Introductory Nanotechnology ~ Basic Condensed Matter Physics...

Department of Electronics

Introductory Nanotechnology ~ Basic Condensed Matter Physics ~

Atsufumi Hirohata

Quick Review over the Last Lecture

Quantum mechanics and classical dynamics

Quantum mechanics Classical dynamics

Schrödinger equation Equation of motion

: wave function A : amplitude

| |2 : probability A2 : energy

Quantum tunneling :

Optical absorption :

( transmittance ) + ( reflectance ) = 1

Abso

rption c

oef

fici

ent

Abso

rption c

oef

fici

ent

Valence band

Conduction band

Direct transition starts

Indrect transition starts

Direct transition starts

Conduction band

Valence band

Contents of Introductory Nanotechnology

First half of the course : Basic condensed matter physics

1. Why solids are solid ?

2. What is the most common atom on the earth ?

3. How does an electron travel in a material ?

4. How does lattices vibrate thermally ?

5. What is a semi-conductor ?

6. How does an electron tunnel through a barrier ?

Second half of the course : Introduction to nanotechnology (nano-fabrication / application)

7. Why does a magnet attract / retract ?

8. What happens at interfaces ?

Why Does a Magnet Attract / Retract ?

• Magnetic moment

• Magnetisation curve

• Origin of magnetism

• Curie temperature

• Types of magnets

• Magnetic anisotropy

• Magnetic domains

How Did We Find a Magnet ?

• 6th century BC, from Magnesia ( ) in Greece

Magnetite (Fe3O4)

• 3th century BC, from Handan area ( ) in China

found by shepherd Magnes ?

220 ~ 265 AD, first compass * http://www.wikipedia.org/

Magnet and Magnetism

Study on magnetism started by William Gilbert :

Magnete, Magneticisque Corporibus,

et de Magno Magnete Tellure (1600)

• The earth is a large magnet (compass).

• Fe looses magnetism by heating.

• N / S poles always appear in pair.

* http://www.wikipedia.org/

Development of Permanent Magnets

Permanent magnets are used in various applications :

* Corresponding pages on the web.

Only 4 elements are ferromagnetic at room temperature !

Which Elements are magnetic ?

In the periodic table,

Magnetic Moment

By dividing a magnet, N (+) and S (-) poles always appear :

* S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

• No magnetic monopole has been discovered !

• A pair of magnetic poles is the minimum unit : magnetic (dipole) moment.

m = m

m

Magnetisation :

• Vector sum of m per unit volume

Coulomb's Law

Force between two magnetic poles, m1 [Wb] and m

2, separated with r [m] is defined as

F =1

4 μ0

m1m2

r 2 [N]

Here, m2 receives magnetic force (= magnetic field) :

m1

m2 r

H

F = m2H H =1

4 μ0

m1

r 2 [N/Wb] = [A/m]

Magnetic flux density is proportional to the magnetic field :

Magnetic flux density at r from a magnetic pole m is B =m

4 r 2By comparing with H, B = μ0H

μ0 : magnetic permeability in a vacuum [H/m]

B

M

Under the presence of magnetisation, B = μ0 H +M( )

If the system is not in a vacuum, B = μH

By assuming ( : susceptibility), M = μ0( )Hμ = μ0 +

Magnetic Dipole Field and Magnetic Flux


Magnetisation Curve


M

Initial permeability : μ i = (1 / μ0) (M / H)initial

Saturation magnetisation : MS

Residual magnetisation : Mr

Coercivity : Hc

Magnetic hysteresis

M

H

+ + +

- - - H

d

Demagnetising field : Hd = -NM

(N : demagnetising factor)

Magnetic Field Induced by an Electrical Current

Biot-Savart Law :

According to the right-handed screw rule,

dH = k1

r 2ids er( )

For an infinite straight wire, H =k

r 2i sin ds i

ds

dH

r (= rer)

a

By substituting = a / r and r = (a2 + s2) 1/2,

H = kia1

r 3ds = kia

1

a2 + s2( )3 2

ds dx

a2 + x 2( )3 2

=x

a2 a2 + x 2( )1 2

= kias

a2 a2 + s2( )1 2

=ki

a

s

a a2 + s2( )1 2

=2ki

a

By taking an integral along H,

Hdl =2ki

adl =

2ki

aad = 2ki d

0

2= 4 ki

Ampère’s law Hdl = i 4 k 1( ) H =i

2 a

Magnetic Field Induced by an Electrical Current (Cont'd)

For a circular current, using the right-handed screw rule,

H = dHx = dH cos

i ds

dH

r

a

By substituting the Biot-Savart law,

H =1

4

ids

r 2sin

2

cos =

i cos

4 r 2ds

=i cos

4 r 22 a =

ia cos

2r 2=ia2

2r 3

H

x

Magnetic Field Induced by a Magnetic Dipole

Potential at point P, which is separated r from the dipole :

=1

4 μ0

m

l1+

m

l2

=

m

4 μ0

l2 l1l1l2 r

Here,

l1 = r 2+

d

2

2

rd cos , l2 = r 2+

d

2

2

+ rd cos

l 1

-m

+m

l 2

P d

For r >> d, d 2 and higher terms can be neglected.

l1 r 1d

rcos r 1

d

2rcos

= r

d

2cos

l2 r +d

2cos

l2 l1 d cos

l1l2 r 2

Therefore, potential is calculated to be =

m

4 μ0

d cos

r 2=m r

4 μ0r3

H = =1

4 μ0

m

r 3r

=

1

4 μ0

m

r 33

r 4m r( )

rr

=

1

4 μ0r3m

3

r 2m r( )r

Magnetic field at P is

Magnetic Field Induced by a Magnetic Dipole (Cont'd)

In H, a component along m is

i ds

dH

r

a

H

x

H =1

4 μ0r3

m3

r 2mrr

=

m

2 μ0r3

Assuming m = μ0iA (A = a2),

H =μ0iA

2 μ0r3=ia2

2r 3

Same as H induced by a circular electrical current

Circular current i holds a magnetic moment of m = μ0iA.

Circular current i is equivalent to a magnetic moment.

Origin of Magnetism

Angular momentum L is defined with using momentum p :

L = r p

z component is calculated to be Lz = xpy ypx

0 p

r

L

In order to convert Lz into an operator, p

h

i q

Lz =h

ixy

yx

By changing into a polar coordinate system,

Lz =h

iSimilarly,

Lx =h

isin + cot cos

, Ly =

h

icos + cot sin

Therefore,

L2 = Lx2

+ Ly2

+ Lz2

= h2 1

sinsin

+

1

sin2

2

2

In quantum mechanics, observation of state = R is written as

L2 = h2 1

sinsin

+

ml2

sin2

R h

2K R( ) = l l +1( )h2

Origin of Magnetism (Cont'd)

Thus, the eigenvalue for L2 is

L2 = l l +1( )h2 L = l l +1( )h l = 1, 2, 3, K( )

azimuthal quantum number (defines the magnitude of L)

Lz = mlh ml = 0, ±1, ± 2, K( )


Similarly, for Lz,

magnetic quantum number (defines the magnitude of Lz)

For a simple electron rotation,

L

Lz

Orientation of L : quantized

In addition, principal quantum number :

defines electron shells

n = 1 (K), 2 (L), 3 (M), ...

For l = 1, ml = 1, 0, -1

Orbital Moments


Orbital motion of electron :

generates magnetic moment

m = μBL h

μB : Bohr magneton (1.165 10-29 Wb m)

Spin Moment and Magnetic Moment

Zeeman splitting :

For H atom, energy levels are split under H

dependent upon ml.


Summation of angular momenta :

Russel-Saunders model J = L + S

Magnetic moment :

ml

2 1 0 -1 -2

1 0 -1

2

1

l

E = h

H = 0 H 0

Spin momentum :

L l ml = l, l 1, K, 0, K, l 2l +1( )

S s ms = s, s s =1

2

2

S = s s+1( )h =1

2

1

2+1

h

m = gμBJ h

g = 1 (J : orbital), 2 (J : spin)

z

S

msh =1

2h

M = Morb +Mspin = μB L+ 2S( ) h = gμBJ h

Exchange Energy and Magnetism

* K. Ota, Fundamental Magnetic Engineering I (Kyoritsu, Tokyo, 1973); ** http://www.wikipedia.org/ & http://www.bradley.edu/.

Exchange interaction between spins :

Eex = 2JexSiS jSi

Sj

Eex : minimum for parallel / antiparallel configurations

Jex : exchange integral

Atom separation [Å]

Exc

han

ge

inte

gra

l J e

x

antiferromagnetism

ferromagnetism

Dipole moment arrangement :

Paramagnetism

Antiferromagnetism

Ferromagnetism

Ferrimagnetism

Heisenberg

Ising

Paramagnetism

Applying a magnetic field H, potential energy of a magnetic moment with is

U = mH = mH cos m rotates to decrease U.

H

Assuming the numbers of moments with is n and energy increase with + d is + dU,

dn

n

1

TdU( ) lnn

U

T+ const. lnn =

U

kBT+ lnn0

n = n0 expU

kBT

Boltzmann distribution

Sum of the moments along z direction is between -J and +J

M = mzn = gμBMJ( )n0 expU

kBT

(MJ : z component of M)

Here, N = n = n0 expU

kBT

n0 = N exp

U

kBT

M =

N gμB MJ( ) expU

kBT

expU

kBT

= NgμB

b exp by( )

exp by( ) b MJ , y

gμBH

kBT

Paramagnetism (Cont'd)

Now, exp by( ) = e Jy + e J 1( )y+L+ eJy = e Jy + e Jyey +L+ eJy =

e Jy eJyey

1 ey

e y 2 exp by( ) =e y 2 e Jy eJyey( )e y 2 1 ey( )

=e Jye y 2 eJyey 2

e y 2 ey 2

Using sinha =1

2ea e a( ) eby =

eJ +1

2

y

eJ +1

2

y

ey 2 e y 2=

sinh J +1

2

y

sinhy

2Using

d

dyln eby( ) =

beby

eby

d

dyln eby( ) =

d

dyln

sinh J +1

2

y

sinhy

2

=d

dylnsinh J +

1

2

y lnsinh

y

2

=1

sinh J +1

2

y

cosh J +1

2

y J +

1

2

y

1

sinhy

2

coshy

2

1

2

= J +1

2

coth J +

1

2

y

1

2coth

y

2

=2J +1

2

coth

2J +1

2

a

1

2coth

a

2J a Jy( )

Paramagnetism (Cont'd)

Therefore,

M = NgμBJ2J +1

2J

coth

2J +1

2J

a

1

2Jcoth

a

2J

= NgμBJBJ a( ) a =

gμBJH

kBT

BJ (a) : Brillouin function

BJ a( ) = 11

Je a J

L 1 M = M 0 = NgμBJ

For a (H or T 0),

For J 0, M 0

For J (classical model),

2J +1

2J1

1

2Jcoth

a

2J=1

2Jcosh

a

2Jsinh

a

2J

1

2J1

a

2J

=1

a

B a( ) = coth a1

aL a( )

L (a) : Langevin function


Ferromagnetism

Ferromagnetism

Weiss molecular field :

In paramagnetism theory, M = NgμBJBJ a( ), a =gμBJH

kBTH

m

Hm = wM (w : molecular field coefficient, M : magnetisation)

Substituting H with H + wM, and replacing a with x,

M = NgμBJBJ x( ), x =gμBJ

kBTH + wM( )

Spontaneous magnetisation at H = 0 is obtained as kBT = gμBJwM

Using M0 at T = 0,

M

M 0

= BJ x( )

M

M 0

=kBTx

Ng2μB2J 2w

For x << 1, BJ x( )J +1

3Jx

Assuming T = satisfies the above equations,

M

M 0

=J +1

3Jx =

kBNg2μB

2J 2wx

=Ng2μB

2J J +1( )w3kB

=Nm2

3kBw

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

(TC) : Curie temperature

Ferromagnetism Paramagnetism

Major Phases of Fe

Fe changes the crystalline structures with temperature / pressure :

-Fe (ferrite)

1184

-Fe (austenite)

1665

-Fe

-Fe

T [K]

p [hPa]

1808

Liquid-Fe

1043

( -Fe)

bcc

fcc

bcc

1

hcp

56 Fe :

Most stable atoms in the universe.

Phase change

Martensite Transformation :

’-Fe (martensite)

Ferromagnetism (Cont'd)

* S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997); ** http://www.wikipedia.org/.

For x << 1, BJ x( )J +1

3Jx

M = NgμBJBJ x( ) = NgμBJ J +1( )x

3

= Ng2μB2J J +1( )

1

3kBTH + wM( )

M = C1

TH + wM( ) C Ng2μB

2J J +1( ) 3kB = Nm2 3kB( )

M = CH

T CwTherefore, susceptibility is

=M

H=

C

T Cw=

C

T

(C : Curie constant)

Curie-Weiss law

Spin Density of States

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

Antiferromagnetism

By applying the Weiss field onto independent A and B sites (for x <<1),

* S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997); ** http://lab-neel.grenoble.cnrs.fr/.

A-site B-site

MA =1

2NgμBJBJ xA( ) =

Nm2

6kBTHA =

C

2THA

M B =1

2NgμBJBJ xA( ) =

Nm2

6kBTHB =

C

2THB

Therefore, total magnetisation is

M = MA + M B =C

2TH wMA w M B( ) + H w MA wMB( )[ ] =

C

2T2H + w + w ( )M[ ]

=M

H=

C

T +C

2w + w ( )

=C

T +

Néel temperature (TN)

Antiferromagnetism Paramagnetism

Magnetic Anisotropy

Magnetocrystalline anisotropy :


Easy axis

Hard axis

Easy axis :

Magnetic anisotropy energy : minimum

Stable direction for spontaneous magnetisation

Hard axis :

Magnetic anisotropy energy : maximum

Unstable direction for spontaneous magnetisation

Magnetostriction

* http://www.gmmtech.co.jp/ ** http://www.ednjapan.com/

Electromagnet

Magnetostrictive material

Change

Flat panel speaker

Magnetic Domain Structures

Stable magnetic domain configuration is defined to minimize total energy :

U = Umag + Uex + Ua


Umag : magnetostatic energy maximum when magnetic poles appear at the edge. minimum when no magnetic poles appear at the edge.

Uex : exchange energy maximum for antiparalell minimum for parallel

Ua : magnetic anisotropy energy maximum for hard axis minimum for easy axis

Magnetic Domain Walls

Bloch wall :

Néel wall :

Domain Wall Evolution with Film Thickness


Magnetic domain walls change the configuration with film thickness :

Bloch wall

Néel wall

Cross-tie wall

Domain Wall Displacement in a M - H Curve

In a magnetisation process, domains are annihilated / nucleated by a field :

* J. Stoör and H. C. Siegmann, Magnetism (Springer, Berlin, 2006).

rotational domain motion

irreversible domain motion

Barkhausen jump

Introductory Nanotechnology ~ Basic Condensed Matter Physics...

Documents

Transcript of Introductory Nanotechnology ~ Basic Condensed Matter Physics...