF) model [3]:

NBN is generated extrinsically while the DW modulated charge passes through the

host lattice sites or its defects.

The Phase Slip Generation (PSG) model [4,5]:

the NBN is generated by the phase slips occurring near injecting contacts.

CDW and SDW order parameters:

h_CDW =h₀cos(Qx+j)

h_SDW =mh₀cos(Qx+j)

m - the unit vector of the staggered magnetization.

Oscillating charge densities:

CDW: r_CDW =|h₀| cos(Qx+j)

SDW: r_SDW =|h₀|² cos(2Qx+2j)

WBF model:

W_CDW =-¶j/¶t=pj

W_SDW =-2¶j/¶t=2p

PSG model:

W_CDW =W_SDW =pj

Difficult to explain the PS regularity.

Problems within WBF model :

The interaction between the rigid DW and the regular host lattice V_host ~cos(nj) would give an e.g. n=4 - fold WBF contrary to experiments!

The interaction between the rigid DW and the host impurities V_imp~cos(j+Qx_i) Ô the positionally random phase shifts ~Qx_i will prevent any coherence in generation.

The only possibility left is to suppose that the DW does not slide at the sample surface Ô the coupling ~cos(j_bulk - j_surf ) would provide a necessary WBF.

We show that the SDWs allow for p PSs forbidden in CDW which do provoke a double frequency. The rich order parameter of Spin Density Waves allows for an unusual object of a complex topological nature: “Hymers” composed by a half - integer dislocation combined with a semi - vortex of the staggered magnetization. They become energetically favorable due to enhanced Coulomb interactions. Their generation changes the NBN frequency.

Principal points:

· Non-local elastic theory due to Coulomb interactions.

· Staggered magnetization vortices and domain walls in SDW

· Combined topological defects in SDW – half dislocation coupled to semi-vortex

· Double core dislocation + magnetic domain wall for the spin-orbital coupling

Three types of self mapping for the SDW order parameter h_SDW ®h_SDW

· normal dislocation, 2p translation:

j®j+2p, m®m

· normal m - vortex, 2p rotation:

m® O_2p m, j®j

· combined object :

j®j+p, m® O_p m = -m

Non-local elastic theory for Density Waves.

W{j} = h v_F S |j_k|² [k_||²+ak_^²+k² k_||²/( k_||²+k_^²+r_scr^-2)]

k² = 8p²g/ a_^²~ w_p²/v_F² ~ 1A

g = e²/(h v_F)

r_scr^-2 = (4pe²/a_^²)( ¶n/¶m) = k² r_n

If there were no Coulomb interactions:

W{j} = h v_F S |j_k|² [k_||²+ak_^²]

a - the interchain coupling parameter

Well screened Coulomb interactions :

W{j} = h v_F S |j_k|² [k_||²k² r_scr^-2+ak_^²]

123

r_n^-1

Elastic theory in space (x, r_^/ (r_n a)^1/2 ).

Stretched coordinates r_^~ x(r_n a)^1/2

Non screened Coulomb interactions (within the screening volume)

W{j} ~ [(a/L_^²)+( +k² L_^² / L_||²)] L_|| L_^² ~ N

Minimum over L_|| ; L_^² ~ L_|| a_^/g

W{j} = g E₀N

Not usual perimetrical (N^1/2lnN) but area law (~N)

At large distances the standard (N^1/2lnN) law is restored but enhanced as ~r_n^–1/2 .

The chemical potential of dislocations m₀ controls its equilibrium with respect to aggregation of electrons to DLs.

Splitting of dislocations in SDW.

Energy of the vortex with the winding number n

W_m~T_cr_sn²

Energy of the dislocation:

W_j~T_c(r_s/r_n)n²

In general if n®2(n/2) then W®W/2

Only smallest n are stable

T~Tc : W_j ~ W_m all energies are comparable

· Normal dislocation

· Half-dislocation combined with semi-vortex

· Normal magnetic vortex

Result depends on numbers.

T<<Tc : W_j >> W_m

· Half-dislocation combined with semi-vortex – obligatory decoupling of the dislocation

n_j=1/2, n_m=1/2

n_j=1 W = ( W_j +W_m)/2~ W_j/2

n_j=1/2, n_m=1/2

Half dislocation + semi-vortex

Spin – orbital coupling (Anisotropy)

Spin anisotropy → the free rotation of spins is prohibited. The two objects will be bound by a string which is the Neel domain wall.

Spin-flop field H_s-f ~ 1T originate the string of the length ~0.1mm.

At higher magnetic fields only a small in-plane anisotropy is left so that the string length may

reach the sample width.

W_m =v_F[(¶_im_a)² +l^-2 m_z² ]

l - the DW width is determined via spin-flop field.

v_F/l ~m_bH_s-f H_s-f ~1T

p semi-vortex ® 180^o domain wall

W_DW ~1K/chain, l~10x₀ ~10⁴A

Splitting of the isolated dislocation line ®

double cores dislocation,

i.e. two half dislocations confined by the string of the180^o domain wall .

r<r_scr :

Energy lost W_m= W_DW N , W_DW~1K/chain

Energy gain W_Disl. = -E₀N/2,

E₀ > W - constant repulsion wins against constant

attraction

r>r_scr :

Energy lost W_m= W_DW N

Energy gain W_Disl. = -(E₀/r_n^1/2)lnN+ W_DW N

Equilibrium distance between half dislocations

N~ E₀/ (r_n^1/2W_DW )

Conclusions

The sliding SDW should generate "hymers":

the combined topological objects where the spin rotations are coupled to the DW displacements.

This combination effectively reduces the SDW period allowing for the twice increase in the NBN frequency. The "chymers" are stable lowering the DL Coulomb energy.

In SDWs a normal dislocation must split into two objects of the combined topology. They will have the same sign of the displacive half-integer winding numbers and opposite signs of the half-integer spin rotation numbers.

In presence of spin anisotropy the free rotation of spins is prohibited at large distances from the DL. Then the two objects are bound by a string which is the Neel domain wall. Usually the spin anisotropy is noticeable only in one direction as characterized by the Spin-Flop field ~1T . Above the spin-flop magnetic field only a small in-plain anisotropy is left so that the string length may reach the sample width.

The interest in such unusual topological objects may go far beyond the NBN generation or the current conversion problem in SDWs.