Basic Condensed Matter Physics ~

Introductory Nanotechnology

~ Basic Condensed Matter Physics ~

Atsufumi Hirohata Department of Electronics

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 :

( transmittance ) + ( reflectance ) = 1

Optical absorption :

Conduction band Conduction

band Valence

band Valence band Absorption coefficient Absorption coefficient Direct Indrect Direct transition transition transition

starts starts starts 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 ?

7. Why does a magnet attract / retract ?

8. What happens at interfaces ?

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

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
found by shepherd Magnes ?

Magnetite (Fe3O4)

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

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.

Which Elements are magnetic ?

In the periodic table,

Only 4 elements are ferromagnetic at room temperature ! Magnetic Moment

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

•
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

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

Coulomb's Law

Force between two magnetic poles, m1 [Wb] and m2, separated with r [m] is defined as 1 m m 1 2 H F = 2 [N] 4
μ0 r r m m1 2

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

1 m1 F = m2H H = 2 [N/Wb] = [A/m] 4
μ0 r

Magnetic flux density is proportional to the magnetic field : m Magnetic flux density at r from a magnetic pole m is B = 4
r 2 By comparing with H, B = μ0H

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

Under the presence of magnetisation, B = μ0()H + M B If the system is not in a vacuum, B = μH

By assuming μ = μ 0 +
(
: susceptibility), M =
μ H ()0 M Magnetic Dipole Field and Magnetic Flux

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

Magnetisation Curve

M Saturation magnetisation : MS

Residual magnetisation : Mr

Initial permeability : μ = (1 / μ ) (M / H) Coercivity : H i 0 initial M c - + - + - Hd +

H Magnetic hysteresis

Demagnetising field : Hd = -NM (N : demagnetising factor)

* S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). Magnetic Field Induced by an Electrical Current

Biot-Savart Law :

According to the right-handed screw rule, dH 1  dH = k 2 ()ids
er ds r (= re ) r r
k a i For an infinite straight wire, H = i sinds 
r 2 2 2 1/2 By substituting  = a / r and r = (a + s ) , 
1
1 dx x  H = kia ds = kia ds =  3  32  32 12 
r 
a 2 s2 a 2 x 2 a 2 a 2 x 2  ()+ ()+ ()+  
 
s  ki s  2ki = kia 12 = 12 = 2 2 2  a 2 2  a a a + s  aa + s   ()
 ()

By taking an integral along H, 2ki 2ki 2
Hdl = dl = ad = 2ki d = 4
ki   a  a 0 i
Ampère’s law Hdl = i () 4
k  1 H =  2
a

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

For a circular current, using the right-handed screw rule, H H =
dH x =
dH cos dH By substituting the Biot-Savart law, x r 1 ids 
 i cos H = sin cos = ds
 4
r 2  2  4
r 2  a i ds i cos ia cos ia 2 =  2
a = = 4
r 2 2r 2 2r 3 Magnetic Field Induced by a Magnetic Dipole

Potential  at point P, which is separated r from the dipole :   1 m m m l2  l1 +m  =  + = l 1 4 l l 4 l l
μ0  1 2 
μ0 1 2 d  r P Here, l -m 2 2 2 2  d  2  d  l = r +  
rd cos , l = r +   + rd cos 1  2  2  2 

2 For r >> d, d and higher terms can be neglected.  d  d  d l1
r 1 cos
r 1  cos = r  cos  r  2r  2 l2  l1
d cos  2 d  l1l2
r l
r + cos 2 2 m d cos m r Therefore, potential is calculated to be  = 2 = 3 4
μ0 r 4
μ0r Magnetic field at P is 1  m  1  m 3 r  1  3  H =   =   3 r  =  3  4 ()m r  = 3  m  2 ()m r r  4
μ0  r  4
μ0  r r r  4
μ0r  r 

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

In H, a component along m is 1  3  m H H = m  mrr = 3 2  3 dH 4
μ0r  r  2
μ0r 2 Assuming m = μ0iA (A =
a ), x r 2  μ0iA ia a H = 3 = 3 i ds 2
μ0r 2r

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 L

z component is calculated to be Lz = xpy
ypx h
In order to convert L into an operator, p
z i
q

h 

 0 r Lz =  x  y  p i 
y
x 
By changing into a polar coordinate system, L = h z i
 Similarly, h 

 h 

 Lx =  sin  + cot cos , Ly =   cos + cot sin i 

  i 

 

Therefore,  2  2 2 2 2 2 1

1
L = Lx + Ly + Lz = h  sin   + 2 2   sin

 sin 
  In quantum mechanics, observation of state  = R
 is written as  2 2 2 1

  ml 2 2 L  = h  sin   + 2 R   h KR() = ll()+1 h  sin

  sin 

Origin of Magnetism (Cont'd)

Thus, the eigenvalue for L2 is 2 2 L = ll()+1 h
L = ll()+1 h ()l = 1, 2, 3, K

azimuthal quantum number (defines the magnitude of L)

Similarly, for Lz,

Lz = ml ()ml = 0, ±1, ± 2, h K
magnetic quantum number (defines the magnitude of Lz) For a simple electron rotation, For l = 1, L z m = 1, 0, -1 L l

Orientation of L : quantized In addition, principal quantum number : defines electron shells n = 1 (K), 2 (L), 3 (M), ...

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

Orbital motion of electron : generates magnetic moment m =
μ L B h -29  μB : Bohr magneton (1.165
10 Wbm)

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

Spin Moment and Magnetic Moment

Zeeman splitting : ml 2 l 1 For H atom, energy levels are split under H 2 0 -1 dependent upon ml. -2
E = h Spin momentum : 1  1 0 L l ml = l, l
1, K, 0, K,
l () 2l +1 -1

  1  H = 0 H
0 S s ms = s,
s s = 2 z   2 S 1  1 
S = ss()+1 h =  +1h 1 2  2  msh = h 2 m =
gμ J B h  g = 1 (J : orbital), 2 (J : spin)

Summation of angular momenta : Russel-Saunders model J = L + S Magnetic moment :

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

* S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). Exchange Energy and Magnetism

Exchange interaction between spins : E =
2J S S ex ex i j Sj Si
Eex : minimum for parallel / antiparallel configurations

Jex : exchange integral

ex J ferromagnetism Dipole moment arrangement :

Paramagnetism antiferromagnetism Exchange integral integral Exchange Antiferromagnetism Atom separation [Å] Ising Ferromagnetism

Ferrimagnetism

Heisenberg

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

Paramagnetism

Applying a magnetic field H, potential energy of a magnetic moment with
is U =
mH =
mH cos
m rotates to decrease U.
Assuming the numbers of moments with
is n and energy increase with
+ d
is + dU, H dn 1
U
U  ()
dU ln n  + const. ln n = + ln n0 n T T kBT 
U  n = n0 exp 
Boltzmann distribution  kBT  Sum of the moments along z direction is between -J and +J  U  M = m n = gμ M n exp (M : z component of M)
z
()B J 0   J  kBT   U   U  Here, N =
n =
n0 exp  n0 = N
exp  kBT   kBT   U  Ng
μB()M J exp b exp by  kBT 
()  gμ H  M = = Ngμ b M , y  B   B J U exp()by  kBT 
exp
 kBT  Paramagnetism (Cont'd) Jy Jy y Jy ()J1 y Jy Jy Jy y Jy e  e e Now, exp by = e + e + + e = e + e e + + e =
() L L y y 2 Jy Jy y 1 e e e  e e Jy y 2 Jy y 2 y 2 ()e e  e e e exp()by = = y 2 y 2
ey 2 1 ey e  e ()  1   1   1   J + y  J + y sinh J + y 1 e 2   e  2   2 Using sinh a = ea
e
a  eby = = ()
y 2 y 2 y 2 e  e sinh beby 2 d by
Using ln e = ()
by dy
e  1   sinh J +  y  d d 2 d  1 y  ln eby =  ln  = lnsinh J + y  lnsinh ()
y    dy dy  sinh  dy  2 2  2  1 1 1 1 y 1 = cosh J +  y  J +  y  cosh  1 2 2 y 2 2 sinh J +  y sinh 2 2 1 1 1 y = J +  coth J +  y  coth 2 2 2 2 2J +1 2J +1 1 a =  coth  a  coth ()a  Jy 2 2 2 2J

Paramagnetism (Cont'd)

Therefore,    2J +1  2J +1 1 a  gμBJH M = NgμBJ coth a
coth = NgμBJBJ ()a a =  2J  2J 2J 2J   kBT

 BJ (a) : Brillouin function For a 
(H 
or T  0),

1
aJ BJ ()a = 1
e
L  1 M = M 0 = NgμBJ J For J  0, M  0 Ferromagnetism

For J 
(classical model), 2J +1
1 2J 1 a 1  a a  1  a  1 coth =  cosh sinh 
1  = 2J 2J 2J  2J 2J  2J  2J  a 1 B
()a = coth a   La() a  L (a) : Langevin function

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

Weiss molecular field : H m = wM (w : molecular field coefficient, M : magnetisation) gμBJH In paramagnetism theory, M = NgμBJBJ ()a , a = kBT Substituting H with H + wM, and replacing a with x, Hm gμBJ M = NgμBJBJ ()x , x = ()H + wM kBT

Spontaneous magnetisation at H = 0 is obtained as kBT = gμBJwM
M  = BJ ()x  M 0 Using M0 at T = 0,   M kBTx = 2 2 2  M 0 Ng μB J w J +1 For x << 1, BJ ()x
x Ferromagnetism Paramagnetism 3J Assuming T =
satisfies the above equations,

M J +1 kB
= x = 2 2 2 x M 0 3J Ng μB J w
(TC) : Curie temperature 2 2 2 Ng μB JJ()+1 w Nm 
= = w 3kB 3kB * H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

Major Phases of Fe

Fe changes the crystalline structures with temperature / pressure :

56 Fe : T [K] Liquid-Fe Most stable atoms in the universe. 1808

-Fe bcc

1665 Phase change

-Fe (austenite) fcc 1184 Martensite Transformation : (-Fe)
’-Fe (martensite) 1043 -Fe

-Fe (ferrite) bcc

hcp

1 p [hPa] Ferromagnetism (Cont'd) J +1 For x << 1, BJ ()x
x 3J x M = NgμBJBJ ()x = NgμBJJ()+1 3 2 2 1 = Ng μB JJ()+1 ()H + wM 3kBT 1 2 2 2 M = C ()H + wM C
Ng μB JJ()+1 3kB = Nm 3kB T () H M = C T
Cw Therefore, susceptibility
is M C C  = = = H T
Cw T
 (C : Curie constant)

Curie-Weiss law

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

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),
1 Nm2 C  M A = NgμBJBJ ()xA = H A = H A  2 6k T 2T  B  1 Nm2 C  M B = NgμBJBJ ()xA = H B = H B  2 6kBT 2T A-site B-site Therefore, total magnetisation is C C M = M A + M B = ()H
wM A
w M B + ()H
w M A
wM B = 2H + ()w + w M 2T []2T [] M C C  = = = H C T T + ()w + w +
2

Néel temperature (TN)

Antiferromagnetism Paramagnetism

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

Magnetic Anisotropy

Magnetocrystalline anisotropy :

Hard axis

Easy axis

Easy axis : Magnetic anisotropy energy : minimum Stable direction for spontaneous magnetisation

Hard axis : Magnetic anisotropy energy : maximum Unstable direction for spontaneous magnetisation

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

Magnetostrictive material

Change

Electromagnet

Flat panel speaker

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

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

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

* S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). Domain Wall Displacement in a M - H Curve

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

rotational domain motion

Barkhausen jump

irreversible domain motion

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