DSpace at VNU: Thickness dependent properties of magnetic ultrathin films

Thickness dependent properties of magnetic ultrathin filmsBach Thanh Conga, Pham Huong Thaoa,b,n a Faculty of Physics, Hanoi University of Science, VNU, 334 Nguyen Trai, Hanoi, Viet Nam b

Thickness dependent properties of magnetic ultrathin films

Bach Thanh Conga, Pham Huong Thaoa,b,n

a

Faculty of Physics, Hanoi University of Science, VNU, 334 Nguyen Trai, Hanoi, Viet Nam

b

Faculty of Physics, Hue College of Education, 32 Le Loi, Thua Thien Hue, Viet Nam

a r t i c l e i n f o

Article history:

Received 27 March 2013

Received in revised form

24 May 2013

Accepted 30 May 2013

Available online 20 June 2013

Keywords:

Thin film

Functional integral method

Magnetic properties

a b s t r a c t The dependence of magnetic properties on the thickness of few-layer thinfilms is investigated at finite temperature using the functional integral method for solving the Heisenberg spin model The temperature dependence of the ultra-thinfilm's magnetization and Curie temperature are calculated

in terms of the meanfield theory and of the Gaussian spin fluctuation approximations It has been shown that both Curie temperature and temperature interval, where the magnetization is non-zero, are strongly reduced with the thickness reduction by using the spinfluctuation approximations in comparison with the meanfield results Curie temperature dependence on the film thickness calculated numerically well agrees with the experimental data for Ni/Cu(1 0 0) and Ni/Cu(1 1 1) ultrathinfilms

& 2013 Elsevier B.V All rights reserved

1 Introduction

Size dependent effects of ferroelectric and ferromagnetic

industrial importance The most important problem is to

deter-mine the size intervals where the long range order parameters of

considerably change, compared with those of the bulk one It was

thickness is reduced to several atomic layers, which has been

observed in the ferromagnetic (FM) transition[1,2], and rare earth

metalfilms[3]

Several theories have been developed to explain the

approximations The analytical methods using famous Heisenberg,

[8,9

In the present paper, we study the magnetic properties of the

been taken into account The FIM has been developed successfully for different bulk spin models[10,11]; hence it would be intrigued

This paper is organized as follows: the general FIM for magnetic films is described in the next section Numerical results produced

comparison between numerical and experimental results

of n spin layers, where the spin numbers N in a layer are practically

infinite (Fig 1) The coordinate system is chosen as follows: Oz is

indexesνj, where ν is the layer (spin plane) index ðν ¼ 1; …; nÞ and

written as

vj

Sz

νj−1

νj;ν′j′

the second term is the usual Heisenberg exchange interaction

Jνν′ðRj−Rj ′Þ denotes the exchange integral between Sνjand Sν′j′spins, which is considered as the matrix element of the matrix JðRj−Rj ′Þ

Physica B

0921-4526/$ - see front matter & 2013 Elsevier B.V All rights reserved.

n Corresponding author at: Faculty of Physics, Hanoi University of Science, VNU,

334 Nguyen Trai, Hanoi, Viet Nam Tel.: +84 976693644.

E-mail address: hthao82@gmail.com (P.H Thao)

thefirst term, we have

v′j′Jνν′ðRj−Rj′Þ〈Sz

ν′j′〉

!

Szνj

þ ∑

νj;ν′jJνν′ðRj−Rj′Þ〈Sz

νj〉〈Sz ν′j′〉−12 ∑

νj;ν′j′

α

Jνν′ðRj−Rj′ÞδSανjδSαν′j′

δSz

νj¼ Sz

νj−〈Sz

νj〉; δSx

νj¼ Sx

νj; δSy

νj¼ Sy

νj; α ¼ x; y; z

Fourier transformations for spin operators, which is

ν;ν′Jνν′ð0Þ〈Sz

ν〉〈Sz

ν′〉−∑

vj

Hint¼ −1

ν; ν′; k

α

Jνν′ðkÞδSα

νðkÞδSα

where

gμhν¼ gμh þ ∑

ν′Jνν′ð0Þ〈Sz

ν′〉

Jνν′ðkÞ ¼ ∑

j −j′

Jνν′ðRj−Rj ′Þexp ½ikðRj−Rj ′Þ

and

δSα

νðkÞ ¼ Sα

νðkÞ ¼p 1ffiffiffiN∑

j

Sανjexp ½−ikRj for α ¼ x; y

δSz

νðkÞ ¼p 1ffiffiffiN∑

j δSz

νjexp ½−ikRj ¼ Sz

νðkÞ−δðkÞN1 =2〈Sz

ð3Þ

In Eqs.(2) and (3), kðkx; kyÞ is the two-component wave vector

relating to the translational symmetry in spin layers All spin sites

in the same plane are equivalent due to this translational

sym-metry, then〈Sz

νj〉 ¼ 〈Sz

ν〉 The statistical operator of the system is given in the interaction representation using the temperature

interacting HintHamiltonians (see Eqs.(2b)and(2c)):

0

HintðτÞdτ

ð4Þ

F0¼ −1βlnðSpe−βH0Þ ¼N

νν′Jνν′ð0Þ〈Sz

ν〉〈Sz ν′〉−NβlnshðS þ 1=2Þyν

ν 0Jνν0ð0Þ〈Sz

ν′〉

ð6cÞ

φαðqÞ ¼ 1

β1=2

Zβ

0 φαðk; τÞexp −iωτ½ dτ; ω ¼ 2πm=β; m ¼ 0; 71; 72; …

ð7Þ

is defined by[10]

ðdφÞ ¼ ∏

αν

−∞

dφα

νð0Þ ffiffiffiffiffiffi

2π

q≠0

−∞

dφα;c

ν ðqÞ ffiffiffi π

−∞

dφα;s

ν ðqÞ ffiffiffi π

φα

νðqÞ are denoted by φα;c

ν ðqÞ and φα;s

ν ðqÞ respectively

φα

νðqÞ ¼ φα;c

ν ðqÞ þ iφα;s

interacting functional which has an explicit form as

ν;ν′;l;qβ1 =2J1=2νν′ðkÞφl

νðqÞδSl

ν 0ðqÞ

0

ð10Þ

〈…〉0¼ Spðe−βH 0…Þ=Spðe−βH 0Þ; l ¼ 7; z;

2 φx

νðqÞ∓iφy

νðqÞ

δS7

ν ðqÞ ¼ S7

ν ðqÞ ¼ ½Sx

νðqÞ7iSy

νðqÞ

δSz

νðqÞ ¼ Sz

νðqÞ−δðqÞN1 =2〈Sz

The logarithmic functional in Eq.(10)is readily presented in the series form which is

−FintðφÞ ¼ ∑∞

m ¼ 2

1

m!ν1;ν′ ∑

1 ;l 1 ;q 1 ;…ν m ;ν ′

m ;l m ;q m

β1 =2J1ν=2

1 ν ′

1ðk1Þ⋯β1 =2J1ν=2

m ν ′

mðkmÞ

φl 1

ν 1ðq1Þ⋯φl m

ν mðqmÞ〈^TδSl 1

ν 0

1ðq1Þ⋯δSl m

ν ′

mðqmÞ〉ir

The symbol〈…〉ir

operators over non-interacting equilibrium statistical ensemble

Fig 2 Here we use the Gaussian approximation, where the functional

FintðφÞ has the simple form as

−FintðψÞ ¼12β ∑

ν;k;ω 1 ;ω 2

fψz

νðk; ω1Þψz

νð−k; ω2Þb′ðyνÞδðω1Þδðω2Þ þ4δðω1þ ω2ÞK−ωðyνÞbðyνÞψ−

νðk; ω1Þψþ

Fig 1 A spin position is defined in a spin lattice by a plane index νðν ¼ 1; 2; :::; nÞ

and by a two-dimensional lattice vector Rj(denoted shortly by νj) J s is the nearest

neighbor (NN) in-plane exchange integral of the surface layers; J are the NN

in-plane exchange integral of interior layers and the NN out of plane exchange

integral of adjacent layers.

νðqÞ ¼ ∑

ν′J1νν′=2ðkÞφl

νðqÞ; KωðyνÞ ¼ 1

where bðyνÞ ¼ ðS þ ð1=2ÞÞcthðS þ ð1=2ÞÞyν−ð1=2Þcthðyν=2Þ and b′ðyνÞ

mean the Brillouin function and its derivative Calculating the

ν;ν′

Jνν′ð0Þ〈Sz

ν〉〈Sz

ν′〉−N

β∑ν lnshðS þ 1=2Þyν shðyν=2Þ

2β∑k

lndet j ^1− ^AðkÞj þ1β∑

k ;ω

^AðkÞ and ^Bðk; ωÞ are n  n matrices

^AðkÞ ¼

βb′ðy1ÞJ11ðkÞ βb′ðy1ÞJ12ðkÞ βb′ðy1ÞJ13ðkÞ ⋯ βb′ðy1ÞJ1nðkÞ

βb′ðy2ÞJ21ðkÞ βb′ðy2ÞJ22ðkÞ βb′ðy2ÞJ23ðkÞ ⋯ βb′ðy2ÞJ2nðkÞ

βb′ðy3ÞJ31ðkÞ βb′ðy3ÞJ32ðkÞ βb′ðy3ÞJ33ðkÞ ⋯ βb′ðy3ÞJ3nðkÞ

βb′ðynÞJn1ðkÞ βb′ðynÞJn2ðkÞ βb′ðynÞJn3ðkÞ ⋯ βb′ðynÞJnnðkÞ

0

B

B

B

@

1 C C C A

ð15Þ and

^Bðk; ωÞ ¼

βbðy 1 ÞJ 11 ðkÞ

y1−iβω βbðyy11−iβωÞJ12ðkÞ βbðyy11−iβωÞJ13ðkÞ ⋯ βbðy 1 ÞJ 1n ðkÞ

y1−iβω βbðy 2 ÞJ21ðkÞ

y2−iβω βbðyy22−iβωÞJ22ðkÞ βbðyy22−iβωÞJ23ðkÞ ⋯ βbðy2 ÞJ2nðkÞ

y2−iβω βbðy 3 ÞJ31ðkÞ

y3−iβω βbðyy33−iβωÞJ32ðkÞ βbðyy33−iβωÞJ33ðkÞ ⋯ βbðy3 ÞJ3nðkÞ

y3−iβω

βbðy n ÞJ n1 ðkÞ

yn−iβω βbðyynn−iβωÞJn2ðkÞ βbðyynn−iβωÞJn3ðkÞ ⋯ βbðy n ÞJ nn ðkÞ

yn−iβω

0

B

B

B

B

B

1 C C C C C

Furthermore, one can introduce the correlation functions

time-ordered spin Green functions):

χz

ν 1 j 1 ;ν 2 j 2ðτ1; τ2Þ ¼ χz

ν 1 ν 2ðRj

1−Rj2; τ1−τ2Þ ¼ 〈^TðSz

ν 1 j1ðτ1Þ−〈Sz

ν 1〉ÞðSz

ν 2 j2ðτ2Þ−〈Sz

ν 2〉Þ〉

χx

ν 1 j1;ν 2 j2ðτ1; τ2Þ ¼ χx

ν 1 ν 2ðRj1−Rj 2; τ1−τ2Þ ¼ 〈^TSx

ν 1 j1ðτ1ÞSx

ν 2 j2ðτ2Þ〉

χy

ν 1 j1;ν 2 j2ðτ1; τ2Þ ¼ χy

ν 1 ν 2ðRj1−Rj2; τ1−τ2Þ ¼ 〈^TSy

ν 1 j1ðτ1ÞSy

ν 2 j2ðτ2Þ〉

ð17Þ The Fourier transformations of the spin correlation functions

have the following form:

χα

ν 1 ν 2ðR; τÞ ¼N1∑

k ;ω~χα

~χα

ν 1 ν 2ðk; ωÞ ¼1βZ β∑χα

R ¼ Rj1−Rj2 The local spinfluctuation is easily derived from Eqs

(17) and (18a)whenτ ¼ 0; R ¼ 0, which is

〈δS2

ν〉 ¼ ∑

αχα

ννð0; 0Þ ¼N1 ∑

α;k;ω~χα

Fourier image of the spin correlation function becomes

~χα

ν 1 ν 2ðk; ωÞ ¼ −1β J−1ν

1 ν 2ðkÞ−∑

νJ−1ν

1 νðkÞCα

νν 2ðk; ωÞ

ð20Þ where Cανν′ðk; ωÞ is the element of the matrix ^Cαðk; ωÞ and

^Cz

ðk; ωÞ ¼ ð ^1− ^AðkÞÞ−1; ^CxðyÞ

If we define mν¼ 〈Sz

ν〉 as a relative magnetization per site of the νth spin layer, then in the MF approximation, we have

mMFν ðβÞ ¼ 〈Sz

is taken into account, we obtain from Eqs.(17),(18a)and(18b)that

mSF

ν ðβÞ ¼ 〈Sz

ν〉 ¼ SðS þ 1Þ−N1 ∑

α;k;ω

ð~χα

ννðk; ωÞÞ

ð23Þ

In order to compare with experiments, an average relative (or

mðβÞ ¼μNnM ¼1

n ∑n

mSF, respectively

3 Numerical calculations

In this part, we use the nearest neighbor (NN) exchange

(Δ ¼ Rj−Rj′means in-plane NN spin lattice vector), and NN out-of-plane exchange integral Jν;νþ1ðΔÞ ¼ Jνþ1;νðΔÞ are nonzero In the

^AðkÞ ¼

βb′ðy2ÞJnðkÞ βb′ðy2ÞJ22ðkÞ βb′ðy2ÞJðkÞ ⋯ 0

0 B B B

@

1 C C C A ð25Þ

Fig 2 Diagram representation for the series (12) Here wiggly lines mean the longitudinal field component φ z

ν ðqÞ, cross lines (++++) and dashed lines (——) correspond to

φ þ

ν ðqÞ and φ −

ν ðqÞ respectively The other graphical elements (ovals, arrows etc.) are described in the Izuymov's monograph [12]

JsðkÞ ¼ 2Js½ cos ðkxaÞ þ cos ðkyaÞ for surface layersν ¼ 1; n

JννðkÞ ¼ 2J½ cos ðkxaÞ þ cos ðkyaÞ for interior layersν ¼ 2; …; n−1

ð27Þ

Jν;νþ1ðkÞ ¼ JðkÞ ¼ JexpðikzaÞ ð28Þ

Jνþ1;νðkÞ ¼ Jn

ν;νþ1ðkÞ ¼ JnðkÞ ¼ Jexp ð−ikzaÞ ð29Þ

b for a face centered cubic (FCC) thinfilm lattice

JsðkÞ ¼ 4Jscos k x a

cos ky a 2

for surface layersν ¼ 1; n

JννðkÞ ¼ 4J cos k x a

cos ky a 2

for interior layersν ¼ 2; …; n−1

ð30Þ

Jν;νþ1ðkÞ ¼ JðkÞ ¼ 2J cos kxa

2

2

exp ikza 2

ð31Þ

Jνþ1;νðkÞ ¼ Jn

ν;νþ1ðkÞ ¼ JnðkÞ ¼ 2J cos kxa

2

þ cos kya 2

exp −ikza 2

ð32Þ where a is a lattice constant of the thinfilm In the following

is used

lattice, in which Tb¼ ðSðS þ 1ÞZJÞ=3kB(Z is the number of NN

spins, for FCC spin lattice, Z¼ 12).ξ ¼ Js=J is the ratio

char-acterizing the difference of the surface and the interior

tempera-ture within the MF (SF) theory is obtained from a solution of

an equation mMFðτCÞ ¼ 0 (mSFðτCÞ ¼ 0), where τC¼ TC=Tb

In the next parts, numerical calculations are performed for

The magnetization is calculated in the approach of the MF

spin latticefilm using Eq.(22)

Fig 3shows the temperature dependence of the

ξ ¼ 1 It is clearly seen that the temperature interval, where the

However, layer magnetizations vanish at the same temperature,

At a given temperature, magnetizations of layers decrease

surfaces (Fig 4) for the cases of no-surface effect (ξ ¼ 1) and the

fi-ciently large (see theξ ¼ 2:2 curve inFig 4) There are two factors

originating this behavior: (i) the number of NN spins at surfaces is

internal exchanges are generally different

Below the Curie temperature, the magnetization increases with

for n ¼2 case)

MF calculation based on FIM is in good agreement with the results

the Curie temperature is subtracted with the deduction of the thin film thickness The analytic expression for TCwas also given in our

Fig 3 Temperature dependence of the magnetization of the FCC films with different thicknesses The surface exchange parameters are chosen as ξ ¼ 1, and S¼ 1.

0.70 0.75 0.80 0.85 0.90 0.95

mν

ξ=0.8 ξ=1 ξ=2.2

n (layer)

Fig 4 Dependence of magnetizations on layer position in the thin films with n¼5, S¼ 1 at τ ¼ 0:53.

0.0 0.2 0.4 0.6 0.8 1.0

mV

τ

ξ=0.8 ξ=1 ξ=2

Fig 5 Temperature dependence of the magnetization of a two-layer FCC film (m MF

1 ¼ m MF

2 ) calculated within the MF approximation for different surface exchange parameters ξ, and S ¼ 1.

Mermin and Wagner's theorem for one- or two-dimensional

For improving our MF results, we examine the temperature

dependence of the magnetization and of the Curie temperature

taking into account the SF, which is not treated in the usual MF

theory but it is quite obvious in the FIM

Curie temperatures are carried out within the Gaussian functional

dependence of magnetization with different thicknesses and

without the surface exchange effect (ξ ¼ 1) It is noticed that the

magnetization at zero temperature in the SF approximation isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

SðS þ 1Þ

p

≈1:4 for S ¼ 1 (Fig 7), which is different from 1 obtained

z spin component is treated

It is clearly seen that temperature ranges of nonzero

magneti-zations and the Curie temperatures obtained in the SF theory are

essentially smaller than those of the MF method (Fig 8)

This strong reduction proves the crucial role of the SF in the

the SF theory with the Gaussianfluctuations gives τSF

theorem This enhanced accuracy is due to the Gaussian

components are involved The higher order terms in the series

exchange

Fig 9shows the temperature dependence of the magnetization for

temperature and the magnetization strongly decrease when the SF

4 Comparisons with experimental results for nickel ultra-thin films

The experimental investigations on the magnetic order of

are carried out in many works[1,15,16] It was observed that the Curie temperature for a given thickness is rather lower for Ni on

validate the SF approximation, the Curie temperature dependence

on the thickness of FCC Ni ultra-thinfilms is plotted inFig 10, in

Cu(1 1 1) surfaces (denoted as Ni/Cu(1 0 0) and Ni/Cu(1 1 1)) are

film cases are well described in the framework of Heisenberg

smaller compared with the interior one Our results also imply

0.2

0.4

0.6

0.8

1.0

1.2

τC

ξ=0.8 ξ=1 ξ=2

n (layer)

Fig 6 The Curie temperatures obtained in the MF approximation versus the film

thickness with different surface exchange parameters ξ Here S ¼ 1.

0.0

0.3

0.6

0.9

1.2

1.5

τ

n=1 n=2 n=3 n=4 n=5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig 7 The magnetizations of the FCC films versus the temperature with various

thicknesses, in which the surface exchange and spin parameters are set as ξ ¼ 1,

and S ¼ 1.

0.0 0.2 0.4 0.6 0.8 1.0

τC

ξ=1 (MF) ξ=1 (SF) ξ=0.5 (MF) ξ=0.5 (SF)

n (layer)

Fig 8 The Curie temperature as a function of layer thickness n obtained within the

SF (solid line) and the MF (dash line) approximations Here S ¼ 1.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

mν

τ ξ=2

ξ=0.8 ξ=1

Fig 9 The magnetization of a two-layer FCC film (m SF

1 ¼ m SF

2 ) calculated within the

SF theory Parameters are the same as given in Fig 5

a lower Curie temperature with regard to smaller surface exchange

parameterξ (Fig 6)

5 Conclusion

The functional integral method is successfully applied for

finite temperature It is shown that the finite size effect

be illustrated by the Heisenberg model with different NN spin

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n (layer) Fig 10 The Curie temperature dependence on the FCC Ni ultra-thin film thickness.

The square (triangular) points are the theoretical results with ξ≈0:75 (ξ≈2:2), S¼2,

and h ¼0 The solid (dash) curve through square (triangular) points is guide to the

eyes The experimental data of Ni/Cu(1 0 0) and Ni/Cu(1 1 1) films are taken from

Ref [1]

However, layer magnetizations vanish at the same temperature,

At a given temperature, magnetizations of layers decrease...

dependence of magnetization with different thicknesses and

without the surface exchange effect (ξ ¼ 1) It is noticed that the

magnetization at zero temperature in the SF approximation... investigations on the magnetic order of

are carried out in many works[1,15,16] It was observed that the Curie temperature for a given thickness is rather lower for Ni on

validate the