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MAGNETIC MATERIALS SD-JIITN-PH611-MAT-SCI-2013 All magnetic phenomena result from forces between electric charges in motion. In an atom, magnetic effect may arise due to: 1. Effective current loop of electrons in atomic orbit (orbital Motion of electrons); 2. Electron spin; 3. Motion of the nuclei. SD-JIITN-PH611-MAT-SCI-2013 FUNDAMENTAL RELATIONS 1. RELATION BETWEEN B, H and M A magnetic field can be expressed in terms of Magnetic field intensity (H) and Magnetic flux density. In free space, these quantities are related as B 0 H (1.1) In a magnetic material, above relation is written as B H (1.2) Here 0 = absolute permeability of free space, = absolute permeability of the medium and / 0 = r = relative permeability of the magnetic material. SD-JIITN-PH611-MAT-SCI-2013 MAGNETIZATION (M) Magnetization is defined as magnetic moment per unit volume and expressed in ampere/ meter. It is proportional to the applied magnetic field intensity (H). M H (1.3) Here, = r – 1 = Magnetic susceptibility (cm-3). Let us consider B H B 0 r H 0 r H 0 H 0 H B (r 1)0 H 0 H B 0 H 0 H B 0 M 0 H B 0 ( M H ) (1.4) SD-JIITN-PH611-MAT-SCI-2013 CLASSIFICATION OF MAGNETIC MATERIALS Diamagnetic: Examples: Materials with –ve magnetic susceptibility . (Au) = - 3.6 cm-3, (Hg) = - 3.2 cm-3 (H2O)) = - 0.2X10-8 cm-3) Paramagnetic: Magnetic materials with +ve and small magnetic susceptibility . Examples: (Al) = 2.2X10-5 cm-3 (Mn)= 98 cm-3 Ferromagnetic: Magnetic materials with +ve and very large magnetic susceptibility . Examples: Normally of the order of 105 cm-3. SD-JIITN-PH611-MAT-SCI-2013 2. A MICROSCOPIC LOOK In an atom, magnetic effect may arise due to: 1. Effective current loop of electrons in atomic orbit (orbital Motion of electrons); 2. Electron spin; 3. Motion of the nuclei. SD-JIITN-PH611-MAT-SCI-2013 MAGNETIC MOMENTS AND ANGULAR MOMENTUM 1. ORBITAL MOTION Consider a charged particle moving in a circular orbit (e.g. an electron around a nucleus), The magnetic moment may be given as IA q r 2 qr 2 2 But L r p r mv 2 L mrv mr Therefore, q L 2m (2.1.1) SD-JIITN-PH611-MAT-SCI-2013 For an electron orbiting around the nucleus, magnetic moment would be given as e L L e 2m 2m Where, B l (l 1) B l (l 1) (2.1.2) e 9.27 x10 24 = Bohr Magneton 2m In the equation (2.1.2) e 2m L L (2.1.3) orbital gyro-magnetic ratio, . SD-JIITN-PH611-MAT-SCI-2013 2. ELECTRON SPIN Electrons also have spin rotation about their own axis. As a result they have both an angular momentum and magnetic moment. But for reasons that are purely quantum mechanical, the ratio between to S for electron spin is twice as large as it is for a orbital motion of the spinning electron: e S S m (2.2.1) 3. NUCLEAR MOTION Nuclear magnetic moment is expressed in terms of nuclear magneton e n 2m p mp is mass of a proton. (2.3.1) What happens in a real atom? In any atom, there are several electrons and some combination of spin and orbital rotations builds up the total magnetic moment. The direction of the angular momentum is opposite to that of magnetic moment. Due to the mixture of the contribution from the orbits and spins the ratio of to angular momentum is neither -e/m nor –e/2m. e gJ J 2m Where g is known as Lande’s g-factor. It is given as gJ 1 J ( J 1) S ( S 1) L( L 1) 2 J ( J 1) SD-JIITN-PH611-MAT-SCI-2013 DIAMAGNETISM Diamagnetism is inherent in all substances and arises out of the effect of a magnetic field on the motion of electrons in an atom. Suppose an electron is revolving around the nucleus in atom, the force, F, between electron and the nucleus is F m 2 r When this atom is subjected to a magnetic field, B, electron also experiences an additional force called Lorentz force FL erB Thus when field is switched on, electron revolves with the new frequency, , given by F erB m '2 r m 2 r erB m '2 r SD-JIITN-PH611-MAT-SCI-2013 m 2 m '2 eB m 2 r erB m '2 r ( 2 '2 ) For small B, eB m ( ' )( ' ) ' eB 2 m and eB m ' 2 eB 2m Thus change in magnetic moment is er 2 er 2 eB 2 2m 2 er 2 [ ] 2 e2r 2 B 4m SD-JIITN-PH611-MAT-SCI-2013 Suppose atomic number be Z, then equation (2.9) may be written as, i z e 2 ri 2 i 1 4m B Where, summation extends over all electrons. Since core electrons have different radii, therefore e2 Z r 2 B 4m If the orbit lies in x-y plane then, r 2 x 2 y 2 If R represents average radius, then for spherical atom R 2 x 2 y 2 z 2 For spherical symmetry, R2 x y z 3 2 2 2 Therefore, r 2 x 2 y 2 2 2 R 2 r 3 Therefore, equation (2.10) may be written as e 2 BZ 2 ( R 2 ) 4m 3 2 2 R Ze B Thus, 4m 3 Ze 2 B 2 R 6m 2 If there are N atoms per unit volume, the magnetization produced would be NZe 2 B 2 M N R 6m 0 NZe2 H 6m R2 Susceptibility, , would be 0 NZe 2 B 2 0 M R B 6mB NZ0e 6m 2 R2 This is the Langevin’s formula for volume susceptibility of diamagnetism of core electrons. Conclusions: Diamagnetic susceptibility 0 NZe2 R 2 M H 6m 1. Since Z, bigger atoms would have larger susceptibility. 2. dia depends on internal structure of the atoms which is temperature independent and hence the dia. 3. All electrons contribute to the diamagnetism even s electrons. 4. All materials have diamagnetism although it may be masked by other magnetic effects. Example: R = 0.1 nm, N = 5x1028/ m3 4 107 5 1028 (1.6 1019 ) 2 (0.1109 ) 2 6 3 3 10 cm 6 9.11031 SD-JIITN-PH611-MAT-SCI-2013 PARAMAGNETISM Paramagnetism occurs in those substances where the individual atoms, ions or molecules posses a permanent magnetic dipole moments. The permanent magnetic moment results from the following contributions: - The spin or intrinsic moments of the electrons. - The orbital motion of the electrons. - The spin magnetic moment of the nucleus. SD-JIITN-PH611-MAT-SCI-2013 Examples of paramagnetic materials: - Metals. - Atoms, and molecules possessing an odd number of electrons, viz., free Na atoms, gaseous nitric oxide (NO) etc. - Free atoms or ions with a partly filled inner shell: Transition elements, rare earth and actinide elements. Mn2+, Gd3+, U4+ etc. - A few compounds with an even number of electrons including molecular oxygen. SD-JIITN-PH611-MAT-SCI-2013 CLASSICAL THEORY OF PARAMAGNETISM Let us consider a medium containing N magnetic dipoles per unit volume each with moment . In presence of magnetic field, potential energy of magnetic dipole V .B B cos Where, is angle between magnetic moment and the field. B =0, M=0 B ≠0, M≠0 V B (minimum) when 0 It shows that dipoles tend to line up with the field. The effect of temperature, however, is to randomize the directions of dipoles. The effect of these two competing processes is that some magnetization is produced. Suppose field B is applied along z-axis, then is angle made by dipole with z-axis. The probability of finding the dipole along the direction is f(θ ) e V kT e μ B co s θ kT f() is the Boltzmann factor which indicates that dipole is more likely to lie along the field than in any other direction. The average value of z is given as z f ( )d f ( )d z Where, integration is carried out over the solid angle, whose element is d. The integration thus takes into account all the possible orientations of the dipoles. SD-JIITN-PH611-MAT-SCI-2013 Let n(θ) be the number of dipoles per unit solid angle at θ, we have n( ) n0 e pE cos kT The number of dipoles in a solid angle d n0 e pE cos kT Note: Here d is calculated as follows: d n0 e pE cos kT 2 sin d Dipole moment of dipoles making angle with the field (along x-axis) is p x p cos Therefore, the dipole moment along the field within angle d n0 e pE cos kT 2 sin d ( p cos ) Now, average dipole moment (Total dipole moment divided by total no. of dipoles) can be written as n e pE cos kT 0 p 2 sin ( p cos )d 0 n e 0 0 pE cos kT 2 sin d n e pE cos kT 0 p 2 sin ( p cos )d 0 n e 0 pE cos kT 2 sin d p p pE cos x kT and p e a e a 1 a a p e e a CASE 2: When a is very low (at high temperature) i.e. a <<1 a L(a ) 3 0 e 0 0 Let e pE cos kT Np 2 E Po 3kT pE a kT sin cos d pE cos kT sin d p a L( a ) p 3 p o E a p p 3 p2 o 3kT p2E p 3kT 1 o T Some of the quantities are replaced and the result is almost same Substituting z = cos and d = 2 sin d z cos 2 sin e 0 2 sin e B cos kT B cos kT d cos sin e 0 sin e d 0 B cos B cos kT kT d d 0 z cos sin e a cos d 0 Let sin e a cos d B kT a 0 Let cos = x, then sin d = - dx and Limits -1 to +1 1 z xeax dx 1 1 ax e dx ea ea 1 z a a a e e 1 SD-JIITN-PH611-MAT-SCI-2013 coth( a ) ea ea 1 1 coth( a ) z a a a e e a z L(a) [a B kT ] Langevin function, L(a) a a3 2 a5 L( a ) 3 45 945 In most practical situations a<<1, therefore, a L(a ) 3 a 2B z 3 3kT The magnetization is given as N 2 B M N z 3kT Variation of L(a) with a. ( N = Number of dipoles per unit volume) SD-JIITN-PH611-MAT-SCI-2013 N 2 B N 2 0 H M N z 3kT 3kT N0 2 M H 3kT This equation is known as CURIE LAW. The susceptibility is referred as Langevin paramagnetic susceptibility. Further, contrary to the diamagnetism, paramagnetic susceptibility is inversely proportional to T Above equation is written in a simplified form as: C T N0 2 where, C 3k Curie constant SD-JIITN-PH611-MAT-SCI-2013 Self study: 1. Volume susceptibility () 2. Mass susceptibility (m) 3. Molecular susceptibility (M) Reference: Solid State Physics by S. O. Pillai SD-JIITN-PH611-MAT-SCI-2013 QUANTUM THEORY OF PARAMAGNETISM Recall the equation of magnetic moment of an atom, i. e. e S - S m e L L 2m e J -g J 2m Where g is the Lande’ splitting factor given as, J ( J 1) S ( S 1) L( L 1) g 1 2 J ( J 1) here, J L S Consider only spin, L0 J S S ( S 1) S ( S 1) g 1 2S ( S 1) e J S -2 S 2m 2 e S - S m SD-JIITN-PH611-MAT-SCI-2013 Consider only orbital motion, S 0 J L J ( J 1) S ( S 1) L( L 1) g 1 2 J ( J 1) L( L 1) L( L 1) g 1 1 2 L( L 1) e J L -1 L 2m e L L 2m Let N be the number of atoms or ions/ m3 of a paramagnetic material. The magnetic moment of each atom is given as, e J -g J 2m In presence of magnetic field, J z MJ according to space quantization. Where MJ = –J, -(J-1),…,0,…(J-1), J i.e. MJ will have (2J+1) values. PARAMAGNETIC SUSCEPTIBILITY IN QUANTUM TERMS j 2 g 2 B 2 J ( J 1) g 2 B 0 N 0 J M N J ( J 1) H 3kT 3kT 2 N0 M 2 2 H 3kT Np 2 eff B 0 where, 3kT C T 2 peff g[ J ( J 1)] C 1 2 Npeff B 0 2 where, C T 2 3k This is curie law. Further, peff B J peff J B Thus Peff is effective number of Bohr Magnetons. C is Curie Constant. Obtained equation is similar to the relation obtained by classical treatment. SD-JIITN-PH611-MAT-SCI-2013 Calculation of peff: 1. Write electronic configuration. Say for 6C 2 2 1s 2s 2 p Partially filled sub-shell 2 2. Find orbital quantum number (l) for partially filled sub-shell. l 1 In the given case: 3. Obtain magnetic quantum number. In the given example: In the given case: ml 1,0,1 4. Accommodate electrons in d sub shell according to Pauli’s exclusion principle. For the given case: ml 1 0 -1 ms SD-JIITN-PH611-MAT-SCI-2013 5. Apply following three Hund’s rules to obtain ground state: (i) Choose maximum value of S consistent with Pauli’s exclusion principle. ml 1 0 -1 In the given example: ms S ms 2 1 1 2 (ii) Choose maximum value of L consistent with the Pauli’s exclusion principle and rule 1. In the given example: L 1 (iii) If the shell is less than half full, J = L – S and if it is more than half full the J = L + S. ml 1 0 -1 ms 5. Obtain J. Since, shell is less than half filled therefore, J L S 1 1 0 6. Obtain g. In the given example: gJ 1 J ( J 1) S (S 1) L( L 1) 2 J ( J 1) Now calculate peff using peff g[ J ( J 1)] 1 2 For 6C, peff = 0, hence it does not show paramagnetism. SD-JIITN-PH611-MAT-SCI-2013 WEISS THEORY OF PARAMAGNETISM Langevin theory failed to explain some complicated temperature dependence of few compressed and cooled gases, solid salts, crystals etc. Further it does not throw light on relationship between para and ferro magnetism. Weiss introduced concept of internal molecular field in order to explain observed discrepancies. According to Weiss, internal molecular field is given as H i M Where is molecular field coefficient. Therefore the net effective field should be H e H M But, we know from classical treatment of paramagnetism that a B M Ms( ) (For a << 1) (a ) 3 kT SD-JIITN-PH611-MAT-SCI-2013 N 2 0 H e a M Ms( ) 3 3kT N 2 0 M ( H M ) 3kT N 2 0 N 2 0 M (1 ) H 3kT 3kT N 2 0 H M N 2 0 3kT (1 ) 3kT M H N 2 0 N 2 0 2 N 0 N 2 0 3kT (1 ) 3k (T ) 3kT 3k C T c Curie-Weiss Law. Paramagnetic N 2 0 curie point where c 3k 2 N 0 and C 3k Curie constant SD-JIITN-PH611-MAT-SCI-2013 FERROMAGNETIC MATERIALS SD-JIITN-PH611-MAT-SCI-2013 FERROMAGNETIC MATERIALS Certain metallic materials posses permanent magnetic moment in the absence of an external field, and manifest very large and permanent magnetization which is termed as spontaneous magnetization. Example: Fe, Co, Ni and some rare earth metals such as Gd. ~ 106 are possible for ferromagnetic materials. Therefore, H<<M and B 0 H 0 M B 0 M Spontaneous magnetization decreases as the temperature rises and is stable only below a certain temperature known as Curie temperature. SD-JIITN-PH611-MAT-SCI-2013 ORIGIN: Atomic magnetic moments of un-cancelled electron spin. Orbital motion also contributes but its contribution is very small. Coupling interaction causes net spin magnetic moments of adjacent atoms to align with one another even in absence of external field. This mutual spin alignment exists over a relatively larger volume of the crystal called domain. Dimension ~ 10-2 cm, No. of atom/ domain 1015 to 1017 The maximum possible magnetization is called saturation magnetization. M s N There is also a corresponding saturation flux density Bs (=0Ms). SD-JIITN-PH611-MAT-SCI-2013 DOMAINS AND HYSTERESIS What happens when magnetic field is applied to the ferromagnetic crystal? According to Becker, there are two independent processes which take place and lead to magnetization when magnetic field is applied. 1. Domain growth: Volume of domains oriented favourably w. r. t to the field at the expense of less favourably oriented domains. 2. Domain rotation: Rotation of the directions of magnetization towards the direction of the field. ORIGIN OF DOMAINS According to Neel, origin of domains in the ferromagnetic materials may be understood in terms of thermodynamic principle that IN EQUILIBRIUM, THE TOTAL ENERGY OF THE SYSTEM IS MINIMUM. Total energy: 1. Exchange energy; 2. Magnetic energy; 3. Anisotropy energy and 4. Domain wall or Bloch wall energy. 1. Exchange Energy Ee 2 J e Si S j It is lowered when spins are parallel. Thus, it favours an infinitely large domain or a single domain in the specimen. 2. Magnetic Energy This arises because the magnetized specimen has free poles at the ends and thus produce external field H. Magnitude of this energy is 1 8 2 H dv Value of this energy is very high and can be reduced if the volume in which external field exists is reduced and can be eliminated if free poles at the ends of the specimen are absent. 3. Anisotropy energy For bcc Fe [100] easy direction [110] medium direction [111] hard direction For Ni [111] easy direction [110] medium direction [100] hard direction The excess energy required to magnetize a specimen in a particular direction over that required to magnetize in the easy direction is called crystalline anisotropy energy. 4. Domain wall or Bloch wall energy Domain wall creation involves energy which is known as domain wall energy of Bloch energy. EXCHANGE INTERACTION IN MAGENTIC MATERIALS Heisenberg (1928) gave theoretical explanation for large Weiss field in ferromagnetic materials. Parallel arrangements of spins in ferromagnetic materials arises due to exchange interaction in which two neighboring spins. The exchange energy of such coupling is Eexch 2 Js1s2 Here J = exchange integral. Its value depends upon separation between atoms as well as overlap of electron charge cloud. J > 0, favors parallel configuration of spins, while for J < 0, spins favors anti-parallel. If there are Z nearest neighbors to a central ith spin, the exchange energy for this spin is Eexch 2 J ij Si .S j 2ZJS 2 j 1 This energy must be equal to K as at , ferromagnetic order is destroyed. Thus, 2ZJ ij S 2 K J ij K 2ZS 2 Thus criteria for ferromagnetism (due to Slater) becomes – atoms must have unbalanced spins and the exchange integral J must be positive. Alloys like Mn-As, CuMn and Mn-Sb show ferromagnetism. Idea of magnetic energy due to domain: Based on the definition of these energies a scheme is drawn below which helps in minimization of energy of the system: Domain closure Single domain Magnetic energy high Domain halved magnetic energy reduced Elimination of magnetic energy by domain closure BLOCH WALL (Due to Bloch) The entire change in spin direction between domains does not occur in one sudden jump across a single atomic plane rather takes place in a gradual way extending over many atomic planes. Bloch Wall Because for a given total change in spin direction, the exchange energy is lower when change is distributed over many spins than when the change occurs abruptly. From the Heisenberg model, exchange energy is Eexch 2 J e Si .S j Eexch 2 J e S 2 cos 0 Substituting cos 0 1 0 2 2 ... Eexch 2 J e S (1 2 (Where 0 is the angle between two spins.) 0 2 2 ...) For small angle , the change in exchange energy when angle between spins change from 0 to is Eexch Eexch ( ) Eexch (0) Eexch 2 J e S 2 (1 0 2 2 ...) (2 J e S 2 ) Eexch 2 J e S 2 (1 0 2 ...) (2 J e S 2 ) 2 2 J e S 2 2 J e S Eexch J e S 20 2 2 0 2 ... 2 J e S 2 2 Thus exchange energy increases when two spins are rotated by an angle from exact parallel arrangement between them. Now suppose the total change of angle between two domains occurs in N equal steps. Thus the change of angle between two neighbouring spins = Eexch J e S 2 2 0 N2 0 N Eexch J e S 2 2 0 N2 Thus total energy change in N equal steps (Eexch )T J e S 2 2 0 N 2 N ( Eexch )T J e S 2 2 0 N Thus total energy change decreases when N increases. Q. Why does not the wall becomes infinitely thick. Ans. Because of increase of the anisotropy energy. Therefore competing claims between exchange energy and anisotropy energy leads to an equilibrium thickness. Exchange energy per unit area (refer to the figure), 2 Eexch J e S 2 0 2 (Where = ) 0 Na Eanis KNa Anisotropy energy is Where K = anisotropy constant, a = lattice constant Thus total wall energy would be Ew J e S 2 0 2 Na 2 dE J e S 2 20 2 Ka dN N a 2 KNa dE 0 J e S 2 20 2 Ka dN N a 2 For minimum E wrt N JeS Thus 2 0 2 2 N a Ka N J e S 2 2 Ew J e S JeS 2 0 2 Na 2 (JeS 2 2 0 2 Ka 3 N (JeS KNa 0 2 2 2 0 Ka 1 2 2 ) a 3 K (JeS 2 0 2 Ka 3 1 2 ) a 2 0 2 Ka 3 ) 1 2 Ew J e S 2 (JeS 0 2 2 2 0 Ka K (JeS 2 1 2 3 ) a 0 0 2 2 Ew J e S 2 1 2 Je S 1 2 K a Ew 1 2 1 2 a Thus Ka KJ e S 3 2 a2 3 1 2 ) a 1 2 K a 1 2 1 2 1 2 0 1 2 J e K S0 1 2 0 2 K Je S 3 2 0 a 1 2 J e K 12 Ew 2 S0 ( ) a This is equation for Block wall energy. a Magnetization by domain rotation Domain growth irreversible boundary displacements. Domain growth reversible boundary displacements. Crystal structure FeO·Fe2O3 (Iron ferrite) Tetrahedral site:Fe ion is surrounded by four oxygens Octahedral site:Fe ion is surrounded by six oxygens Applications of Magnetism Applications of Magnetic Induction • Tape / Hard Drive • Tiny coil responds to change in flux as the magnetic domains (encoding 0’s or 1’s) go by. • Credit Card Reader –Must swipe card generates changing flux –Faster swipe bigger signal Capacity of the disk: Sectors and tracks • The road map consists of magnetic markers embedded in the magnetic film on the surface of the disk. • The codes divide the surfaces of the disk into sectors (pie slices and tracks (concentric circles). • These divisions organize the disk so that data can be recorded in a logical manner • and accessed quickly by the read/write heads that move back and forth over the disk as it spins. • The number of sector and tracks that fit on a disk determines the disk’s capacity. • A typical track is shown in yellow; a typical sector is shown in blue. A sector contains a fixed number of bytes -- for example, 256 or 512. Either at the drive or the operating system level, sectors are often grouped together into clusters. Basic principles of magnetic recording 1. The drive channel electronics receive data in binary form from the computer and converts them into a current in the head coil. 2. The current in the coil reverses at each 1 and remains the same at each 0. 3. This current interaction with the media results in magnetization of the media, which direction depends on the current direction in the coil. Magnetic heads: MR sensors 1. Today's magnetic head typically consists of an MR (magneto-resistive) or GMR (giant MR) reading head and a thin-film inductive write head. 2. MR head design is based on the ability of metals to change their resistivity in the presence of a magnetic field. This effect was first found in 1857. 3. The alloy of Ni and Fe (81%/19%) is widely used in MR heads and is called Permalloy. 1. MR heads are suitable for extremely high bit density and have superior signal-to-noise ratio when compared to inductive read heads. Multiple platters and heads are commonly used to increase the amount of information • How many platters in this hard disk? 3 • How many read/write heads? 6 Physical organization of the disk Little bit about RAM • Basic Terms: RAM: Random Access Memory (volatile in case of semiconductor RAM so we need to reboot the system after every occasion of power-cut) SRAM: Static RAM (Used mostly for Cache Memory, Static RAM holds information in memory as long as the power is on. It doesn't have to be constantly refreshed, like standard Dynamic RAM (DRAM). Static RAM is faster than DRAM but it's more expensive and takes up more space.) DRAM: Dynamic RAM SDRAM: Short for Synchronous DRAM, a type of DRAM that can run at much higher clock speeds than conventional memory. SDRAM actually synchronizes itself with the CPU's bus and is capable of running at 133 MHz, about three times faster than conventional FPM RAM, and about twice as fast EDO DRAM and BEDO DRAM. DDRAM: Double Data Rate RAM. Simply, this is RAM which handles data at twice the speed of the old SDR (Synchronous Dynamic RAM.) DDR RAM typical operates at DDR speeds of 266MHz, 333MHz, and 400MHz (actual speeds are 133, 166 and 200 respectively.) MRAM: Magnetic RAM • It has magnetic memory which is nonvolatile=>you don’t need to reboot the system after every power cut. There are three technologies: 1. Hybrid semiconductor/magnetic devices 2. Magnetic tunnel junction 3. All metal spin transistors The leakage field from the magnetic elements can be detected by a hall effect sensor and from the sign of the hall voltage the orientation of the memory bit can be determined. Other applications: • Ground fault interrupter • Electric guitar http://entertainment.howstuff works.com/electricguitar1.htm • Magnetic detector at LRC • Metal detector • Generator • …