Soluções tipo-vórtice de spins na fita de Möbius

(1)

(2)

WALTER ANDRADE DE FREITAS

SOLUÇÕES TIPO-VÓRTICE DE SPIN NA FITA DE MÖBIUS

Dissertação apresentada à Universidade Federal de Viçosa, como parte das exigências do Programa de Pós-Graduação em Física Aplicada, para obtenção do título de Magister Scientiae.

APROVADA: 31 de março de 2009.

Prof. Afrânio R. Pereira (Coorientador)

Prof. Luiz Cláudio (Coorientador)

Prof. Ricardo Reis Cordeiro Prof. Daniel Heber Theodoro Franco

(3)

(4)

(5)

(6)

(7)

(8)

Q= +1

NbSe3

ABCD

AB CD

AB z

φ

2 φ z

(9)

Φ(φ) φ Φ(φ) = φ

Asinφ

Φ(φ) φ

Φ(φ) = φ + π/2

L = 0.1

R = 10

Φ(φ) φ

Φ(φ) = φ + π/2

L= 6 R = 104

Φ(φ) φ

Φ(φ) =φ r

−1000 1000 800

Φ(φ) φ

r ₋1000 1000

Φ(φ) φ

Φ(φ) =φ ₋3 3

R = 7

Φ(φ) L = 15 R =

(10)

Φ(φ) L= 15 R =

8 R = 12 R= 20

Φ r

L = 15 R = 8

R = 12 R = 20

R/r = 1

R = 8 R = 18 R= 28 R= 38

i

k r φ L = 8R = 10 √1

k

r φ L= 8 R= 10

k r φ R = 8000

1/√k r φ R = 8000

1/√k r

1/√k φ φ

1/√k r

r 1/√k φ

θ φ θ φ_∈[0,2π]

(11)

(12)

(13)

(14)

H B M

B =µ0H,

µ0

B H J

v/c_≪1

B(x) = µ0 4π

J(x′₎_×₍_x₋_x′₎

|x₋x′_|3 d

3_x′_,

M B H

M = lim

∆V→0

1 ∆V

i

µi.

µi

F =< H >₋T S

Tc

(15)

µl =−gl

e 2mL.

S

µs =−gs

e 2mS.

gl = 1 g

gs ≈2 g

J =L +S.

J µ

µ=gµBJ,

µB =

e 2m

g

g = 1 + J(J+ 1) +S(S+ 1)−L(L+ 1) 2J(J+ 1) .

χ

χ= µ0|M|

(16)

(a)

(b)

(17)

(18)

(a)

(19)

TN

H =₋J 2

<i,j>

(Si.Sj)

J < 0 J > 0

(20)

i

σ

f(r)

(21)

s(x) = s(cosθ(x),sinθ(x))

θ(x)

θ(x) =φ+θ0,

θ0 x= (r, φ)

∇θ= 1/r

θ 2π

Q θ 2Qπ

Q

Q = 1

Q

Q= +1 Π1

(22)

Q= +1

θ(x) = Qφ+θ0

dθ =

C

dθ

dsds= 2Qπ, Q= 0,±1,±2, ...

s(x)

θ(x) 2Qπ

θ(x) = Qφ+ sinφ θ(x) = Qφ+ tanh(rcosφ) cos 7φ

(23)

Q= +1

Q > 1

Ed

Ec Eel

Ed=Eel+Ec.

∂2_φ

∂t2 −

∂2_φ

∂x2 +

1

b2sin(bφ) = 0

(24)

Q= +1

(25)

(26)

(27)

NbSe3

50µm NbSe3

NbSe3

(28)

(29)

J

J =ES−ET.

ES ET

Ho =−J′S1.S2,

S1 S2 1 2

H=₋J ′ 2

i

(Si.Si+1+Si.Si−1+Si.Si+2+Si.Si−2),

1 2

i i i+1 i₋1 i+2 i₋2

i

H =₋J ′ 2

<i,j>

Si.Sj.

(30)

H=₋J ′ 2

<i,j>

(Si1Sj1+Si2Sj2+Si3Sj3).

λ

H =₋J ′ 2

<i,j>

(S_i1S_j1+S_i2S_j2+ (1 +λ)S_i3S_j3).

a

Si i

H =J

2 i,j=1 3 a,b=1

gij_h

ab(1 +λδa3)

∂Sa

∂ηi

∂Sb

∂ηj

|g_|dη1dη2,

J _≡ J₂′ _|g_|dη1dη2 η1 η2 δa3

gij _h

ab

Θ Φ S = (Sx_{, S}y_{, S}z₎ _≡

(sin Θ cos Φ,sin Θ sin Φ,cos Θ)

Θ = Θ(η1, η2) Φ = Φ(η1, η2)

σ

J > 0

J < 0

S n = 1₂(S1 −S2)

(31)

A D B C 180o

S1 _x2 ₊_y2 ₌_R2 _AB _xy

|x_|< r z= 0 y= 0 c AB S1

φ AB φ₂ c

S1 _AB

ABCD

AB CD

R

r

f

f/2 -r

Z

Y

X

L=2r

AB z φ₂

(32)

−R2y+x2y+y3₋2Rxz₋2x2z₋2y2z+yz2 = 0,

x= (R+rcos(φ/2)) cos(φ),

y= (R+rcos(φ/2)) sin(φ),

z =rsin(φ/2),

R z r _∈ [₋L₂,L₂] L

φ_∈[0,2π]

gij

ds gij

ds2 =gijdxidxj.

gij =

∂r ∂qi ·

∂r ∂qj

,

∂r

∂qi r = (x, y, z) qj (q1, q2, q3)

(R, r, φ)

R

gij =

∂x ∂qi ∂x ∂qj + ∂y ∂qi ∂y ∂qj + ∂z ∂qi ∂z ∂qj .

q1 =r q2 =φ

(33)

(gij) =

⎡

⎣ 1 0

0 3r2+4R2+2r2cos₄ φ+8rRcosφ/2

⎤

⎦,

gij

gik_g

kj =δij

δij δij = 1 i = j δij = 0 i = j

(gij) =

⎡

⎣ 1 0

0 4

3r2 +4R2

+2r2

cosφ+8rRcosφ/2

⎤

⎦.

ds2 ₌_g

ijdxidxj =dr2+ [R2+ 2Rrcos(

φ 2) +

r2

4(3 + 2 cosφ)]dφ

2_,

− →

∇ =qi

1 hi ∂ ∂qi , h2

i =gii qi k=R2+ 2Rrcos(φ₂) +r

2 4(3 +

2 cosφ)

− →

∇ =r∂r+φ

1

√

k∂φ. k

k(r, φ) = k(₋r, φ+ 2π).

k = (rcos(φ/2) +R)2+ r

2

(34)

-4 -2 0 2 4 0 2 4 6 50 100 150 200 L r k

k r φ L= 8R = 10

G=₋ 4R

2

(4R2_{+ 3}_r2_{+ 2}_r₍₄_R_cos_φ/_{2 +}_r_cos_φ₎₎2 =−

1 4(R/k)

2_.

φ r

G

k2

G(r, φ) =G(₋r, φ+ 2π)

k r_→0

φ r = 0

−2R1

(35)

S2 ₌ _S2

x +Sy2 = 1

hab =δab

H =J

l

−l

2π

0

(3r2_{+ 4}_R2_{+ 2}_r2_cos_φ_{+ 8}_rR_cosφ 2) 1 2 2 [( ∂Sx ∂r )

2_{+ (}∂Sy

∂r )

2_{+ (1 +}_λ₎₍∂Sz

∂r )

2_]

+ 2

(3r2_{+ 4}_R2_{+ 2}_r2_cos_φ_{+ 8}_rR_cos φ 2) 1 2 [(∂S x ∂φ )

2_{+ (}∂Sy

∂φ )

2_{+ (1 +}_λ₎₍∂Sz

∂φ )

2_]_drdφ.

S = (sin Θ cos Φ,sin Θ sin Φ,cos Θ)

Θ Φ k =R2_{+ 2}_Rr_cos(φ 2) +

r2

4(3 + 2 cosφ)

H =J

l

−l

2π

0 √

(36)

+_√1

k[(1 +λsin

2_Θ)(_∂

φΘ)2+ sin2Θ(∂φΦ)2]drdφ,

∂θ ≡

∂

∂θ ∂φ≡ ∂ ∂φ Θ Φ ∂_L ∂ϕ − ∂ ∂xµ ∂_L ∂(∂µϕ)

= 0.

ϕ

xµ= (x1, x2)

= (r, φ).

L

H =π(q,q˙) ˙q₋L_⇒H =₋L,

q π(q,q˙) H L

h=√k[(1 +λsin2Θ)(∂rΘ)2+ sin2Θ(∂rΦ)2] +

1

√

k[(1 +λsin

2_Θ)(_∂

φΘ)2+ sin2Θ(∂φΦ)2].

Θ Φ

∂h ∂Φ−

∂ ∂r[

∂h ∂(∂rΦ)

]₋ ∂ ∂φ[

∂h ∂(∂φΦ)

] = 0,

∂h ∂Θ −

∂ ∂r[

∂h ∂(∂rΘ)

]₋ ∂ ∂φ[

∂h ∂(∂φΘ)

(37)

Θ(r, φ) Φ(r, φ)

sin Θ cos Θ√k[λ(∂rΘ)2+ (∂rΦ)2] +

1

√

k[λ(∂φΘ)

2_{+ (}_∂ φΦ)2]

=∂r[ √

k(1 +λsin2_Θ)_∂

rΘ] +∂φ[

1

√

k(1 +λsin

2_Θ)_∂ φΘ],

∂r[ √

ksin2Θ(∂rΦ)] +∂φ[

1

√

ksin

2_Θ(_∂

φΦ)] = 0.

1

√

k s

R+ssinθ 1 Ssinθ

1

ρτ φ

R s S

ρτ

Θ Φ

λ = ₋1 Θ = π

2 Φ

(38)

Φ Θ

λ=₋1 Θ = π 2

HM RP =J

l

−l

2π

0

[√k(∂rΦ)2+

1

√

k(∂φΦ)

2_]_drdφ.

∂r[ √

k(∂rφ)] +∂φ[

1

√

k(∂φΦ)] = 0

√

k

(39)

(˜x,y,˜ z˜)

˜ z

(40)

r _→0

Φ : Φ = Φ(φ)

HM RP =J

l

−l

2π

0

[√1

k(∂φΦ)

2_]_drdφ.

∂φ[

1

√

k∂φΦ] = 0. limr→0√1_k = _R1

Φ(φ) =ζφ+φ0, ζ ∈Z;

ζ φ0

R _{→ ∞}

R_≫r

Φ = Φ(φ)

∂φ[

1

√

(41)

R_≫r R √1

k

1

√

k∂φ[∂φΦ] = 0,

Φ(φ) =ζ′_φ₊_φ

1, ζ′ ∈Z.

ζ′ _φ

1

r_{→ ∞}

Φ = Φ(φ)

∂_φ2Φ₋ 1

2k∂φk∂φΦ = 0

r_{→ ∞}

∂_φ2Φ₋ 1/2 sinφ

3/2 + cosφ∂φΦ = 0,

Φ(φ) =q1E(

φ 2,

4

5) +φ0,

E(φ, m) =₀φ[1₋msin2θ]1/2_dθ

q1 = q_E(π4 5)

τ = 1 2π

(42)

C −→_∇

dl

τ = 1 2π

−→

∇Φ_·dl= 1 2π

∂φΦdφ=q ∈Z.

q = +1

1 2 3 4 5 6 Φ 1 2 3 4 5 6 Φ

Φq1EΦ 2,

4

5 ΦΦ

Φ(φ) φ Φ(φ) =φ

Asinφ

R_→0

r _{→ ∞}

(43)

∂_r2Φ + 1 k∂

2 φΦ +

1 2k

∂k ∂r∂rΦ−

1 2k2

∂k

∂φ∂φΦ = 0

Φ(r, φ) φ r

∂rΦ =

Φi,j+1−Φi,j−1

2p ,

∂φΦ =

Φi+1,j−Φi−1,j

2h ,

∂2 rΦ =

Φi,j+1−2Φi,j+ Φi,j−1

p2 ,

∂_r2Φ = Φi+1,j−2Φi,j+ Φi−1,j h2 ,

p h r φ i

φ_∈[0,2π] j r_∈[₋L₂,L₂]

Φ(r, φ)

(44)

L_→0

r _→ 0

Φ(φ) =ζφ+φ0

1 2 3 4 5 6 Φ

1 2 3 4 5 6 7 8

Φ

Φ(φ) φ

Φ(φ) =φ+π/2

L= 0.1 R= 10

L_→0

R_≫r

Φ(φ) =ζ′_φ₊_φ

0

(45)

1 2 3 4 5 6 Φ 1

2 3 4 5 6 7 8

Φ

Φ(φ) φ

Φ(φ) =φ+π/2

L= 6 R= 104

r >> R r _{→ ∞}

r_∈(250,750) r

r = 0

Φ(φ)

(46)

1 2 3 4 5 6 Φ 3

4 5 6 7 8

Φ

Φ(φ) φ

Φ(φ) = φ r

−1000 1000 800

r >> R

Φ

E(φ₄,4 5)

(47)

2.25 2.75 3.25 3.75 4Φ

3.5 4.5

5 5.5

6

FHΦL

Φ(φ) φ

(48)

Φ

Φ = φ +Asin(φ) A r

r _→ 0 A _→ 0

Φ =ζφ+φ0

L_≈R

L_≈20µm R_≈25µm

L= 6u.c R = 7u.c

Φ

r φ

1 2 3 4 5 6 Φ

1 2 3 4 5 6 7 8

Φ

Φ(φ) φ Φ(φ) = φ

(49)

Φ

1 2 3 4 5 6 Φ

1 2 3 4 5 6 7 8 Φ

1 2 3 4 5 6 Φ 2 4 6 8 10 Φ

Φ(φ) L = 15 R =

(50)

1 2 3 4 5 6Φ

2.6 2.8 3.2 3.4 Φ

1 2 3 4 5 6Φ

2.6 2.8 3.2 3.4

Φ

1 2 3 4 5 6Φ

2.6 2.8 3.2 3.4 Φ

Φ(φ) L = 15 R =

8 R= 12 R= 20

Φ r

(51)

(52)

Φ(φ) =qnφ+φ0

τ = 1 2π

C −→

∇Φ_·dl= 1 2π

C

∂φΦdφ=qn.

Φ =ζφ+φ0

EM RP =

2π 0 L/2 −L/2 1 √

k(∂φΦ)drdφ

EM RP =ζ2

2π 0 L/2 −L/2 1 √ kdrdφ φ

R R _{→ ∞}

R₋πr R+πr

R_{→ ∞}

(53)

10 20 30 40 50 60 70 r 6

8 10 12 14 16

EnergiaRr1

2 4 6 8 10

R

r 2

4 6 8 10 Energia

L

R/r = 1

L

(54)

R _≫r L_≈0

R_{→ ∞} r_→0

(55)

(56)

H =₋J′

<i,j>

Hi,j =−J′

<i,j>

(S_ixS_jx+S_iyS_jy+ (1 +λ)S_jzS_jz).

Si i i+ 1 i+ 2

i₋1 i₋2 J =J′_/₂

H =₋J[

i

α=x,y,z

S_iα(S_i+1α +S_i+2α +S_iα₋₁+S_iα₋₂)

+λ

i

S_iz(S_i+1z +S_i+2z +S_iz₋₁+S_iz₋₂)].

(57)

Si+1α =Siα+a∂xSiα+

a2

2∂

2

xSiα+...,

S_iα₋₁ =S_iα₋a∂xSiα+

a2

2∂

2

xSiα−...,

S_i+2α =S_iα+a∂ySiα+

a2

2∂

2

ySiα+...,

S_iα₋₂ =S_iα₋a∂ySiα+

a2

2∂

2

ySiα−...,

a

H =₋J

i

α=x,y,z

S_iα[4S_iα+a2(∂

2_Sα i

∂x2 +

∂2_Sα i

∂y2 )]

−Jλ

i

Siz[4Siz+a2(

∂2_Sz i

∂x2 +

∂2_Sz i

∂y2 )].

i

dxdy a2

H =₋4J S

2

a2 dxdy−J

α=x,y

Sα[∂

2_Sα

∂x2 +

∂2_Sα

∂y2 ]dxdy

−J(1 +λ) Sz[∂

2_Sz

∂x2 +

∂2_Sz

∂y2 ]dxdy.

H =₋J

α=x,y

Sα[∂

2_Sα

∂x2 +

∂2_Sα

∂y2 ]dxdy−J(1 +λ) S z_[∂2Sz

∂x2 +

∂2_Sz

∂y2 ]dxdy.

H =J

α=x,y,z

(1 +δα3λ)[(

∂Sα

∂x )

2_{+ (}∂Sα

∂y )

2_]_dxdy,

H =J

2 i,j=1 3 a,b=1

δijhab(1 +δa3λ)(

(58)

δij 1 i = j 0 hab

hab =δab

dxdy= ∂x ∂µ1 ∂x ∂µ2 ∂y ∂µ1 ∂y ∂µ2

dµ1dµ2.

H =J

2 i,j=1 3 a,b=1

δijhab(1 +δa3λ)(

∂Sa ∂xi )(∂S b ∂xj ) ∂x ∂µ1 ∂x ∂µ2 ∂y ∂µ1 ∂y ∂µ2

dµ1dµ2.

δij

gij

H =J

2 i,j=1 3 a,b=1

δijgijhab(1 +δa3λ)(

∂Sa

∂xi

)(∂S

b

∂xj

)_|g_|dµ1dµ2,

|g_{| ≡}

(59)

ϕ(x) x

X ϕ

Y ϕ Y = S1

ϕ ϕ =ϕ1xˆ+ϕ2yˆ

|ϕ2_| _{= 1} _Y ₌ _S2 _ϕ

ϕ=ϕ1xˆ+ϕ2yˆ+ϕ3zˆ |ϕ2|= 1

X 0 _≤ x _≤ 2π

ϕ(x) xi ∈X ϕ(xi) Y x

0 2π Y ϕ(0)

ϕ(2π) ϕ(0) =ϕ(2π) = ϕ0

ϕ0

Y ϕ0

Y

(60)

•

Y =E2

xy y0

y(x0) y0

Y = E2 ₋₍₀_,₀₎

y0 Y

y0

m

y0_{, y}1_{, y}2_{, ...} _ym

m

Y Π1(Y) 1 X

S1

Π1[E2−(0,0)]

Z_∞

Π1(E2−(0,0)) =Z∞.

(61)

y0 X

S2 _Π

2(Y)

Π1(Y)

Π2(Y) X

(62)

k

k =R2+ 2Rrcos(φ 2) +

r2

4(3 + 2 cosφ) = (rcos(φ/2) +R)

2 ₊r2

2 >0

R/r >>1

-4 -2 0 2 4 0 2 4 6 50 100 150 200 (a) r L k -4 -2 0 2 4 0 2 4 6 0.08 0.1 0.12 0.14 0.16 (b) r L

1/ k

Ö

k r φ L = 8R = 10 _√1 k

r φ L= 8R = 10

(63)

r

L

r

L

k r φ R = 8000

(64)

-2 -1 1 2 r 4·10-7

6·10-7 8·10-7 1·10-6 1.2 ·10-6

1

????

_k

(a)

1 2 3 4 5 6 Φ

4·10-7 6·10-7 8·10-7 1·10-6 1.2 ·10-6

1 _k

(b)

1/√k r

1/√k φ φ

1/√k r r

1/√k φ

k

k(r, φ) =k(₋r, φ+ 2π)

k2 =₋

(65)

f(x₋h)

f(x+h) h x

f(x₋h) =f(x)₋hf′₍_x_{) +} h

2

2 f

′′₍_x₎₋h

3

3!f

′′′₍_x_{) +} h

4

4!f

′′′′₍_x₎₋_...,

f(x+h) =f(x) +hf′₍_x_{) +} h

2

2 f

′′₍_x_{) +}h

3

3!f

′′′₍_x_{) +} h

4

4!f

′′′′₍_x_{) +}_....

f(x+h)₋f(x₋h) h

f′₍_x₎ f(x+h)−f(x−h)

2h ,

f(x+h) +f(x₋h)

f′′₍_x₎ f(x+h)−2f(x) +f(x−h)

(66)

f

x, y

∂xf =

fi,j+1−fi,j−1

2p ,

∂yf =

fi+1,j−fi−1,j

2h ,

∂2 xf =

fi,j+1−2fi,j +fi,j−1

p2 ,

∂_y2f = fi+1,j−2fi,j +fi−1,j h2 .

θ φ

x= (R+rsinθ) cosφ,

y= (R+rsinθ) sinφ,

z=rcosθ,

θ φ_∈[0,2π] r, R

(gij) =

⎡

⎣ (R+rsinθ)

2 ₀

0 r2

⎤

(67)

q f F(q,f) 2p 2p 0

θ φ θ φ_∈ [0,2π]

(gij) =

⎡

⎣

1

(R+rsinθ)2 0

0 _r12

⎤

⎦

HRP =J

π

−π

2π

0

[κ′₍_∂

θΦ)2 +

1 κ′(∂φΦ)

2_]_dφdθ.

κ′ ₌ R+rsinθ

r

Φ

∂θ[κ′(∂θΦ)] +∂φ[

1

κ′(∂φΦ)] = 0

∂_θ2Φ + 1 κ′2∂

2 φΦ +

1 κ′

∂κ′ ∂θ∂θΦ−

1 κ′3

∂κ′

∂φ∂φΦ = 0,

θ _→r κ′2 _→_k

Φ(θ, φ)

Φ(θ, φ) θ φ

∂θΦ =

Φi,j+1−Φi,j−1

2p ,

∂φΦ =

Φi+1,j−Φi−1,j

2h ,

∂_θ2Φ = Φi,j+1−2Φi,j+ Φi,j−1 p2 ,

(68)

p h θ φ i

φ j θ

θ φ [0,2π]

Φ

R

Φ(0, φ) = Φ(2π, φ) =φ+π/2 Φ(θ,0) =π/2

Φ(θ,2π) = 5π/2

Φ φ

1 2 3 4 5 6 Φ 1

2 3 4 5 6 7 8 Φ

Φ φ

Φ φ Φ = qφ+φ0

(69)

(70)

(71)

(72)

(73)

(74)

(75)