MSE 7025 Magnetic Materials (and Spintronics)

Lecture 4: Category of Magnetism

Chi-Feng Pai [email protected]

Course Outline • Time Table

Week Date Lecture 1 Feb 24 Introduction 2 March 2 Magnetic units and basic E&M 3 March 9 Magnetization: From classical to quantum 4 March 16 No class (APS March Meeting, Baltimore) 5 March 23 Category of magnetism 6 March 30 From atom to atoms: Interactions I (oxides) 7 April 6 From atom to atoms: Interactions II (metals) 8 April 13 Magnetic anisotropy 9 April 20 Mid-term exam 10 April 27 Domain and domain walls Course Outline • Time Table

Week Date Lecture 11 May 4 Magnetization process (SW or Kondorsky) 12 May 11 Characterization: VSM, MOKE 13 May 18 Characterization: FMR 14 May 25 Transport measurements in materials I: Hall effect 15 June 1 Transport measurements in materials II: MR 16 June 8 MRAM: TMR and spin transfer torque 17 June 15 Guest lecture (TBA) 18 June 22 Final exam Hund’s Rules

• So, how do we determine the ground state?

– For a given atom with multiple electrons, the total orbital angular momentum L and spin angular momentum S can have (2l+1)(2s+1) combinations.

(j)

Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules

Coulomb interaction

Spin-orbit interaction

Note: Apply strictly to atoms, loosely to localized orbitals in solids, not at all to free electrons Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules

• So, how do we determine the ground state?

Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules

• Why Hund’s rules are important?

Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules

• Why Hund’s rules are important?

Hund’s Rules

• Why Hund’s rules are important?

eff  eff g B j( j  1)

z  zgm j  B

Blundell, Magnetism in Condensed Matter (2001) Magnetization “M” and magnetic susceptibility “χ”

• Magnetic susceptibility

Ferromagnetism

  0

  0 Magnetization “M” and magnetic susceptibility “χ”

• Magnetic susceptibility

Hummel, Electronic Properties of Materials (2000) Magnetization “M” and magnetic susceptibility “χ”

• Magnetic susceptibility (T-dependence)

Diamagnetism Category of magnetism

Different behaviors of susceptibility  1 

Coey, Magnetism and Magnetic Materials (2009) Category of magnetism

• Diamagnetism

Lenz law

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism

• Paramagnetism

Ferromagnetic material above TORD (Tc) becomes paramagnetic

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism

• Ferromagnetism

M s J  0

Ferromagnetic material below TORD (Tc)

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism

• Antiferromagnetism

J  0

 p

Antiferromagnetic material below TORD (TN)

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism

• Ferrimagnetism (“Inverse” spinel)

M s

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism

• Ferrimagnetism (tetrahedral) (octahedral)

M s

J  0

C. M. Hurd, Contemporary Physics, 23:5, 469 (1982) Category of magnetism

• Ferrimagnetism

– Iron garnets, R3Fe2(FeO4)3

– Yttrium Iron Garnet (YIG), Y3Fe2(FeO4)3 – “Fruit-fly for magnetic studies” Charles Kittel

Category of magnetism

• Ferro, antiferro, and ferrimagnetism Exchange interaction

J  0

J  0 Category of magnetism

• Magnetic states of different elements at room temp.

Diamagnetism

• Classical diamagnetism – Diamagnetism is present in all matter, but is often obscured by paramagnetism or ferromagnetism. – Classically can be viewed as a manifestation of Lenz law. – However, the classical picture is not accurate (Bohr-van Leeuwen theorem). – Still, let’s look at the classical (orbital) picture of diamagnetism, which is related to the Larmor precession

LI

ddLII    Larmor

e e e2 d dL   m x2  y 2  B   x 2  y 2 B e      2me 2 m e 4 m e Diamagnetism

• Classical diamagnetism

e e e2 d dL   m x2  y 2  B   x 2  y 2 B e      2me 2 m e 4 m e

22 x2 y 2  x 2  y 2  z 2  r 2 33 

er22 dia dB   6me

22 nv e0  r dia M H  00 M B   n v  dia B   6me

nv  N/ V  Diamagnetism

• Quantum diamagnetism – Consider the Hamiltonian operator (total energy operator) and momentum operator…

2 ˆ pA e  p  i  H  2me

22ie e HAAAˆ  22  2me 2 m e 2 m e

BA  Consider magnetic field only along z-direction

yB xB A  , ,0 A  0 22 Diamagnetism

• Quantum diamagnetism – Consider the Hamiltonian operator (total energy operator) and momentum operator…

2  pA e L  i x  y ˆ   p  i  z  H  yx 2me

2 2 2 ˆ 2ie B e B 2 2 H     x  y  x  y  2me 2 m e y x 2 8 m e Kinetic energy (Orbital) paramagnetism Diamagnetism

2 2 2 2 e B2 2 e B 2 Edia  x  y  r 8mmee 12 Diamagnetism

• Quantum diamagnetism – Consider the Hamiltonian operator (total energy operator) and momentum operator…

e2 B 2 e 2 B 2 E x2  y 2  r 2 dia   8mmee 12

E eB2   dia   r 2 Bm6 e

22 nv e0  r 2 dia    Zreff (indep. of T) 6me Paramagnetism

• Classical paramagnetism – Consider an ensemble of atoms (moments) – Statistical approach

E EUB  μ B  cos P exp(probability) kTB

cos  exp B cos  k T d   B  z  cos  expB cos k T d   B 

dsin d  d  (solid angle) Paramagnetism

• Classical paramagnetism – Consider an ensemble of atoms (moments) – Statistical approach

cos  exp B cos  k T sin  d     B z expB cos  k T sin  d    B  cos  exp B cos  k T d cos    B expB cos  k T d cos    B  1 xexp sx dx 1  1 s  B kB T expsx dx 1 x  cos Paramagnetism

• Classical paramagnetism – Consider an ensemble of atoms (moments) – Statistical approach

1 xexp sx dx s es e s  e s  e s    1    z 1 ss expsx dx s e e  1

1 z   coths     L ( s ) (Langevin function) s

At low field and/or high T B s  0 L( s )  s / 3  3kTB Paramagnetism

• Classical paramagnetism – Curie’s law

2 nv0 C para M H  0 n v  z B   3kB T T

Langevin function Paramagnetism

• Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j

z gm j B

E U  μ B   gmjB B

j  mjexp gm j B B / k B T  mjj  M nv g m j B  n v g B  j  expgmj B B / k B T  mjj  Paramagnetism

• Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j

M nv g B j  B j()() x  M s B j x

(Saturation x gBB jB/ k T Ms n v g B j magnetization) Brillouin function

2j 1 2 j 1  1  x  Bj ( x ) coth x  coth   2j 2 j  2 j  2 j  Paramagnetism

• Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j

Brillouin function 2j 1 2 j 1  1  x  Bj ( x ) coth x  coth   2j 2 j  2 j  2 j 

j  j 1/ 2

B( x ) tanh( x ) Bj ()() x L x j1/2 Langevin function (spin-1/2 system) Paramagnetism

• Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j

Brillouin function 2j 1 2 j 1  1  x  Bj ( x ) coth x  coth   2j 2 j  2 j  2 j 

 jx1 3 Bj () x   O x  when x 1 3 j

 j11 x j gB jB M nv g B j   n v g B 33j kB T Paramagnetism

• Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j

2 j j1 gB  B Mn v 2 3kTB eff

2 2 nvB0 j j1 g  nv0 C   para  para-J 3k T T 3kTB B (Curie’s law)

eff gB j( j 1) Paramagnetism

• Quantum paramagnetism – Discrete possible states – Consider the total angular momentum quantum number j – Curie constant C for quantum scenario

C   T

2 n j j1 g  n 2 C vB0 v 0 eff 33kkBB Paramagnetism

• Quantum paramagnetism – Brillouin function

when x 1

Ms n v g B j

when x 1 Paramagnetism

• Quantum paramagnetism

when x 1

Ms n v g B j when x 1 Paramagnetism

• Pauli paramagnetism – Free electron model – Band structure (Fermi energy)

http://nptel.ac.in/courses/115103038/module2/lec7/ Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field

  2000 HME   (Remember Hw2?)

MHHHHM   applied E    applied   

CM  THMapplied   MCC     HTCTTapplied  c Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field

  2000 HME   (Remember Hw2?)

– Connection to “exchange interaction” and total Hamiltonian

ˆ H  JijSSSBSBB i  j  g B  i   g B  i  mf   i, j i i

BBHHHMmf  0 E    0     Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field

MCC     HTCTTapplied  c Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field

CC   TCTT c

• Curie Temperature Tc

2 nv 0  eff TCc  3kB Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field – Again, borrow from theory of quantum paramagnetism

MngjBxMBxMv B  j( )  s j ( )  s tanh x

Ms n v g B j

xgjBkTgjBBBB//   0 H applied   MkT

11kBB T   k T  M x  Happlied    x  H applied  gj      BB00j  1/ 2   Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field – Again, borrow from theory of quantum paramagnetism – Find M(x) by plotting these two functions!

MM/ s

tanhx

M/ Ms  tanh x

1 kTB M/ Ms  x Happlied nv  B  B 0

Happlied

nvB Ferromagnetism

• Curie-Weiss law – Weiss internal field, Weiss exchange field, Weiss molecular field – Again, borrow from theory of quantum paramagnetism

MM/ s Ferromagnetism

• Curie-Weiss law – Curie temperature of various ferromagnetic materials