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 Wide Bandgap Semiconductor Spintronics editors Preben Maegaard Anna Krenz Wolfgang Palz Vladimir Litvinov The Rise of Modern Wind Energy Wind Power for the World CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20160308 International Standard Book Number-13: 978-981-4669-71-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my wife, Valeria, and my children, Natasha and Vlady This page intentionally left blank Contents Preface GaN Band Structure 1.1 1.2 1.3 1.4 Symmetry Hamiltonian Valence Band Structure Linear k-Terms in Wurtzite Nitrides Rashba Hamiltonian xi 1 13 16 21 2.1 Bulk Inversion Asymmetry 2.2 Structure Inversion Asymmetry 2.3 Microscopic Theory of Rashba Spin Splitting in GaN 22 24 29 3.1 Spontaneous and Piezoelectric Polarization 3.2 Remote and Polarization Doping 3.3 Rashba Interaction in Polarization-Doped Heterostructure 3.4 Structurally Symmetric InxGa1–xN Quantum Well 3.4.1 Rashba Coefficient in Ga-Face QW 3.4.2 Rashba Coefficient in N-Face QW 3.4.3 Inverted Bands in InGaN/GaN Quantum Well 3.5 Experimental Rashba Spin Splitting 38 41 44 50 53 54 57 58 4.1 Double-Barrier Resonant Tunneling Diode 4.1.1 Current–Voltage Characteristics 4.1.2 Spin Current 64 64 66 Rashba Spin Splitting in III-Nitride Heterostructures and Quantum Wells Tunnel Spin Filter in Rashba Quantum Structure 37 63 viii Contents 4.1.3 Tunnel Transparency 4.1.4 Polarization Fields 4.1.5 Spin Polarization 4.2 Spin Filtering in a Single-Barrier Tunnel Contact 4.2.1 Hamiltonian 4.2.2 Boundary Conditions and Spin-Selective Tunnel Transmission Exchange Interaction in Semiconductors and Metals 5.1 5.2 5.3 5.4 Direct Exchange Interaction Indirect Exchange Interaction Three-Dimensional Metal: RKKY Model RKKY Interaction in One and Two Dimensions 5.4.1 1D-Metal 5.4.2 2D Metal 5.5 Exchange Interaction in Semiconductors 5.6 Indirect Magnetic Exchange through the Impurity Band 5.7 Conclusions Ferromagnetism in III-V Semiconductors 6.1 Mean-Field Approximation 6.2 Percolation Mechanism of the Ferromagnetic Phase Transition 6.3 Mixed Valence and Ferromagnetic Phase Transition 6.3.1 Magnetic Moment 6.4 Ferromagnetic Transition in a Mixed Valence Magnetic Semiconductor 6.4.1 Hamiltonian and Mean-Field Approximation 6.4.2 Percolation 6.5 Conclusions 68 72 74 76 76 78 85 86 88 92 95 96 98 100 103 105 109 110 115 118 118 125 126 130 132 Contents Topological Insulators 7.1 7.2 7.3 7.4 7.5 135 Bulk Electrons in Bi2Te3 Surface Dirac Electrons Effective Surface Hamiltonian Spatial Distribution of Surface Electrons Topological Invariant 137 140 145 150 154 8.1 Spin-Electron Interaction 8.2 Indirect Exchange Interaction Mediated by Surface Electrons 8.3 Range Function in Topological Insulator 8.4 Conclusions 160 Magnetic Exchange Interaction in Topological Insulator Index 159 165 172 176 179 ix Indirect Exchange Interaction Mediated by Surface Electrons Figure 8.1 s–d exchange parameters vs z-position of a magnetic impurity Figure 8.2 The spin-flip exchange parameter vs the z-position of a magnetic impurity 8.2  Indirect Exchange Interaction Mediated by Surface Electrons We deal with the Hamiltonian (8.8) where the s–d interaction part is given by Eq (8.12) As shown in Chapter 7, diagonalization of the second term in Eq (8.8) gives an experimentally observable energy spectrum of surface electrons: H S0 + H S1  H = diag(E v , E c , Ev , E c ), E c,v (k ) = E0 + VS + Dk ± E c,v (k ) = E0 + VS + Dk ±  S 2 – Bk  + (  A2k – Vas )2 ,     S 2 – Bk  + (  A2k + Vas )2 , (8.15)    165 166 Topological Insulators Linear spin splitting near the -point E = ± aRk determines the Rashba parameter aR =  A2V , V = V as D2S + 4V as (8.16) The term, proportional to D in Eq (8.15), makes the effective masses of electrons and holes different It has a numerically small effect on the indirect exchange interaction and will be neglected Transition to the representation which diagonalizes the free-electron Hamiltonian also transforms s-d interaction (8.12) Eigenvectors (7.36) (see Chapter 7) diagonalize ​H​S1​ ​ and deliver spectrum (8.15) So, the s-d interaction in this representation consists of matrix elements of ​H​ Ssd​  ​Eq (8.12) taken with eigenvectors Eq (7.36) (Chapter 7): H Ssd =  Q (Z ) exp(i(k – k¢)R0||)ai+k a jk , nA ki,k¢j ij where (8.17) Qij (Z ) = i |W (Z )| j , i , j = 1(V ), 2(C), 3(V ), 4(C) (8.18) Matrix Qij describes the s–d interaction of observable surface electrons with a magnetic atom The interaction depends on the energy of the incoming electron as well as on the position of the localized spin with respect to both surfaces We use the approximate Qij matrix by taking it at k  0, corresponding to low-energy electrons: Q11 = Q44 = –Q22 = –Q33 = gS J  [S × n]z , Q21 = Q12 = –Q43 = –Q34 = J  |V |[S × n]z , Q = Q * = S (2 J V – J ) – ig J (Sn ), 13 31 z 12 z S  Q14 = Q = Q23 = Q = sgn[V](– J 12 gS S z – 2iV J  (Sn )), * Q24 = Q42 = –S z (2 J 12V + J z ) + igS J  (Sn ), * 41 * 32 gS = D S (D2S + 4V as )–1/2, n = k/k (8.19) The spin structure of the interaction is determined by vector and scalar products of impurity spin and the unit vector along the electron in-plane momentum Indirect Exchange Interaction Mediated by Surface Electrons Interaction (8.17) generates the indirect exchange between two magnetic atoms separated by the vector R|| = R||1 – R||2 (see Chapter 5): H int = T 2(2p)4 n2  dk dk¢ wn exp[iR||( k – k¢)]Tr {Q(Z1 , n )G(k , wn) Q (Z 2, n¢)G(k¢, wn )}, (8.20) where G(k, wn) is the Green function, G(k , wn ) = (i wn – H + m)–1 = diag[(i wn – E v + m)–1 , (i wn – E c + m)–1 , (i wn – E v  + m)–1 , (i wn – E c + m)–1 ] (8.21) m is the chemical potential, wn = (2n + 1)pT is the Matsubara frequency, T is the temperature in energy units The trace runs over four quantum numbers: bands and spins After frequency summation in Eq (8.21), the nonzero contributions at T = come from the terms in which the product of the two Green functions comprises the one of initial and the other of the final electron state with energies on both sides of m After trace calculation the structure of Eq (8.20) depends on the position of the chemical potential If the chemical potential lies in the energy gap of the surface spectrum, m < √ ​ ​Ds​ ​ ​/4 + V ​  2 ​ s  ​  ​ ​, a there are no carriers in the surface bands at T = 0, and the indirect exchange stems from interband excitations across the gap: Hcv = T 2(2p)4 n2  dk dk¢exp[iR (k – k¢)] × wn || {8V M(Z1 , Z )[Gv ( k , wn )Gc (k¢, wn ) + Gv (k , wn) Gc (k¢, wn )][S1 × n]z [S2 × n¢]z + 8V M(Z1 , Z )[Gc (k , wn ) Gv (k¢, wn ) + Gv (k , wn )Gc (k¢, wn ](S1n )(S2n¢) + 2gS2 K (Z1 , Z )[Gc (k , wn ) Gv (k¢, wn ) + Gv (k , wn ) Gc (k¢, wn )]S1z S 2z }, M(Z1 , Z2 ) = J (Z1 ) J  (Z2 ), K (Z1 , Z2) = J 12 (Z1 ) J 12 (Z2 ) (8.22) 167 168 Magnetic Exchange Interaction in Topological Insulator In a degenerate slab ( m > D2s /4 + Vas ), the leading interaction term is of the RKKY-type which originates from excitations around the Fermi energy in the surface conduction band  = m – D2 /4 + V as F s Conduction Green functions contribute to the exchange of excitations around the Fermi energy, so the intraband indirect exchange is expressed as Hcc = T  dk dk¢exp[iR (k – k¢)] × 2(2p)4 n2 wn {gS2 M(Z1 , Z )[Gc (k , wn ) Gc (k¢, wn ) + Gc (k , wn ) Gc (k¢, wn )] {[S1 × n]z [S2 × n¢]z + 2gS2 M(Z1 , Z ) Gc (k , wn ) Gc (k¢, wn )(S1 n )(S2 n¢) + 2L(Z1 , Z ) Gc (k , wn ) Gc (k¢, wn ) S1z S 2z}, L(Z1 , Z ) = J z (Z1 ) Jz (Z ) + 2V [ J 12 (Z1 )J z (Z ) + J (Z )J (Z )] + 4V 2K (Z , Z ) 12 z 1 (8.23) (8.24) To find out the spin texture of the indirect exchange we perform integration over angles in Eqs (8.22) and (8.23): I1  = S1z S 2z  exp[iR (k – k¢)]d 1d 2 2p 2p 0 =  d 1exp[ikR cos(1 )] d 2exp[–ik¢Rcos(2 )] = 4p2 J0 (kR ) J0 (k¢R )S1z S 2z , I2 =  (S1n )(S2n¢) exp[iR (k – k¢)]d 1d 2 = S1 S = I3 × 2p 2p  cos(a2 ) exp[–ik¢Rcos(2 )] 4p2 J (kR ) J1 (k¢R )(S1R  )(S2R  ), R2 = [S1 × n]z [S2 × n¢]z exp[iR  (k – k¢)]d 1d 2 = (8.25)  d  cos(a )exp[ikR cos( )]  d 4p2 J (kR )J (k¢R )[S1 × R  ]z [S2 × R  ]z , R 2  (8.26) (8.27) Indirect Exchange Interaction Mediated by Surface Electrons where J0,1(x) are the Bessel function of the first kind Calculations were performed making use a1,2 expressed in terms of angles g1,2 between spin vectors and the vector R|| that connects them, as illustrated in Fig 8.3: cos(a1 ) = cos(g1 ) cos(1 ) – sin(g1 )sin (1 ) (8.28) cos(a2 ) = cos(g2 ) cos(2 ) – sin(g2 )sin (2 ) Figure 8.3 Angles in integrals (8.26) and (8.27) After integration over these angles the interband indirect exchange from Eq (8.22) can be written as Hcv = × 4p2n2 4   V M (Z1 , Z )[Fv1c + Fv1c ] [S1 × R  ]z [S2 × R  ]z   R + where  V M (Z1 , Z ) [Fc1v + Fv1c ] (S1R  ) (S1R  )  R   + gS2 K (Z1 , Z )[Fc0v + Fv0c ] S1z S 2z ,  (8.29) 0,1 0,1 0,1 Fisjs ¢ = T Gis (wn , R|| )G js¢ (wn , R|| ), wn  Gis0,1 (wn , R|| ) =  kdk J0,1 (kR|| )(i wn – E is (k ) + m)–1 Intraband interaction (8.23) has the form: (8.30) 169 170 Magnetic Exchange Interaction in Topological Insulator Hcc = 1   gS M (Z1 , Z )[Fc1c + Fc1c ][S1 × R|| ]z [S2 × R|| ]z 2 8p n   R  2 g M(Z1 , Z )Fc1c (S1R || )(S2R ||)  S R   + 2L (Z1 , Z )Fc0c S1z S 2z  + (8.31) Some conclusions on possible magnetic phases can be drawn even without actual calculation of the range functions Coefficients K, M and L carry the dependence of the indirect exchange on positions of magnetic impurities along the z-direction The coefficient M = J  (Z1 )J  (Z ) is the product of two factors each of which tends to zero in the vicinity of either of the surfaces as shown in Fig 8.2 So, if magnetic atoms are located on either of two surfaces, the terms proportional to M are negligible and the interaction has the Ising-type spin texture favoring magnetic ordering perpendicular to the film surfaces (last terms in Eqs (8.29) and (8.31)) M-terms come into play if the magnetic atoms are close to the middle of the film in the z-direction, and magnetic ordering caused by M-terms has a complicated spin pattern depending on angles between the vector connecting two impurities and their spin vectors For magnetic atoms located at the surfaces the interband indirect exchange (8.29) is fully determined by the K-term whose sign depends on the mutual positions of interacting magnetic atoms The function J12(Z) that enters the coefficient K = J 12 (Z1 )J 12 (Z ), is illustrated in Fig 8.4 It follows from the spatial dependence shown in Fig 8.4 that the sign of the K-term is positive if interacting spins belong to the same surface and negative if they lay near opposite surfaces Away from the surfaces, the Ising term in (8.29) tends to zero and a weak interspin coupling is described by the first and second terms So, the gate-bias-induced complex planar spin texture described by non-Ising terms appears in the middle of the slab As the positions of spins move toward the surfaces, the non-Ising terms fade out In intraband interaction (8.31) the Ising term is governed by the coefficient L which is switchable by an external voltage applied across the film The voltage dependence of the interaction Indirect Exchange Interaction Mediated by Surface Electrons is illustrated in Fig 8.5 The numerical example implies that one of the impurity spins is located close to the top of the slab (Z1 = –L/2) and Jz = J||  J Figure 8.4 Interband exchange parameter J 12 (Z )/J z vs Z-position of a magnetic impurity in the slab If the applied voltage changes its sign, the maximum interaction amplitude is switchable between opposite surfaces as shown in Fig 8.5 For a small bias (​| Vas |​ and w < The poles  (i w + F )  – aR ± a2R + 4B k1,2 =   n  2B  (8.36) are shown in Fig 8.6 Figure 8.6 Poles of the integrand in (8.35) on the complex k-plane Finally, the integral (8.35) is found by calculating the residue pi –R + R + ix + F (1) Ic (wn ) = –  × H0  R || –R + R + ix + F  , 2B R + ix + F  aR  w ; F = F ; x = n R=  2B 2B 2B Calculating the integral Ic(wn ) in a similar way we obtain  (8.37) 173 174 Magnetic Exchange Interaction in Topological Insulator  p2  R2 Ic (wn )Ic (wn ) = – 1 – H (1) R (–R + R + ix + F )   R + ix + F   || 4B   × H0(1)  R ||(R + R + ix + F ), (8.38) The range function V(R), (8.32) contains the sum over Matsubara frequencies and at T  the sum can be replaced with the integral (see Chapter 5): S M = T  Ic (wn ) Ic (wn ) = – wn i wn  w + i sign(w) i 2p I  c (w) Ic (w)d w, (8.39) Contour  = 1 + 2, shown in Fig 8.7, avoids the branch point x0 = –R2 – F and the cut line determined by the square root in Eq (8.38) Figure 8.7 Integration contour in Eq (8.39) The integral over the circle around the pole tends to zero when the radius decreases, and SM can be expressed via integration over the upper and lower sides of the path 2: ip SM =  4B (1) R2 + F   R2  ydy1 –  {H0(1)[R || (–R + y )]H0(1)[R || (R + y )] y   –H [R || (– R – y )]H0(1)[R || (R – y )]}, y = R2 + x + F (8.40) Hankel function ​H(1) ​0​  ​ has a cut on the negative part of the real axis To avoid the integration of ​H(1) ​0​  ​ in this region we use (2) the relation ​H(1) ​ ​  ​(–y) = – ​H​0​  ​(y), so (8.40) takes the form Range Function in Topological Insulator SM = ip × 4B   R2   R  yh1( y )1 – dy + y2      R R2F   R  yh2 ( y )1 – dy , y    h1( y )= H0(2)[R ||(R + y )]H0(1)[R||(R – y )] – H0(2)[R||(R – y )]H0(1)[R||(R + y )], h2( y )= H0(1)[R ||( y – R )]H0(1)[R||( y + R )] – H0(2)[R||( y + R )]H0(2)[R||( y – R )] (8.41) Changing the variable to s = yR|| and using notations (8.37) we get the final result for the range function: V (R ) = – L( Z1 , Z )m pn2h2R||2  R|| kR  R || kR2  kF2 × 0 s ( s )P( s )ds + R k s( s )Q( s )ds , || R   ( s )= – (R ||kR )2 / s 2; kR = aR m / 2 ; kF = 2mF / 2, P( s )= J0 (R ||kR + s )N0 (R ||kR – s ) – J0 (R ||kR – s )N0 (R||kR + s ) Q( s )= J0 ( s – R ||kR )N0 ( s + R||kR )+ J0 ( s + R ||kR )N0 ( s – R||kR ) (8.42) where N0(S), is the Newmann function (Bessel function of the second kind), kR and kF are the Rashba and Fermi momentum, respectively In Eq (8.42) we used the identities ​H1​0​(​  s) = J0(s) + iN0(s), ​H2​0​(​  s) = J0(s) – iN0(s) If the bias and then the parameter kR ~ Vas tends to zero, the second term in Eq (8.42) becomes a standard oscillating 2DRKKY range function (see Chapter and Refs [11, 12]) Under an applied voltage the range function contains additional beatings, with a tunable period proportional to 1/kR The result is to be expected from the qualitative standpoint as the beating is a consequence of two distinct Fermi momenta in the spin-split Rashba electron gas The first integral is an additional contribution to the RKKY range function that presents the signature of Rashba spin splitting It should be noted that spin-splitting-related features in RKKY may exist even without external bias as films are grown 175 176 Magnetic Exchange Interaction in Topological Insulator on a substrate and the built-in electric field near broken surfaces makes them non-equivalent, thus violating inversion symmetry 8.4  Conclusions In this chapter we consider the indirect exchange interaction mediated by degenerate surface electrons In the nondegenerate state the chemical potential is placed within the surface energy gap and interband terms (8.29) determine the magnetic ordering in the slab Based on qualitative considerations one may predict some features of the magnetic interaction in this case If BDS < the surface spectrum has a direct gap and the range function should fade out exponentially with R|| as it happens in an ordinary non-degenerate semiconductor (see Chapter 5) In the topologically non-trivial phase, BDS > 0, the spectrum is inverted (see Fig 7.8, Chapter 7) Two types of virtual electronhole transitions contributing to the indirect exchange become possible: 1) vertical transitions across the minimum gaps at ± ​ (DS⁄2B) ​  , and 2) transitions across the gap between k0, k0 = √ minima at different momenta with momentum transfer K ≈ 2k0 The range function oscillates with a period proportional to K –1 and exponentially decaying amplitude ~exp(–R||/r0), where 3/4  –1/2 –1/4 if VAS = (see Problem 1) A similar type r0  B ( A2 ) (2D S ) of range function appears in indirect-gap semiconductors [13], graphene with the energy gap induced by spin-orbit interaction [14], and excitonic insulators [15] So, the signature of the nondegenerate topological phase is oscillating indirect exchange with exponentially decaying amplitude whereas in a trivial insulator the non-degenerate surface states would mediate a monotonically decreasing exponential range function In conclusion, the low-energy effective s-d interaction model in TI has been developed by projecting the bulk s-d interaction onto surface states It is shown that magnetic atoms interact with the surface electrons through position-sensitive s-d interaction that can be controlled by gate bias The range function of the indirect exchange oscillates in a degenerate sample and, in addition to main oscillations determined by a finite Fermi momentum, it acquires zero magnetic field beatings with a period related to the magnitude of the Rashba spin splitting References Problems 8.1 Estimate the period of oscillations and the decrement of exponential decay of the range function mediated by non-degenerate surface electrons See Section 8.4 and Fig 7.8 of Chapter 8.2 Estimate the gate voltage which turns off exponential distance dependence in the range function mentioned in Problem 8.1 References Henk J, Flieger M, Maznichenko IV, Mertig I, Ernst A, Eremeev SV, Chulkov EV (2012) Topological character and magnetism of the Dirac state in Mn-Doped Bi2Te3, Phys Rev Lett, 109, 076801 Zhang J, Chang C-Z, Tang P, Zhang Z, Feng X, Li K, Wang L-l, Chen X, Liu C, Duan W, He K, Xue Q-K, Ma X, Wang Y (2013) Topology-driven 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P, Butch NP, Paglione J, Efimov IS, Damascelli A (2011) Rashba spin-splitting control at the surface of the topological insulator Bi2Se3, Phys Rev Lett, 107, 186405 10 Litvinov VI (2014) Magnetic exchange interaction in topological insulators, Phys Rev B, 89, 235316 177 178 Magnetic Exchange Interaction in Topological Insulator 11 Fischer B, Klein MW (1975) Magnetic and nonmagnetic impurities in two-dimensional metals, Phys Rev B, 11, 2025–2029 12 Litvinov VI, Dugaev VK (1996) RKKY interaction in one- and twodimensional electron gases, Phys Rev B, 58(7), 3584–3585 13 Abrikosov AA (1980) Spin-glass with a semiconductor as host, J Low Temp Phys, 39(1/2), 217–229 14 Dugaev VK, Litvinov VI, Barnas J (2006) Exchange interaction of magnetic impurities in graphene, Phys Rev B, 74, 224438 15 Litvinov VI (1985) Spin glass in excitonic insulator, Sov Phys Solid State, 27(4), 740–741 [...]... ferromagnetic phase transition is described, as is the percolation picture of phase transition in certain systems, for example, wide bandgap semiconductors, for which mean-field theory breaks down The electronic properties of topological Bi2Te3 insulators are discussed in Chapter 7, where the semiconductor is taken as an example Topological insulator film biased with a vertical voltage presents a system with... (1974) Symmetry and Strain Effects in Semiconductors, Wiley, New York 2 Anselm A (1981) Introduction to Semiconductor Theory, Prentice-Hall, New Jersey 3 Chen GD, Smith M, Lin JY, Jiang HX, Wei SH, Asif Khan M, Sun CJ (1996) Fundamental optical transitions in GAN, Appl Phys Lett, 68, 2784–2786 4 Chuang SL, Chang CS (1996) k . p method for strained wurtzite semiconductors, Phys Phys, B54(4), 2491–2504... Chapter 4 Chapters 5 and 6 are devoted to a detailed theoretical description of mechanisms of ferromagnetism in magnetically doped semiconductors, specifically in the III-V Nitrides These chapters discuss the indirect exchange interaction in metals of any dimension and in semiconductors Emphasis is placed on the specific feature of the indirect exchange interaction in a one-dimensional metal Also, the... explored in various directions One of them, semiconductor spintronics, is of particular recent interest since materials developed for electronics and optoelectronics are gradually becoming available for spinmanipulation-related applications, e.g., spin-transistors and quantum logic devices allowing the integration of electronic and magnetic functionalities on a common semiconductor template The scope of... y(r) of the translation operator: y(r + R ) = y(r )exp(ikR ), (1.1) where exp(ikR) is the eigenvalue of the translation operator, and R is the arbitrary lattice translation This condition is the Wide Bandgap Semiconductor Spintronics Vladimir Litvinov Copyright © 2016 Pan Stanford Publishing Pte Ltd ISBN  978-981-4669-70-2 (Hardcover),  978-981-4669-71-9 (eBook) www.panstanford.com  GaN Band Structure... allowing the integration of electronic and magnetic functionalities on a common semiconductor template The scope of this book is largely concerned with the spintronic properties of III-V Nitride semiconductors As wide bandgap III-Nitride nanostructures are relatively new materials, particular attention is paid to the comparison between zinc-blende GaAsand wurtzite GaN-based structures where the Rashba... dipole spin resonance (EDSR) that is induced by ac-electric field optical spin-flip transitions between two spin states [2, 3] The spin-splitting terms in the Hamiltonian will be considered below Wide Bandgap Semiconductor Spintronics Vladimir Litvinov Copyright © 2016 Pan Stanford Publishing Pte Ltd ISBN  978-981-4669-70-2 (Hardcover),  978-981-4669-71-9 (eBook) www.panstanford.com 22 Rashba Hamiltonian... of the topics discussed in this book and Toni Quintana for carefully reading and correcting the text Chapter 1 GaN Band Structure To deal with the spin and electronic properties of wurtzite IIInitride semiconductors and understand the specific features that differentiate them from zinc blende III-V materials, one has to know the energy spectrum The energy spectrum gives us all necessary information... transitions and gain in group-III nitride quantum wells, J Appl Phys, 88, 5814–5820 8 Lew Yan Voon LC, Willatzen M, Cardona M, Christensen NE (1996) Terms linear in k in the structure of wurtzite-type semiconductors, Phys Rev, B53 (16), 10703–10714 9 Kim K, Lambrecht WRL, Segall B, van Schilfgaarde M (1997) Effective masses and valence-band splitting in GAN and AlN, Phys Rev, B56(12), 7363–7375 10... energy surfaces in crystals with wurtzite symmetry, Phys Rev Lett, 5(8), 371–373 13 Cardona M, Christensen NE, Fasol G (1988) Relativistic band structure and spin-orbit splitting of zinc-blende-type semiconductors, Phys Rev, B38, 3, 1810–1827 19 This page intentionally left blank Chapter 2 Rashba Hamiltonian Two types of linear-k terms in the Hamiltonian were discussed in Chapter 1 One stems from