Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University

MAGNETIC PROPERTIES OF MATERIALS B.Sc. 3rd Year: Modern Physics Course & M.Sc. 4th Sem.: Condensed Matter Physics Course of DDUGU

Presented by,

Dr. Nikhil Kumar (Ph.D., IIT Kanpur) Assistant Professor Department of Physics DDU Gorakhpur University, Gorakhpur, Uttar Pradesh-273009 email: [email protected] [email protected] Website: https://nikjnu.wixsite.com/physics

E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Plan of the Lecture

➢ Basics of Magnetism ➢ Types of Magnetism (a) Diamagnetism *Langevin (Classical) Theory of Diamagnetism *Quantum Theory of Diamagnetism (b) Paramagnetism *Langevin (Classical) Theory of Paramagnetism: Curie Law *Quantum Theory of Paramagnetism (c) Ferromagnetism *Domain Theory and Hysteresis loop in Ferromagnetism *Weiss Molecular Field Theory of Ferromagnetism (d) Antiferromagnetism and Ferrimagnetism (e) Comparison Table for various types of magnetic materials ➢ References ➢ Assignment for Students ➢ Advertisement of Experimental Condensed Matter Physics Group, DDUGU 2

E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Basics of Magnetism

➢ Magnetism is a physical property of a material by virtue of which it attracts or repels another material.

➢ The ability to get magnetized in presence of magnetic

field is known as Magnetization (M). B = µ0 (H+M) It is also defined as Magnetic Dipole moment per unit volume.

➢ Magnetic Susceptibility (χm ) is the quantitative major of the extent up to which the material may be magnetized for a given ꞇ= µ x B µ=I A applied magnetic field. χm = M/H ; χm = µr – 1 ➢ Magnetic Dipole Moment (µ) is a quantity that represents the magnetic strength and orientation of a magnet or any other object that produces a magnetic field. The movement of electrons orbiting around the nucleus as current loop creates a south pole and north pole resulting in atom’s behavior as a magnetic dipole.

➢ Basic source of magnetism is because of magnetic moments arising due to motion of electrons around the nucleus, which constitutes orbital motion and spin motion around their own axis. ➢ Only specific materials are magnetic in nature which have 3 unpaired electrons. Figures: Internet Source E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Diamagnetism

➢ When a material is placed in an external magnetic field and very small induced magnetic moment is originated in opposite direction of the applied field because of change in the orbital motion of electrons. This phenomenon is known as Diamagnetism. ➢ It is a nonpermanent and very weak form of magnetism which persists only in the presence of an external field. ➢ Magnetic Susceptibility is negative (-10-6 to -10-5) and temperature independent. (Refer Table 1) ➢ Ex: Atoms with paired electrons such as Bi, Si, Cu, Ag, water, Inert gases and Superconductors below Tc.

Meissner Effect: Superconductors

below Tc are diamagnetic and χm = -1 4

Figure Source: Taken from Internet E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Langevin Theory of Diamagnetism (Classical Electron Theory;1905)

Z B L v -e FL +ze Fc -e Y μ X L

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E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Quantum Theory of Diamagnetism

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E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Paramagnetism

➢ In a paramagnetic substance, the orientation of magnetic dipoles in absence of external magnetic field are random, leading to no net magnetization. However, when an external magnetic field is applied, magnetic dipoles align along the direction of magnetic field inducing net positive magnetization in the sample. Materials which exhibit small positive magnetization and susceptibility in the presence of magnetic filed are known as paramagnetic materials and the effect is termed as Paramagnetism.

➢ The effect is a nonpermanent form of magnetism which persists only in the presence of an external field. Since dipoles do not interact with each other, extremely large magnetic filed is required to align all the magnetic dipoles.

➢ Magnetic Susceptibility is slightly positive (10-5 to 10-3) and varies inversely with

temperature (Curie Law: χm ~ 1/T).

➢ Ex: Atoms with unpaired electrons such as Al, Pt, Mn, Ca, Na, Nb, O2, MnBi etc.

Figures: Internet Source

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E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Langevin Classical Theory of Paramagnetism (Curie Law)

μ

θ

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E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Quantum Theory of Paramagnetism ➢ The components of magnetic moment of an atom are as following:

1.) Orbital magnetic moment µL = - (e/2m)L 2.) Spin magnetic moment µS = -(e/m)S ➢ The total magnetic moment of atom µJ = -g(e/2m)J, where, g is the Lande’ splitting factor given by, g = 1+[J(J+1)+S(S+1)-L(L+1)]/2J(J+1)

➢ Here, J = L+S; In presence of magnetic field, according to space π

quantization. Jz = MJ h/2π h/2 J

where, MJ = –J, -(J-1),…,0,…(J-1), J i.e. MJ will have (2J+1) values. J

➢The magnetic moment of an atom along the magnetic field corresponding M = z to a given value of MJ is (µJ)z = - MJ g µb J ➢If dipole is kept in a magnetic field B then potential energy of the dipole

would be U = -µJ•B = -(µJ)zB so, U = - MJ gµBB ➢By using Boltzmann factor the magnetization: M = N<µ>

where, <µ> = ∑ MJ gµBBexp(MJ gµBB/kbT)/∑exp(MJ gµBB/kbT)

➢ Case (1): when MJ gµBB/kbT >> 1 then we get, M = NgµBJBJ(a) where, BJ(a) = Brillouin function and a = gJµBB/kbT M = MsBJ(a), where, Ms = NgµBJ = Saturation value of magnetization For J = ½, M/Ms = tanh(a); For J = ∞, M/Ms = Coth(a) – 1/a = L(a) = Langevin function ➢ Case (2): when M gµ B/k T >> 1 then we get, Figure: Internet Source J B b 9 M = Ng²µB²BJ(J+1)/3kbT; M = Ng²µB²µoHJ(J+1)/3kbT χ = M/H = NµoµB²/3kbT = C/T ; Where, C = NµB²µo/3kb = Curie constant . E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Ferromagnetism

➢ Ferromagnetic materials are those materials which when placed in an external magnetic field are strongly magnetized in the direction of the field and exhibit a spontaneous net magnetization at the atomic level, even in the absence of the external magnetic field. Ferromagnetic materials are strongly attracted to a magnet.

➢ The effect is a permanent form of magnetism which persists even in the absence of an external field. This is the result of permanent unpaired dipoles formed from unfilled energy levels. Due to exchange interaction or mutual reinforcement these dipoles align with the imposed magnetic field.

➢ Magnetic susceptibility of Ferromagnetic materials such as Fe, Co, Ni, Gd etc. is of the order of 102 -106.

➢ Ferromagnetic materials follow Curie-Weiss law. Above the Curie temperature, ferro- magnetic materials behave as para-magnetic materials and their susceptibility is given by:

[χm ~ 1/(T-Tc)], where Tc is the curie temperature.

➢ Trace out of M vs H curve is a loop known as Hysteresis loop. This is because of the existence of magnetic domains in the ferromagnetic material. Once the magnetic domains are reoriented along the filed direction, it takes some energy to turn them back again. 10 This property of ferromagnetic materials is useful as a magnetic "memory".

E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Hysteresis Loop and Domain Theory

Figures: Nanoscale Research Letters 14(1), 2019 Saturation Magnetization (Ms): If we start with non-magnetized specimen at zero magnetic field. With increasing magnetic field magnetization of the specimen increases and attains a maximum, such that there is no increment in magnetization if we increase magnetic field further. This maximum magnetization is known as saturation magnetization (Ms). At this point all domains are aligned in the direction of magnetic field. Remanant/ Residual Magnetization (Mr): After attaining saturation magnetization in one direction, when the magnetic field is removed, magnetization will not relax back to zero magnetization value but some finite remaining magnetization known as Remanent or Residual magnetization (Mr). Thus it is a measure of remaining magnetization when the driving field is dropped back to zero. Coercivity: It is a measure of the reverse field needed to drive the magnetization to zero after being saturated. Thus the amount of reverse magnetic field required to demagnetize the specimen is known as Coercivity. Further increasing magnetic filed in reverse direction magnetization is saturated in reverse direction. With alternating 11 magnetic field Magnetization will trace out a loop, known as Hysteresis loop .

E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Weiss Molecular field Theory of Ferromagnetism (Curie-Weiss Law) ➢A molecular field tends to produce a parallel alignment of the atomic dipoles despite effect of thermal energy. This internal magnetic field Hm is proportional to the magnetization M of a domain i.e. Hm = λM , where, λ is independent of temperature, called molecular field constant.

➢Effective field experienced by each dipole would be then, He = H + λM ➢Let us consider a ferromagnetic solid containing N number of atoms per unit volume, then magnetization due to spins (J=1/2) can be given as

M = NµBtanh[µoµBHe/kBT] ⇒ M = NµBtanh[µoµB[H +λM]/kBT]

➢At sufficiently high temperature, µoµB[H +λm]<< 1 Then, tanh{µoµB[H +λM]/kBT} ≈ µoµB[H +λM]/kBT 2 Therefore, M = NµoµB [H +λM]/kBT ⇒ M[1 - NµoµB²λ/kBT] = NµoµB²H/kBT ⇒ χ = M/H = [NµoµB²/kBT]/[1 - NµoµB²λ/kBT]

⇒ χ = C/(T – θ) where, C = NµoµB ²/kB and θ = NµoµB²λ/kB = λC This is Curie-Weiss Law.

➢For spontaneous magnetization H = 0 , so we have M = Mstanh{µoµB λM/kBT} 12 ⇒ M/Ms = tanhα where, α = µoµB λM/kBT Figure: Internet Source E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University

Antiferromagnetism and Ferrimagnetism

S. Antiferromagnetism Ferrimagnetism No. 1. Magnetic dipoles with similar In some ceramic materials the dipoles of a cation may align with moment align but alternatively in the magnetic field, while dipoles of other cation may not. These opposite directions, so the net ceramics are called ferrites, and the effect is known as ferri- magnetization is zero. magnetism 2. Magnetic susceptibility of Magnetic susceptibility of ferrimagnets is quite large (> 100). antiferromagnets is positive but Ferrimagnets also follow Curie-Weiss Law. quite small (10-5-10-3) 3. Antiferromagnets attain maximum Ferri-magnetism is similar to anti-ferro-magnetism in that the susceptibility at a temperature spins of different atoms or ions line up anti-parallel. However,

known as Neel temperature (TN), the spins do not cancel each other out, and a net spin moment above which it become exist. Below TN Antiferromagnets behave similar to paramagnetic. Ferromagnets and above it just like paramagnets.

4. Ex: Cr, MnO, FeO etc. Ex: Fe3O4, MnFe2O4, NiFe2O4 etc.

13 Figure Source: Wikipedia

E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Comparison Table for Various Types of Magnetic Materials

14 Table 1: Roberto Nisticò et.al. Inorganics 2020, 8(1), 6 E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University References

➢ Charles Kittel , Introduction to Solid State Physics, 8th Edition,

➢ Aschroft and Mermin, Solid State Physics.

➢ B. D. Cullity and Charles D. Graham Jr, Introduction to Magnetic Materials.

➢ M.A. Wahab, Solid State Physics: Structure and Properties of Materials.

➢ Roberto Nisticò et.al. Inorganics 2020, 8(1), 6

➢ Nanoscale Research Letters 14(1), 2019

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E-Pathshala, DDU Gorakhpur University (http://182.18.165.51/epathshala_content.aspx) Dr. Nikhil Kumar, Department of Physics, DDU Gorakhpur University Assignment for Students

1. Using the ground state wavefunction of Hydrogen atom, calculate the diamagnetic susceptibility of a mole of Hydrogen atom.

2. Consider a helium atom in its ground state (1s). The mean radius of the atom may be approximated by Bohr radius 0.529Å. The density of helium is 0.178 kg/m3. Calculate the diamagnetic susceptibility of helium atom using classical Langevin formula.

3. Estimate the diamagnetic susceptibility of Copper and paramagnetic susceptibility of Oxygen using classical and quantum theories.

4. Discuss classical and Quantum mechanical motion of a charged particle in an electromagnetic field.

5. Estimate the value of Bohr Magneton. Iron has a saturation magnetization of 1.71x106 A/m. What is the average number of Bohr magnetons per atom that contribute to this magnetization? Iron has the BCC crystal structure with a = 0.287 nm. 16

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