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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] 4μ0 r r m m1 2 Here, m2 receives magnetic force (= magnetic field) : 1 m1 F = m2H H = 2 [N/Wb] = [A/m] 4μ0 r Magnetic flux density is proportional to the magnetic field : m Magnetic flux density at r from a magnetic pole m is B = 4r 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 = 4ki a a 0 i Ampère’s law Hdl = i ()4k 1 H = 2a 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 4r 2 a i ds i cos ia cos ia 2 = 2a = = 4r 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 4μ0 r 4μ0r 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 4μ0 r 4μ0 r r r 4μ0r 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 4μ0r r 2μ0r 2 Assuming m = μ0iA (A = a ), x r 2 μ0iA ia a H = 3 = 3 i ds 2μ0r 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 = mlh ()ml = 0, ±1, ± 2, 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 = μBL h -29 μB : Bohr magneton (1.16510 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 ea 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.
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