Spin modes in electron Fermi liquid of organic conductors

The propagation of spin waves in Q2D layered conductors placed in a magnetic field is studied. It is shown, that at certain orientations of the magnetic field with respect to the layers the collisionless absorption is absent and weakly damping spin waves can propagate even under the strong spatial...

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Datum:	2007
Hauptverfasser:	Peschansky, V.G., Stepanenko, D.I.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Schriftenreihe:	Физика низких температур
Schlagworte:	International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application"
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spelling	irk-123456789-1209352017-06-14T03:06:36Z Spin modes in electron Fermi liquid of organic conductors Peschansky, V.G. Stepanenko, D.I. International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" The propagation of spin waves in Q2D layered conductors placed in a magnetic field is studied. It is shown, that at certain orientations of the magnetic field with respect to the layers the collisionless absorption is absent and weakly damping spin waves can propagate even under the strong spatial dispersion. We have analyzed the spectrum of spin modes at an arbitrary form of Landau–Silin correlation function. 2007 Article Spin modes in electron Fermi liquid of organic conductors / V.G. Peschansky, D.I. Stepanenko // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1027–1031. — Бібліогр.: 9 назв. — англ. 0132-6414 PACS: 72.15.Nj http://dspace.nbuv.gov.ua/handle/123456789/120935 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application"
spellingShingle	International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Peschansky, V.G. Stepanenko, D.I. Spin modes in electron Fermi liquid of organic conductors Физика низких температур
description	The propagation of spin waves in Q2D layered conductors placed in a magnetic field is studied. It is shown, that at certain orientations of the magnetic field with respect to the layers the collisionless absorption is absent and weakly damping spin waves can propagate even under the strong spatial dispersion. We have analyzed the spectrum of spin modes at an arbitrary form of Landau–Silin correlation function.
format	Article
author	Peschansky, V.G. Stepanenko, D.I.
author_facet	Peschansky, V.G. Stepanenko, D.I.
author_sort	Peschansky, V.G.
title	Spin modes in electron Fermi liquid of organic conductors
title_short	Spin modes in electron Fermi liquid of organic conductors
title_full	Spin modes in electron Fermi liquid of organic conductors
title_fullStr	Spin modes in electron Fermi liquid of organic conductors
title_full_unstemmed	Spin modes in electron Fermi liquid of organic conductors
title_sort	spin modes in electron fermi liquid of organic conductors
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2007
topic_facet	International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application"
url	http://dspace.nbuv.gov.ua/handle/123456789/120935
citation_txt	Spin modes in electron Fermi liquid of organic conductors / V.G. Peschansky, D.I. Stepanenko // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1027–1031. — Бібліогр.: 9 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT peschanskyvg spinmodesinelectronfermiliquidoforganicconductors AT stepanenkodi spinmodesinelectronfermiliquidoforganicconductors
first_indexed	2025-07-08T18:53:33Z
last_indexed	2025-07-08T18:53:33Z
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fulltext	Fizika Nizkikh Temperatur, 2007, v. 33, No. 9, p. 1027–1031 Spin modes in electron Fermi liquid of organic conductors V.G. Peschansky and D.I. Stepanenko B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: stepanenko@ilt.kharkov.ua Received January 18, 2007 The propagation of spin waves in Q2D layered conductors placed in a magnetic field is studied. It is shown, that at certain orientations of the magnetic field with respect to the layers the collisionless absorp- tion is absent and weakly damping spin waves can propagate even under the strong spatial dispersion. We have analyzed the spectrum of spin modes at an arbitrary form of Landau–Silin correlation function. PACS: 72.15.Nj Collective modes (e.g., in one-dimensional conductors). Keywords: magnetic field, spin waves, quasi-two-dimensional electron energy spectrum. A large family of tetrathiafulvalene-based ion-radical salts of the form (BEDT- TTF) X2 (X stands for a set of various anions) possess layered structure with a pro- nounced anisotropy of electrical conductivity. Observa- tion of Shubnikov–de Haas magnetoresistance oscilla- tions [1,2] in a magnetic field about 10 T, prove that the free path time � in these layered conductors can be suffi- cient for charge carriers to manifest their dynamic proper- ties and their cyclotron frequency�B may exceed signifi- cantly ��1. Under that condition, varios type of weakly attenuating collective mode may exist in Fermi liquid of conduction electrons. At present paper we study the col- lective modes arising from the oscillations of the spin density in layered structures. The paramagnetic spin waves in quasi-isotropic metals were predicted by Silin [3] and observed in alkaline metals by Dunifer and Schultz [4]. The wave processes in layered conductors are characterized by a number of features associated with the quasi-two-dimensional electron energy spectrum. The electrons energy �( )p depends weakly on the momentum projection p z � pn on the normal n to the layers and can be represented in the Fourier series with respect to p z : � � � �( ( , ) ( , , ) cosp) � � � � � � �0 0 p p p p np p x y n n x y z . (1) Here � � � �n x y n x yp p p p� ��1( , , ) ( , , ). The coefficient at the first harmonic � �1( , , )p px y is of the order of �� F F�� where �F is the Fermi energy, � is the parame- ter of the quasi-two-dimensionality of the spectrum, � is the Planck constant. The ratio of the conductivity across the layers to the in-plane conductivity in the absence of a magnetic field, is about �2. We assume that in an external magnetic field B 0 0� � �( sin , , cos )B B0 0 cross-sections S pF B( , )� of the Fermi surface by the plane pB � � �( )pB 0 /B0 const are closed for � �/2 �� . Kinetic properties of the system of fermions should be described by means of the kinetic equation for the density matrix �� and the Maxwell equation. In the quasiclassical case when ��B � T F�� the quantization of the charge carriers energy in the magnetic field does not affect essen- tially the magnetization M (T is the temperature). Under these conditions the density matrix can be presented as an operator in the space of spin variables and as a function depending on coordinates and momentum. The Fermi liq- uid interaction between electrons can be described with the aid of the Landau-Silin correlation function [5,6] �( , � , , � ) ( , ) ( , ) � �p p p p p p� � �� L S , (2) where � are Pauli matrices. The second term on the right-hand part of (2) corresponds to the exchange inte- raction of electrons. The closed electron orbits in momentum space almost the same for different values of the momentum projection pB . So the area S pF B( , )� of the section of the Fermi sur- face by the plane pB � const and the components vx and vy of the velocity of conduction electrons in the plane of the layers, depends weakly on pB , with the order of © V.G. Peschansky and D.I. Stepanenko, 2007 smallness � tan �. This results that the electrons energy and the Landau correlation function can be expanded into the asymptotic series about �, the leading term of the ex- pansions being not dependent on pB . In the main approxi- mation in the small parameter � the functions L( , )p p� and S( , )p p� is independent of pB and can be presented as � � L L S S n n F in n n F i ( , ) ( ) , , ( ) ( ') p p p p' � � � � �� e e n( ') .� �� (3) We have chosen the integrals of motion of an electron in a magnetic field � and pB and the phase� �� B t1 at its orbit in the magnetic field as variables in the p space. Here t1 is the time of motion along the trajectory � �( ) ,p � F pB � const. Because of the symmetry of the function �( , � , , � )p p� �� with respect to its arguments, the coefficients in (3) satisfy the condition L Ln n� � , S Sn n� � . Allowing for next-order terms of the expansion for the correlation function about � does not lead to the noticeable correction of the results. The paramagnetic spin waves represent the high-fre- quency collective modes for which� � �� 1 1 2 1, where �1 and � 2 are the relaxation times for the electron momen- tum and spin density respectively. They result from the oscillations of the electron spin density g r p( , , )t � � Tr� �( � � )� . The function g should be presented as a sum of the equilibrium spin density g B0 0 0� � � �� ( )f / and the nonequilibrium correction � �� ( ) , ,f / t0 � � r p , where f 0( )� is the the Fermi function, � �� 0 01/ S( ), �0 is the magnetic momentum of an electron, S Sn F n � � � �( ) , � �( )F is the density of states at the Fermi level. The components � � �( ) � x yi 1 of the re-normal- ized nonequilibrium correction � � � �� ! " #S � 2 2 3d p' � � �$� % � � � � � � � � f S t0( ' ) ' ( , ) ( , , ) � � &p p r p to the spin density satisfy the integral equation [7,8] � '( ) exp (~ ( ) �� %d i d p B B� � � � � ( � � � kv � % ) ) � � � � � �� i p i B B B p p p ip� � � � � * �0 ( )B ekv( ' , ) ( ) '' � � , (4) where � � �x x z1 � � � �cos sin , the x1 axis is orthogonal to both the y axis and vector B 0 1, ( )~ ~* p p pS / S� � , � �p ip /! " ! "e – � 1 , ~ , ( )� � �� i / Ss0 1 0' , �s � � �2 0 0� B /� is the frequency of spin paramagnetic reso- nance v p p� � ��( ) / is the electron velocity. The wave pro- cess is supposed to be harmonic, and we present the coor- dinate and time dependencies of all variable quantities in the form exp ( )i i tkr – � . For the frequencies �� kc, the alternating magnetic field B B iBx y � ~ 1 , produced by the spin oscillations is determined from the equation B k M k k kM k ~ ~~( , ) [ ( , ) ( ( , ))]� � � �� 4 2k , (5) where M k p k ~ ( , ) ( , , )� � �� ! "0 � is the high-frequency magnetization, c is the velocity of light. After multiplying the equation (4) by e�in� and then in- tegrating with respect to+ � �p /pB 0 cos and �, we obtain the infinitesimal set of linear equations for the coeffi- cients � �n Fd d( ) ( ) ( ) ( , , ) � � % % 1 2 2 0 2 � � + � + � � � � , * � � + +np p B np p pf� ! " � � � � � �� ( ) � � � � � � % � � � � � + � � � � � � 0 1 0 2 1 1 2 B i d d in i B ~ ( ( , ) ) exp ~ kv � � ' ' B B iR i F � � � � � � � � 1 1 0 2 1 2 � - . / 0 1 2 � ! � " - . / 0 1 2 # % ( , ) exp ~ �' kv n , (6) f i d d i p n ip i iR np B ( ) exp ( ) ~ ( + � � � � � � � � � � � � � � �% 1 2 1 0 2 1 1 �' , ) exp ~ � � � � � � 1 0 2 1 2 - . / 0 1 2 � ! � " - . / 0 1 2 % i B �' kv . (7) 1028 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 V.G. Peschansky and D.I. Stepanenko Here R d B ( , ) ' ( , ' ) ,� � � � + � � � � 1 1 1 # � % kv ! " � ! " �% % � ... ..., ... ...,� 3� + � � � � � + 1 2 1 2 0 d d , np is the Kroneker symbol. The dependence of the cyclo- tron frequency on pB should be taken into account in the expression k v /x x B� in the exponent only provided that �kvF � �B where vF is the Fermi velocity. The coefficients of the Fourier series for the smooth function S ~( , )p p� decrease significantly with their num- ber increasing, so we will restrict ourselves to account of a finite number of terms. Making use of this equation it is easy to obtain the magnetic susceptibility, taking into ac- count time and spatial dispersion 4 � � �� ( , ) ( , ) ( ) ( , )~ ~ ( ) ~ k M B B F k k� 0 � � � ! " � � 4 , , * � � + , , * + 0 0 01det [ ( ( ) )( )] det [ p n np p B np p np F f p B npf � � + +! "( ) ] . (8) The properties of the spin waves are determined by the magnetic susceptibility tensor 4 �ik k( , ). For the frequen- cies, that do not coincide with the frequency of eigen-os- cillations of the spin density, the components 4 �ik k( , ) is of the order of the static paramagnetic susceptibility 4 � � �0 0 2 610� ( ) ~F � . For this reason, in order to find the spectrum of the spin waves, it suffices to use the homoge- neous system of equations corresponding to (6). In the ex- pression (6) we can neglect the small non-uniform correc- tion proportional to �0B ~ which allows for influence of the self-coordinated field B ~. The dispersion equation for «free» oscillations of the spin density is of the form D fnp p B np( , ) det ( )� , * � � + +k # � ! " - . / 0 1 2 � 0 . (9) The frequency of eigen-oscillations of the magnetization to within the terms proportional to 4 � � �0 0 2~ ( )F coin- cides with the frequency of the spin density «free» oscil- lations. At this frequency the magnetic susceptibility has a sharp maximum, and the determinant D( , )� k is of the order of 4 0. The collisionless absorption of the spin waves is ab- sent if the following inequality is satisfied 5 5� � �� ! "n B �' max \| \|kv . (10) In the opposite case the integrand in the formula (7) has a pole and after integration with respect to pB the dispersion equation acquires an imaginary part responsible for the strong absorption of the wave. In a layered conductor the velocity v vB � ! " � of drift along the magnetic field of con- duction electrons is an oscillatory function of the angle � between B0 and the normal to the layers. For certain values of the angle � � � i the velocity v B is close to zero. At � � � i the collisionless absorption is absent and existence of collective modes is possible [7,8] even under the condi- tion �kvF � �B . For � and k such that kvF B�� ( , )� � , the solution of the dispersion equation (9) takes the form � � � � �� n nB B1 1 0 1 2' 6 6, , , , � (11) The correction to the resonance frequency may be written as 6 ' � � � 7� n k r B x 1 0 , (12) where r v /F B0 # � and 7 i are roots of the equation 5 5det ( ), * 7 + +np p npI� ! " ��1 0 , (13) I iR i n p ip np ( ) ( ) exp ( , ) ( )( ) ( ) ( ) ( + 8 � � � � 9 9 9 9 9 9 � � � � � � � 1 1 ) ( ) ( ) \|det ( ( , ))\| �- ./ 0 12 �� i s R � � �� 9 9 4 1 1 . (14) The summation in the formula (14) should be carried out over all stationary points � ( ) ( ) ( ) ( , )9 9 : 9� �� which are de- termined from the equations v vx x( ) , ( )� � �� 0 01 . Here 8 9( )( )� �1if the stationary point is inside the domain of in- tegration 0 2 0 2 1 � � � �� 9 9( ) ( ) , and 8 9( )( )� � 1 2/ if it is located on a boundary of the domain, s R� �� sign �� 9 9� � 1 1 ( , )( ) ( ) � �� ( ) ( ),R R 1 1 �� ( )��R 1 and � �� ( )R 1 are the numbers of positive and negative eigenvalues of the matrix Spin modes in electron Fermi liquid of organic conductors Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1029 �� # � � � R R �� 9 9� � � �1 2 1 1 ( , )( ) ( ) , respectively [9]. At kr kvF0 1�� , ,� ' the phase of exponent in (7) has no stationary points. Integrating (7) by parts and sub- stituting the result (9) we obtain follow asymptotic ex- pression for the spectrum of spin mode � 7� i Fkv , (15) where 7 i are the root of equation det ( , ) ( ) ( ) , , * 7 + � � + � np p i n p F/ kv � � - . / / 0 1 2 2 � � � e 1 0 1 kv , ! " � � %%� �+� � �� + 1 2 2 0 2 ( ) .d d Confining ourselves to account of two terms in the for- mula (1) and neglecting the anisotropy in the layers plane, we have � 0 2 2 2( ) ( )p � �p p / mx y , � �1 0( )p � v pF , where v /mF F 2 2� � , m is the electron effective mass in the layers plane. Assuming that the wave vector k � ( , , )k kx z0 is ori- ented in the xz plane, one can obtain, at kr0 1�� , �� kvF , following asymptotic expression for the coefficients f np f k r n pnp x B ( ) ( ) cos [( ) ]cot ~ ( ) + � , � � � + � � � � ! � "� 1 10 2 �' kv � � � ; < = >= � ! � " � � � %sin ( ) ~ ( ) ~ ( ) ( 1 2 � + � � � � + , � � + , , � B B B d nkv + kv� �' ' � - . / / 0 1 2 2 ! � "� � � ? @ = A= �p B ) sin ~ ( ) , � � � + �1 �' kv . (16) Here , � �� arccos (~ ) ,� ' / k rB x 0 � � �� (~ ) ,� ' / k rB x 0 � + � � 9 +B B( ) ( tan ( ) cos )� � �1 1J is the cyclotron frequency of quasi-particles in the first order approximation in � , J n ( )9 is the Bessel function. For the direction of magnetic field in which 9 � ( ) tanmv /pF 0 � is equal one of the zeros of J 0( )9 , the average ! " � � �kv � � 9 +v k kF x zJ 0( ) ( tan ) sin (17) is of the order of �2 and the Landau absorption is absent. Under condition � � ' �� k vx F , the spin waves with the fre- quencies close to the resonance frequencies � �r Bn� ' can propagate in layered conductors at an arbitrary orienta- tion of the wave vector and selected directions of the external magnetic field. The solution of (9) can be represented in the form (11),(12) where 7 should be determinated from equation det cos ( ) ( ) sin ( ) ( ), * 7 � � np p n n p R n p� � � � � � �� 1 2 1 2 1 + � � 0 , R d H ( ) ( ) ( ) / / � � � % 1 2 2 � + � � � � kv . (18) Let us consider the propagation of spin waves along the magnetic field direction in the case when the magnetic field B 0 00 0� ( , , )B is orthogonal to the conducting lay- ers. The integral equation (4) takes form � ' ' ( ) ~ ( ) ~ � �� 0B k v k v z z z z � � � � � �� * � � � p p ip z z Bp k v p � ' ( ) ~ e , (19) and we can obtain the simple analytical expression for spin wave spectrum at an arbitrary values of �k vz F . De- termine ! "�� ( ) ,+ � from (19), we find the high-frequency magnetic susceptibility 4 � 4 � � � � � � � ( , ) ( ) ( ) ( ) ( ) ( ) k � � � � � � � 0 2 2 2 ' ' ' k v k v z F z F sign 2 0� �* � �sign ( )' . (20) The frequency of spin magnetization oscillations are given by � * � * * � � � � � ' '0 2 2 2 0 2 0 2 1 1 ( ) ( ) ( ) k vz F . (21) In long-wave-length limit the frequency of spin waves co- incides with frequency of spin paramagnetic resonance�s . The specifics of Q2D electron energy spectrum of layered conductors lead that in main approximation in the para- meter of the quasi-two-dimensionality �, the Lan- dau–Silin correlation function can be presented as Fou- 1030 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 V.G. Peschansky and D.I. Stepanenko rier series (3), the coefficients of which are independent of the momentum projection pB on the magnetic field direction. This circumstance simplifies essentially the in- tegral equation for spin density and makes it possible to obtain the dispersion equation for rather general form of the correlation function. 1. M.V. Kartsovnik, Chem. Rev. 104, 5737 (2004). 2. J. Singleton, Rep. Prog. Phys. 63, 1111 (2000). 3. V.P. Silin, Zh. Eksp. Teor. Fiz. 35, 1243 (1958). 4. S. Schultz and G. Dunifer, Phys. Rev. Lett. 18, 283 (1967). 5. L.D. Landau, Zh. Eksp. Teor. Fiz. 30, 1058 (1956) [Sov. Phys. JETP 3, 920 (1956)]. 6. V.P. Silin, Zh. Eksp. Teor. Fiz. 33, 495, (1957) [Sov. Phys. JETP 6, 387 (1958)]. 7. D.I. Stepanenko, Fiz. Nizk. Temp. 31, 115 (2005) [Low Temp. Phys. 31, 90 (2005)]. 8. O.V. Kirichenko, V.G. Peschansky, and D.I. Stepanenko, Zh. Eksp. Teor. Fiz. 126, 1435 (2004) [JETP 99, 1253 (2004)]. 9. M.V. Fedoruyk, Asymptotic. Integral and Series, Nauka, Moskow (1987) [in Russian]. Spin modes in electron Fermi liquid of organic conductors Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1031

Spin modes in electron Fermi liquid of organic conductors

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