Unformatted text preview: Chapter 12. Ferromagne1sm and An1ferromagne1sm Diana Huﬀaker 3/1/10 Quiz • Calculate poten1al energy, U of an electron with magne1c moment μ in a ﬁeld B. • What is the force, F on the spinning electron with magne1c moment μ in a ﬁeld B. Last Lecture • Finished paramagne1cs This Lecture •
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• Introduc1on to ferromagne1c materials Mean ﬁeld approxima1on
below and above Tc. Heisenberg model – calculates exchange energy and integral Spontaneous and satura1on magne1za1on Magnons and spin waves Homework – due Thursday 3/5 Bring a manuscript using key words: • Spin wave • Exchange Interac1on Be prepared to discuss: What materials? What measurements? What applica1ons? Homework – due Tuesday 3/8 12.1 – Magnon dispersion rela1onship – derive equa1on (24). 12.2 – Use magnon dispersion to calculate heat capacity. ;) 12.6 – derive satura1on magne1za1on below Curie temperature. Due Thursday 3/10 Submit a ﬁnal exam ques1on – Chapter 12, 14. 5 ques1ons of varying diﬃculty. If your ques1on is selected – you get full points! Final will cover chapter 11, 12, 14. Ferromagne1c Order • A ferromagnet has a spontaneous magne1c moment – even in zero applied magne1c ﬁeld • Existence of a spontaneous moment: electron spins and magne1c moments are arranged in a regular manner • All of the spin arrangements except the “simple an1ferromagnet ” have a spontaneous magne1c moment: satura1on moment Simple ferromagnet Simple an1ferromagnet Ferrimagnet • All of these alignment eﬀects only occur at temperatures below a certain cri1cal temperature, called the Curie temperature (for ferromagnets and ferrimagnets) or the Néel temperature (for an1ferromagnets). Ferromagne1sm
origin • The property of ferromagne1sm is due to the direct inﬂuence of two eﬀects from quantum mechanics: spin and the Pauli exclusion principle.[5] • The spin of an electron, combined with its electric charge, results in a magne1c dipole moment and creates a magne1c ﬁeld • Only atoms with par1ally ﬁlled shells (i.e., unpaired spins) can have a net magne1c moment, so ferromagne1sm only occurs in materials with par1ally ﬁlled shells. • When magne1c dipoles are aligned in the same direc1on, their individual magne1c ﬁelds add together to create a measurable macroscopic ﬁeld. • These unpaired dipoles (spins and angular momentum) tend to align to an external magne1c ﬁeld as paramagne1cs • . Ferromagne1sm involves an addi1onal phenomenon, however: the dipoles tend to align spontaneously, without any applied ﬁeld. This is a purely quantum
mechanical eﬀect. Exchange Interac1on
deﬁni1ons • According to classical electromagne1sm, two nearby magne1c dipoles will tend to align in opposite direc1ons. However in ferromagne1c, they tend to align in the same direc1on due to the exchange interac1on • The Pauli exclusion principle says that two electrons with the same spin cannot also have the same "posi1on” • Under certain condi1ons, when the orbitals of the unpaired outer valence electrons from adjacent atoms overlap, the distribu1on of their electric charge in space is further apart when the electrons have parallel spins than when they have opposite spins. • This reduces the electrosta1c energy of the electrons when their spins are parallel compared to their energy when the spins are an1
parallel, so the parallel
spin state is more stable, and the spins of these electrons tend to line up. This diﬀerence in energy is called the exchange energy. Exchange interac1on
deﬁni1on • The exchange interac1on is also responsible for the other types of spontaneous ordering of atomic magne1c moments occurring in magne1c solids, an1ferromagne1sm and ferrimagne1sm. • In most ferromagnets the exchange interac1on is much stronger than the compe1ng dipole
dipole interac1on. For instance, in iron (Fe) the exchange interac1on between two atoms is about 1000 1mes stronger than the dipole interac1on. Spontaneous Magne1za1on and Tc • Ferromagne1c materials • Ferrimagne1c materials Ferromagne1c Materials Consider Fe and its ion Fe2+. What is the electronic conﬁgura1on for ion Fe2? [Ar] 3d6 Calculate g, S for Fe2+: S=2, L=2, J=4, g=3/2 What is the electronic conﬁgura1on for ion Fe2? [Ar] 3d6 4s2 starts with [Ar] Periodic Table d
shell ground states • Angular momentum quenching aﬀect. • L=0 is experimentally determined as ground state. • Aﬀects value of gJ, magne1sm, dipole moment. Ferromagne1c Order • Consider N ions each with spin S. • S spins will align spontaneously based on an exchange ﬁeld, BE. – Treat as equivalent to a magne1c ﬁeld BE BE simulates a real magne1c ﬁeld in the expressions for the energy –μ BE and the 1me rate of spin change = torque μ x BE on a magne1c moment μ Not a real magne1c ﬁeld – does not enter into the Maxwell equa1ons Curie Point and the Exchange Integral – Assume BE is propor1onal to the magne1za1on M Mean
ﬁeld approxima1on: BE = λ M (1) λ is a constant, independent of temperature. Each spin sees the average magne1za1on of all the other spins Exchange ﬁeld can be destroyed by thermal agita1on. Paramagne1c Phase and Curie Point • Curie temperature Tc: – The temperature above which the spontaneous magne1za1on vanishes. – T>Tc : disordered paramagne1c phase – T<Tc : ordered ferromagne1c phase • In the paramagne1c phase – An applied ﬁeld Ba will cause a ﬁnite magne1za1on B = χ M
A
p
– M will cause a ﬁnite exchange ﬁeld BE M = χ p ( Ba + BE ) (2) The paramagne1c suscep1bility Curie Point and Suscep1bility • Paramagne1c suscep1bility given by Curie law (Chapter 11): C : the Curie constant substituting (1) into (2) MT = C ( Ba + λ M ) – The suscep1bility χ has a singularity at T= Cλ A ﬁnite M can exist for Ba=0 if Χ is inﬁnite. – At this temperature (and below), there exists a spontaneous magne1za1on, Ms Curie
Weiss law: Describes the observed suscep1bility varia1on in the paramagne1c region above the Curie point. Curie Point • Curie constant and Bohr magneton (Chapter 11): • Mean ﬁeld constant λ: C= 3k B 2
Np 2 µ B p = g 2 J ( J + 1) for iron: Tc ≈ 1000 K [Ar] 3d6 4s2 Calculate λ: g ≈ 2, S ≈ 1 ∴ λ ≈ 5000
BE ≈ λ M ≈ 107 G = 103 T with M ≈ 1700 BE >> B due to other magne1c ions in the crystal. B for a single ion ≅ µB
a3 ≈ 103 G = 0.1T at nearest neighbor site.
How does µ B exert a B field?. Exchange Interac1on – Heisenberg model • Exchange ﬁeld – approximate quantum mechanical exchange interac1on • Heisenberg model deﬁnes interac1on energy between two atoms. U = −2 J ex Si ⋅ S j
The energy of interac1on of atoms i, j Exchange coeﬃcient (6)
Electron spins Related to the overlap of the charge distribu1ons of atoms i, j • Depends on rela1ve orienta1on of the spins – an1parallel or parallel. • Model considers direct coupling between two spin states – no momentum component. The Heisenberg Exchange Interac9on • In 1928 Heisenberg applied QM to the magne1c problem, which yields the interac1on energy, Vex is given by: Vex or U = −2 J ex Si ⋅ S j Exchange coeﬃcient or Exchange constant Spins of atoms i and j • Jex is strongly related to the overlap of the two atoms and hence to the inter
nuclear distance. The Heisenberg Exchange Interac9on The Heisenberg Exchange Interac9on Interac1on Energy Exchange integral is given by: e2
J ex = ∫ d 3r1 ∫ d 3r2ψ i ( r1 )ψ j ( r1 )
ψ i ( r2 )ψ j ( r2 )
r1 − r2
Exchange charge VI(r1,r2) = e2 /  r1 − r2  Where ψi(r1) is the real wave func1on for the orbital associated with site i and e
with posi1on coordinate r1 • The exchange energy arises from atomic orbital energy calcula1ons. • Can apply to single atoms or adjacent atoms. • For single atom case, energy does not exist for an1parallel spins. Only for parallel spins. • Always posi1ve for orthogonal orbitals, lowers energy of two electrons with parallel spins.is lowest for parallel but orthogonal orbits – i.e. Hund’s rule. • For two adjacent atoms either parallel or an1
parallel spins are valid. • In inter
atomic case, the exchange energy may be + or
due to the non
orthogonality of the orbitals and is sensi1ve to the inter
nuclear separa1on between the atoms The Heisenberg Exchange Interac9on • Exchange energy can be either + or – • The signs are picked out according to equa1on related to Vex : Si and Sj are parallel, Vex < 0 and Jex is posi1ve Si and Sj are an1
parallel, Vex > 0 and Jex is nega1ve • Realis1c calcula1on of Jex and its distance dependence are diﬃcult to perform • Qualita1ve value of Jex goes from – to + as r increases, as non
orthogonality becomes less important, and then becomes very small. The Heisenberg Exchange Interac9on • Fe, Co, Ni sa1sfy the criterion for ferromagne1c behavior (Jex is posi1ve) • Cr and Mn does not sa1sfy the criterion irrespec1ve of the large unpaired spin in Mn • However, MnAs and MnSb are ferromagne1c • The approximate values for the exchange constant can be obtained in terms of Tc by: • Hence high Tc values in the table will have large Jex values Table: (a) Ferro magne1c materials (b) An1ferromagne1c materials (c) Ferrimagne1c materials The Heisenberg Exchange Interac9on • Tc can be calculated by considering the central atom as the ith atom under the inﬂuence of an eﬀec1ve ﬁeld BE due to an average value of the spins of the z
neighboring atoms <Sj> • The steps in obtaining Tc are explained here: Using Eqs. 15
9, 15
10 and 15
14d we obtain: • Here by assuming that the eﬀects are due to spins only and taking a non
zero Jex only between an atom its z
nearest neighbors Tc can be obtained. • Weiss inner ﬁeld at 0 ˚K is : (since the Brillouin func1on (Eq. 15
9) is 1 at low temperature) The Heisenberg Exchange Interac9on • Exchange ﬁeld and exchange integral can be obtained in terms of Tc . • Consider the central atom as the ith atom under the inﬂuence of an eﬀec1ve ﬁeld BE due to an average value of the spins of the z
neighboring atoms <Sj> • Write interac1on energy in terms of exchange interac1on. U = 2∑ j J ex ( ri j ) S j ⋅ Si = − µi ⋅ BE = g µ B Si ⋅ BE M gµB S j N / V C= µ = g µ B Si from 11.11 NV 2 2
g µ B 2 S ( S + 1)
3k B 2 zJ ex S ( S + 1)
Tc = λC ≈
3k B
• Here by assuming that the eﬀects are due to spins only and taking a non
zero Jex only between an atom and its z
nearest neighbors Tc can be obtained. • Weiss inner ﬁeld at 0 ˚K is : BE (0) = λ M s (0) = λ ( N V ) g J µ B J (since the Brillouin func1on (Eq. 15
9) is 1 at low temperature) λ ≈ 5x103 for Iron Curie Point and the Exchange Integral • Connec1on between the exchange integral J and the Curie temperature Tc – Suppose the atom has z nearest neighbors, each connected with the central atom by the interac1on Jex – From the mean ﬁeld theory: J ex = 3k BTc
2 zS ( S + 1) – For sc, bcc, fcc structures with S= ½, k BTc / zJ ex = 0.500
– From be}er sta1s1cal approxima1ons: k BTc / zJ ex = 0.28; 0.35; 0.346.
– If iron is represented by Heisenberg model with S=1, the observed Curie temperature corresponds to Jex=11.9eV – Later, we will see connec1on between Jex and BE. Magne1za1on in Ferro
phase • Use mean ﬁeld approxima1on to ﬁnd the magne1za1on below the Curie temperature. no applied field: B = B E • Use Brillouin expression for the magne1za1on rather than Curie Law: Brillouin func1on M = N µ tanh( µ B / k BT ) from 11.1720 assumes
J=1/2, g=2 considering B E = λ M : M = N µ tanh( µλ M / k BT ) ...
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 Winter '11
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