Modeling and Simulation of Electron Transport through Nanoscale Heterojunctions Argo Nurbawono BEng. Hons. University of Bristol A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements I am mostly indebted to Dr. Chun Zhang, whose labour assistance has made this work possible. He is the man who helped me to tackle numerous dirty details which otherwise look too impossible to solve without resorting to his vast experience. I sincerely wish to thank Prof. Yuan Ping Feng for his kind assistance through out the years. I also thank him to let me access the computing clusters and to introduce me to various people on the first place. I thank everyone from various corners of the department who helped me in many occasions, the staff, friends and for those who taught me many advanced topics in physics. Last but not least, I thank Prof. Chong Kim Ong and Prof. Dagomir Kaszlikowski for their kind recommendations to the department during my initial PhD scholarship applications. Many thanks to all of you! Argo Nurbawono Singapore, 2011. i ii Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of tables and figures . . . . . . . . . . . . . . . . . . . . . . . . . viii Literature reviews 1.1 1.2 Nanoscale heterojunctions . . . . . . . . . . . . . . . . . . . . . 1.1.1 Weakly correlated nano heterojunction . . . . . . . . . . . 1.1.2 Strongly correlated nano heterojunction . . . . . . . . . . An overview on quantum transport theory . . . . . . . . . . . . . Theoretical formalisms 2.1 2.2 2.3 2.4 11 Green’s function for quantum transport . . . . . . . . . . . . . . . 11 2.1.1 Equilibrium formalisms . . . . . . . . . . . . . . . . . . 11 2.1.2 Nonequilibrium formalisms . . . . . . . . . . . . . . . . 15 Resonant tunneling model . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Current equation . . . . . . . . . . . . . . . . . . . . . . 21 Combining NEGF with DFT . . . . . . . . . . . . . . . . . . . . 23 2.3.1 General methods . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 DFT in LCAO basis sets . . . . . . . . . . . . . . . . . . 24 2.3.3 Density matrix and Hamiltonian of the centre region . . . 28 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Cooper pairing of the condensate . . . . . . . . . . . . . 31 2.4.2 Mean field BCS theory . . . . . . . . . . . . . . . . . . . 34 2.4.3 Josephson effects . . . . . . . . . . . . . . . . . . . . . . 36 iii Optical nanowelding of CNT-metal contacts 3.1 3.2 3.3 3.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 CNT for transistors . . . . . . . . . . . . . . . . . . . . . 41 3.1.2 Schottky barrier in CNT devices . . . . . . . . . . . . . . 42 3.1.3 Optical nanowelding . . . . . . . . . . . . . . . . . . . . 44 Model and simulation methods . . . . . . . . . . . . . . . . . . . 46 3.2.1 Simulation procedures . . . . . . . . . . . . . . . . . . . 46 3.2.2 Geometric set ups . . . . . . . . . . . . . . . . . . . . . . 48 Numerical results and discussions . . . . . . . . . . . . . . . . . 49 3.3.1 Interface properties . . . . . . . . . . . . . . . . . . . . . 49 3.3.2 Current-voltage (I-V) characteristics . . . . . . . . . . . . 51 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Conductance anomaly in SNS heterojunctions 4.1 4.2 4.3 4.4 57 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 Andreev reflections . . . . . . . . . . . . . . . . . . . . . 57 4.1.2 Prior theoretical models . . . . . . . . . . . . . . . . . . 60 Model for SNS junctions . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 Hamiltonian and potential symmetries . . . . . . . . . . . 62 4.2.2 Time dependent current formulation . . . . . . . . . . . . 64 4.2.3 Average current formulation . . . . . . . . . . . . . . . . 67 Numerical results and discussions . . . . . . . . . . . . . . . . . 69 4.3.1 General features of the supercurrent . . . . . . . . . . . . 69 4.3.2 The differential conductance anomaly . . . . . . . . . . . 71 4.3.3 S-N-N systems . . . . . . . . . . . . . . . . . . . . . . . 76 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 SNN heterojunction under radiations iv 41 79 5.1 Interactions with radiations . . . . . . . . . . . . . . . . . . . . . 79 5.2 Quantum dot under semiclassical fields . . . . . . . . . . . . . . 84 5.2.1 Hamiltonian in Floquet-Fourier basis . . . . . . . . . . . 84 5.2.2 Floquet quasienergies . . . . . . . . . . . . . . . . . . . . 88 5.2.3 Time averaged current . . . . . . . . . . . . . . . . . . . 90 5.3 Numerical results and discussions . . . . . . . . . . . . . . . . . 92 5.3.1 5.3.2 5.3.3 5.4 For A > and B = . . . . . . . . . . . . . . . . . . . . 92 For A = and B > . . . . . . . . . . . . . . . . . . . . 94 For both A, B > . . . . . . . . . . . . . . . . . . . . . 99 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Conclusions and future works 103 6.1 Optical Nanowelding of CNT-metal contacts . . . . . . . . . . . . 103 6.2 Superconducting nano heterojunctions . . . . . . . . . . . . . . . 104 A Abbreviations and Symbols 107 B Derivations of relevant equations 111 B.1 BCS free propagators . . . . . . . . . . . . . . . . . . . . . . . . 111 B.2 Time dependent current formulation . . . . . . . . . . . . . . . . 114 B.3 Equation of motion in Nambu space . . . . . . . . . . . . . . . . 115 B.4 Selfenergy in time domain . . . . . . . . . . . . . . . . . . . . . 117 B.5 Selfenergy in frequency domain . . . . . . . . . . . . . . . . . . 120 C Publications 125 References 127 v Summary The thesis discusses some transport aspects of nanoscale heterojunctions, the archetypal devices which constitute a considerable part of the rapidly growing nanotechnology and nanoscience today. It discusses two distinct types of nanoscale heterojunctions, namely weakly correlated and strongly correlated heterojunctions. The weakly correlated heterojunction employs normal metal for its leads, where the particles behave like typical 3D electron gas and the standard density functional theory allows rigorous ab initio analysis for such systems. On the other hand, the strongly correlated heterojunction employs superconductors for its leads, therefore appropriate models need to be used to describe the essential physics from which the transport properties are derived. The weakly correlated heterojunction consists of two normal metals and a carbon nanotube (CNT) in between, a ubiquitous system in nanoscale experimental devices. Despite of all its novel and great promises, a full exploitation of the device has so far been hindered by various problems, and one of them is the interface problems with the metal probes which typically produce considerable resistance. Schottky barriers formed at CNT-metal contacts have been well known to be crucial for the performance of CNT based field effect transistors (FETs). Through an extensive first principles calculations we show that an optical nanowelding process can drastically reduce the Schottky barriers at CNT-metal interfaces, resulting in significantly improved conductivity. Results presented may have great implications in future design CNT-based nanoelectronics. The strongly correlated heterojunction consists of two superconducting leads and a quantum dot in between. A phenomenon of so-called differential conductance anomaly is predicted to occur in such devices at high bias when the transport is theoretically linear. The phenomenon is caused by the potential symmetry which affects the pinning mechanisms of the localized level by the superconducting gaps of the leads. Due to this, we anticipate a counter intuitive phenomenon where the linear conductivity may be increasing as the coupling strength between the leads and the quantum dot is reduced. The phenomenon can be used to investigate the symmetry across the quantum dot which would otherwise be impossible to probe using other methods. A recent experiment may already indicate the exisvi tence of such effects. We then consider another hybrid superconducting system and study the effect of electron tunneling under external microwave radiations. The microwave radiations stimulate interlevel quantum transitions on the multilevel quantum dot. We develop a method to combine Floquet theory and nonequilibrium Green’s function in order to describe supercurrent tunneling process through the heterojunction. We find that the effect of transition amplitude or the coupling between levels is reflected at the current-bias (I-V) curves only at Rabi frequency. The radiation splits the dc resonance and the separation between each splits is proportional to the coupling between the localized levels. The observation provides a possibility for an experimental inference of the interlevel coupling from simple time averaged measurements. In all parts of the transport analysis we employ nonequilibrium Green’s function method which is considered to be the most rigorous and systematic way to treat most quantum transport problems. Some other secondary and on going works are not included in this thesis in order to maintain a coherent picture of the presentation. vii List of tables and figures Tables: • Table 3.1 on page 51: Table of Schottky barrier reductions for various welding temperatures. Figures: • Figure 1.1 on page 8: Two dimensional electron gas diagram. • Figure 2.1 on page 16: Keldysh time contour integral. • Figure 2.2 on page 20: Resonant tunneling diagram. • Figure 2.3 on page 25: ATK model diagram. • Figure 2.4 on page 33: Photon and phonon mediated interactions. • Figure 2.5 on page 34: Cooper pair diagram in k-space. • Figure 3.1 on page 46: Optical nanowelding diagram. • Figure 3.2 on page 47: Approximate temperature profile for the welding simulation. • Figure 3.3 on page 48: Three different configurations for CNT attachments. • Figure 3.4 on page 49: Dipole moment at the interface before and after welding. • Figure 3.5 on page 50: Optimized structures and charge transfer diagrams of CNT devices. • Figure 3.6 on page 52: Plane average potential along transport axis. • Figure 3.7 on page 53: I-V curves for the system with aluminium and palladium leads. • Figure 3.8 on page 56: Comparison of the I-V curves for all three attachments. viii APPENDIX B. DERIVATIONS OF RELEVANT EQUATIONS i d ˆ ΞLi,k (t, t1 ) = − dt1 ˆ r (t, t1 )tˆ⋆ (t1 ) − Ξ ˆ Li,k (t, t1 ) G Li,l Ll l ǫk↑ −∆ ∆ −ǫk↓ Thus they are equivalent. So, finally we have gathered all the terms we need, ˆ r (t, t1 ) = g ˆjr (t, t1 ) + G i,j ˆ Li,k (t, t1 )tˆLj (t1 )ˆ Ξ gjr (t, t1 ) k ˆ Ri,k (t, t1 )tˆRj (t1 )ˆ Ξ gjr (t, t1 ) + k ˆjr (t, t1 ) + = g ˆ r (t, t2 )tˆ⋆ (t2 )ˆ G gkr (t2 , t1 )tˆLj (t1 )ˆ gjr (t, t1 ) i,l Ll dt2 l k ˆ r (t, t2 )tˆ⋆ (t2 )ˆ G gkr (t2 , t1 )tˆRj (t1 )ˆ gjr (t, t1 ) i,l Rl dt2 + l k ˆjr (t, t1 ) + = g ˆ r (t, t2 ) G i,l dt2 k l + ˆ r (t, t2 ) G i,l dt2 l ˆjr (t, t1 ) + = g ˆ jr (t, t1 ) tˆ⋆Ll (t2 )ˆ gkr (t2 , t1 )tˆLj (t1 ) g ˆjr (t, t1 ) tˆ⋆Rl (t2 )ˆ gkr (t2 , t1 )tˆRj (t1 ) g k ˆ r (t, t2 )Σ ˆ r (t2 , t1 )ˆ G gjr (t, t1 ) i,l Ll,j dt2 l + ˆ r (t, t2 )Σ ˆ r (t2 , t1 )ˆ G gjr (t, t1 ) i,l Rl,j dt2 l ˆjr (t, t1 ) + = g ˆ r (t, t2 )Σ ˆ r (t2 , t1 )ˆ G gjr (t, t1 ) i,l l,j dt2 l ˆr = Σ ˆ r +Σ ˆ r . For the lesser self-energy where the total selfenergy is simply Σ l,j Ll,j Rl,j < ˆ Σ , the process is essentially the same but would end up with (tˆ⋆ gˆ< tˆ) instead of (tˆ⋆ gˆr tˆ). This simple expression for the selfenergy is easily obtained because we have a non-interacting quantum dot. B.5 Selfenergy in frequency domain The explicit expressions for the selfenergies in frequency domain in chapter three can be obtained using resonant tunneling model, valid for non-interacting 120 B.5. SELFENERGY IN FREQUENCY DOMAIN quantum dot, r/< r/< ΣL/Rij (t, t1 ) = t∗L/Ri gL/Rk (t, t1 ) tL/Rj (B.25) k Using the expression for the BCS free propagator, the (11) component of the left retarded selfenergy (VL = 0) is, Σr(11)L;m,n (ǫ) = dtei(ǫ+mω)(t−t1 ) (−i)θ(t − t1 )ΓL βL (ǫ) dt1 ei(m−n)ωt1 ∞ dtei(ǫ+mω)(t−t1 ) (−i)ΓL βL (ǫ) dt1 ei(m−n)ωt1 = t1 ∞ d(t − t1 )ei(ǫ+mω)(t−t1 ) (−i)ΓL βL (ǫ) dt1 ei(m−n)ωt1 = = −i i(m−n)ωt1 dt1 e ∞ d(t − t1 )ei(ǫ+mω)(t−t1 ) ΓL βL (ǫ) −∞ i = − ΓL δm,n βL (ǫ + mω) i = − ΓL δm,n βL (ǫm ) where we introduce notation ǫm = (ǫ + mω). The (22) component is identical, while the (12) component is, Σr(12)L;m,n (ǫ) = i = dt1 ei(m−n)ωt1 dtei(ǫ+mω)(t−t1 ) θ(t − t1 )ΓL βL (ǫ) ∆L −iφL e ǫ ∆L −iφL i e ΓL δm,n βL (ǫm ) ǫm the (21) component is similar. For the right retarded self-energy we set VR = −V , and make use of the substitution V = ω/2 (from the Josephson frequency), Σr(11)R;m,n (ǫ) = dt1 ei(m−n)ωt1 dtei(ǫ+mω)(t−t1 ) (−i)θ(t − t1 )ΓR βR (ǫ)e−iV (t1 −t) 121 APPENDIX B. DERIVATIONS OF RELEVANT EQUATIONS ∞ dtei(ǫ+(m+ )ω)(t−t1 ) ΓR βR (ǫ) i(m−n)ωt1 = −i dt1 e t1 ∞ i = − d(t − t1 )ei(ǫ+(m+ )ω)(t−t1 ) ΓR βR (ǫ) i(m−n)ωt1 dt1 e −∞ i = − ΓR δm,n βR (ǫ + (m + )ω) 2 i = − ΓR δm,n βR (ǫm+1/2 ) The (12) component can be derived using a similar way, Σr(12)R;m,n (ǫ) = i dtei(ǫ+mω)(t−t1 ) θ(t − t1 )ΓR βR (ǫ) dt1 ei(m−n)ωt1 ∞ = i dtei(ǫ+(m+1/2)ω)(t−t1 ) ΓR βR (ǫ) i(m−n+1)ωt1 dt1 e ∆R −iφR iV (t1 +t) e e ǫ ∆R −iφR e ǫ t1 = i ∆R −iφR e ΓR δm,n−1 βR (ǫm+1/2 ) ǫm+1/2 and the rest of equation can be derived using the same way. The lesser selfenergy is given by, Σ< L/R;ij (t, t1 ) = i dǫ ˜ L/R ΓL/R fL/R (ǫ)Re[βL/R (ǫ)]e−iǫ(t−t1 ) Σ 2π The (11) component of left lesser selfenergy is, Σ< (11)L;m,n (ǫ) = dt1 ei(m−n)ωt1 dtei(ǫ+mω)(t−t1 ) iΓL fL (ǫ)Re[βL (ǫ)] = iΓL δm,n f (ǫm )Re[β(ǫm )] and for example the (12) component of the lesser right self-energy, Σ< (12)R;m,n (ǫ) = 122 dt1 ei(m−n)ωt1 dtei(ǫ+mω)(t−t1 ) iΓR fR (ǫ)Re[β(ǫ)] − ∆R −iφL iV (t1 +t) e e ǫ B.5. SELFENERGY IN FREQUENCY DOMAIN dtei(ǫ+mω)(t−t1 ) eiω(t1 +t−t1 +t1 )/2 iΓR fR (ǫ)Re[β(ǫ)] − = dt1 ei(m−n)ωt1 = dt1 ei(m−n+1)ωt1 dtei(ǫ+(m+1/2)ω)(t−t1 ) iΓR fR (ǫ)Re[β(ǫ)] − = iΓR δm,n−1 f (ǫm+1/2 )Re[β(ǫm+1/2 )] − ∆R ǫm+1/2 ∆R −iφL e ǫ ∆R −iφL e ǫ e−iφL Collecting all the terms we have for the SNS system selfenergies, i ΣrL;m,n (ǫ) = − ΓL δmn βL (ǫm ) ΣrR;m,n (ǫ) i = −i ΓR − ∆ǫmL e−iφL − ∆ǫmL eiφL −∆R −iφR e δm,n−1 βR (ǫm+1/2 ) ǫm+1/2 δmn βR (ǫm+1/2 ) −∆R iφR δm,n+1 βR (ǫm−1/2 ) ǫm−1/2 e Σ< L;m,n (ǫ) = iΓL δmn fL (ǫm )Re[βL (ǫm )] (B.26) − ∆ǫmL eiφL δmn βR (ǫm−1/2 ) (B.27) − ∆ǫmL e−iφL (B.28) Σ< R;m,n (ǫ) = iΓR δmn fR (ǫm+1/2 )Re[βR (ǫm+1/2 )] −∆R −iφR e δm,n−1 fR (ǫm+1/2 )Re[βR (ǫm+1/2 )] ǫm+1/2 −∆R iφR δm,n+1 fR (ǫm−1/2 )Re[βR (ǫm−1/2 )] ǫm−1/2 e δmn fR (ǫm−1/2 )Re[βR (ǫm−1/2 )] (B.29) 123 APPENDIX B. DERIVATIONS OF RELEVANT EQUATIONS 124 Appendix C Publications 1. Optical nanowelding of CNT-metal contacts: An effective way to control Schottky barrier and performance of CNT-based transistors, A. Nurbawono, A. Zhang, Y. Cai, et al., submitted to ACS Nano - under review. 2. Manipulating absorption and diffusion of hydrogen atom on graphene by mechanical strain, M. Yang, A. Nurbawono, C. Zhang, et al., submitted to American Institute of Physics (AIP) Advances 1, 032109 (2011). 3. C-doped ZnO Nanowires: electronic structures, magnetic properties, and a possible spintronic device, Z. Dai, A. Nurbawono, A. Zhang, et al., Journal of Chemical Physics 134, 104706 (2011). Selected for Vir. J. Nan. Sci. & Tech. 23, 11 (2011). 4. The roles of potential symmetry in transport properties of superconducting quantum point contacts, A. Nurbawono , Y. P. Feng and C. Zhang, Journal of Computational and Theoretical Nanoscience 7, 2448 (2010). 5. Electron tunneling through a hybrid superconducting-normal mesoscopic junction under microwave radiation, A. Nurbawono A, Y. P. Feng and C. Zhang, Physical Review B 82, 014535 (2010). Selected for Vir. J. Appl. Supercond. 19, (2010). 6. Two-dimensional graphene superlattice made with partial hydrogenation, M. Yang, A. Nurbawono, C. Zhang, et al., Applied Physics Letters 96, 125 APPENDIX C. PUBLICATIONS 193115 (2010). 7. Differential conductance anomaly in superconducting quantum point contacts, A. Nurbawono, Y. P. Feng, E. Zhao, et al., Physical Review B 80, 184516 (2009). Selected for Vir. J. Appl. Supercond. 17, 11 (2009). 126 REFERENCES References [1] J. Kong et al., Science 287, 622 (2000). [2] P.G. Collins et al., Science 287, 1801 (2000). [3] M. F. Islam et al., Phys. Rev. Lett. 93, 037404 (2004). [4] D. A. Stewart and F. L´eonard, Phys. Rev. Lett. 93, 107401 (2004). [5] D. S. Hecht et al., Nano Letters 6, 2031 (2006). [6] H. Sugie et al., Applied Physics Letters 78, 2578 (2001). [7] J. Zhang et al., Applied Physics Letters 86, 184104 (2005). 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(bio)chemical sensors [1,2] , optoelectronic devices [5,3,4] , field emission devices [6,7] , electromechanicals [8] and electronic devices [9,10] An overview of its basic properties is available in many good review articles and books [11] , and we 2 1.1 NANOSCALE HETEROJUNCTIONS shall only discuss some relevant aspects related to electronic devices The simplest realization of a CNT electronics is perhaps a... 4 1.1 NANOSCALE HETEROJUNCTIONS that individual transmissions of a quantum contact were ever determined experimentally, since its prediction fifty years ago by Landauer [109] Since then, the microscopic Hamiltonian theory is becoming the mainstream in the subsequent development of superconducting transport A typical nanoscale contact consists only of a small number of eigenchannels and each of them... second type of nanoscale heterojunction is a hybrid superconductingnormal nanoscale junction, where one or both of the leads are superconducting Superconductivity is the result of the instability of the Fermi surface from which a completely new phase of the system appears under the influence of strong correlations between the particles The quasiparticles are called Cooper pairs, which are pairs of electrons... review of the literature on the recent developments in experimental and theoretical works of nanoscale heterojunctions, especially for carbon nanotube (CNT) and superconducting point contacts A list of abbreviations and symbols used in the thesis can be found in appendix A 1.1 Nanoscale heterojunctions While the efforts to miniaturize electronic devices continue in order to cram ever more components... polarization of the normal leads The BTK theory basically extends the Bogoliubov de Gennes equation and adapts certain boundary conditions at the interface between superconductor and normal metal Derivations of the BTK formalisms and its experimental applications can also be found in some review papers [21] An important landmark in the development of quantum mechanical theory of superconducting transport. .. density of integrated circuits doubles every two years 1 CHAPTER 1 LITERATURE REVIEWS regions with the desired optical properties In this thesis we are going to discuss specifically about nanoscale heterojunctions, a topic of quite significant portion in the development of nanotechnology and nanoscience today The relevant physical scales for the heterojunctions is determined by the phase coherence and it... The transport analysis in this system may be conveniently simulated with the available ab initio or first principle method based on density functional theory (DFT) It basically relies only on a few assumptions such as Born-Oppenheimer and pseudopotentials, and it computes the electronic properties directly based on the detailed atomic character of the constituents Most of the nano heterojunctions are of. .. Hamiltonians such as systems under the influence of external field or with dissipative process From the early works of Kadanoff, Baym [28,29] and Keldysh [30] , the NEGF method has made substantial contributions to the development of quantum kinetics of interacting many-body systems in general Along with important contributions from Martin and Schwinger [31] , and also Kubo [32] , the early NEGF method successfully... approach [34] Problems related to dissipations and memory effects have since been overcome, and the NEGF provides satisfactory information on the statistical and dynamical behaviours for systems with inherent quantum behaviours Applications of NEGF in the area of quantum transport typically focus on the regimes of extremely short length scales (∼1 nm) and extremely short time scales (∼1 fs) where the... nanotubes, they can be hundreds of nanometers and electrons travel effectively under ballistic transport throughout the entire tubes, and decoherence by scattering only takes place at the very end at 7 CHAPTER 1 LITERATURE REVIEWS 6 2 Vg metal QPC Conductance (2e /h) metal QPC 2DEG 2DEG 4 2 Vg 0 metal QPC metal QPC 0 2 4 6 8 Gate voltage Figure 1.1: Example of two dimensional electron gases (2DEG) on a . Modeling and Simulation of Electron Transport through Nanoscale Heterojunctions Argo Nurbawono BEng. Hons. University of Bristol A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT. subsequent development of superconducting transport. A typical nanoscale contact consists only of a small number of eigenchannels and each of them is characterized by a transmission coefficient τ n . Each of them. electromechanicals [8] and electronic devices [9,10] . An overview of its basic properties is available in many good review articles and books [ 11] , and we 2 1.1. NANOSCALE HETEROJUNCTIONS shall