Master Thesis Quantum transport calculations using wave function propagation and the Kubo formula Troels Markussen (s001477) [email protected] Supervisors: Mads Brandbyge and Antti-Pekka Jauho MIC – Department of Micro and Nanotechnology Technical University of Denmark 9th March 2006 Preface This thesis is submitted in candidacy for the Master of Science degree in Engi- neering Physics from the Technical University of Denmark (DTU). The project has been carried out at MIC - Department of Micro and Nanotechnology in the period from February 1st 2005 to February 1st 2006 with Mads Brandbyge and Antti-Pekka Jauho as supervisors. I would like to thank my supervisors for their enthusiasm and constructive criticism. I have enjoyed the many fruitful discussions and in general working in the Theoretical Nanotechnology group. In particular I thank Magnus Paulson for help with obscure Linux problems and Python programming. I would like to thank Oticon Fonden for their generous financial support. This enabled me to visit N. Lorente and R. Rurali in Toulouse, France, to discuss collaborative work on the metallic silicon nanowires and learn about the siesta- program. The grant also made it possible to participate in the conference Elec- Mol’05 in Grenoble, France, where some of the results from the project was presented on a poster. I thank the people at MIC for making it a pleasant place to spend a lot of time. I also thank the guys in room 119 for fruitful and indispensable discussions concerning anything but my project. Finally, I thank my girlfriend Wiebke for her patience and support during the project, and for reminding me that there are other things in life than physics. Troels Markussen MIC – Department of Micro and Nanotechnology Technical University of Denmark 9th March 2006 Abstract This project is concerned with modelling the electrical properties of silicon nanowires. The fabrication of nanowires can be controlled with high accuracy, making them very promising candidates for future electronic components with the capability of exceeding conventional technologies. Also, it has been demon- strated that nanowires as chemical and biological sensors can be very sensitive with capability of single virus detection. Calculations on nanowires with randomly placed dopants or defects, require an atomistic model since quantum effects are important due to the small diameter, which can be down to 2 nm. The length of the wires can, however, be several µm and the numerical method should be able to treat more than 104 atoms, implying that (N) scaling is required. O Two (N) methods, both using a tight-binding model based on ab initio calcu- lations,O have been applied in this project. The first approach, derived from the Kubo-Greenwood formula, uses time propagation of wave packets to estimate the conductance. This method has until now only been applied by a single group and primary to model carbon nanotubes. One of the main objectives in this work has been to understand, implement and test this relatively new method. The second approach is based on the Landauer formula and the conductance is found by recursive calculations of Green’s functions. Comparison of the two approaches revealed that the Green’s function method is the preferred choice for modelling thin nanowires. It is more rigorous than the Kubo method and does not suffer from having many convergence parameters. The transport properties in both pure silicon wires and wires passivated with hydrogen is shown to be strongly affected by surface disorder. Randomly missing H atoms causes the electrons to localize, and both the elastic mean free paths and localization lengths scales linearly with the average distance between H vacancies. Comparison with estimated electron-phonon interactions indicates that impurities might be the dominant scattering mechanism for thin wires even at room temperature. Resum´e Dette projekt omhandler modellering af elektriske egenskaber i silicium nano- tr˚ade. Fremstillingen af nanotr˚ade kan kontrolleres med stor nøjagtighed, hvilket gør nanotr˚ade til lovende kandidater for fremtidige elektroniske komponenter med mulighed for at overg˚akonventionelle teknologier. Det er ligeledes blevet demonstreret, at nanotr˚ade som kemiske og biologiske sensorer kan være meget følsomme med mulighed for at detektere enkelte virusmolekyler. Modellering af nanotr˚ade med tilfældigt fordelte doteringer eller defekter kræver en atomar model, da kvanteeffekter er væsentlige p˚agrund af den lille diameter, som kan være ned til 2 nm. Længden af nanotr˚adene kan til gengæld være adskillige µm, og den numeriske metode skal derfor være i stand til a behandle mere end 104 atomer, hvilket betyder, at (N) skalering er nødvendigt. O Der er i projektet blevet anvendt to (N) metoder, der begge benytter en tight-binding model baseret p˚a ab initioO beregninger. Den første metode, der er udledt fra Kubo-Greenwood formlen, bruger tidsudvikling af bølgepakker til at estimere konduktansen. Denne metode er indtil nu kun blevet anvendt af en enkelt gruppe til at studere kulstof nanorør, og et af hovedform˚alene i dette arbejde har derfor været at forst˚a, implementere og teste denne relativt nye metode. Den anden fremgangsm˚ade er baseret p˚aLandauer formlen og konduktansen findes ved rekursive beregninger af Greens funktioner. En sammenligning af de to metoder gav, at Greens funktions metoden er det foretrukne valg til at modellere tynde nanotr˚ade. Den er mere stringent end Kubo-metoden, og har ingen konvergens parametre. Det er vist, at transportegenskaberne i b˚ade rene siliciumtr˚ade og hydrogen passiverede tr˚ade, er stærkt p˚avirket af overfladedefekter. Tilfældigt fjernede H atomer bevirker, at elektronerne lokaliserer, og b˚ade den elastiske fri mid- delvejlængde og lokaliseringslængden skalerer lineært med den gennemsnitlige afstand mellem H vakancerne. Sammenligning med estimerede elektron-phonon vekselvirkninger indikerer, at defekter kan være den væsentligste sprednings- mekanisme selv ved stuetemperatur. Contents List of symbols xi 1 Introduction 1 1.1 Semiconducting nanowires . 1 1.1.1 Growth of silicon nanowires . 2 1.1.2 Electronic applications . 3 1.1.3 SiNWsassensors....................... 4 1.1.4 Critical issues and possible theoretical help . 5 1.2 Theoretical techniques . 6 1.2.1 Atomic and electronic structure . 6 1.2.2 Transport calculations . 7 1.2.3 Real-space Kubo-method . 7 1.2.4 Applications of the real-space Kubo method . 8 1.3 Motivation and outline . 10 1.3.1 Outline ............................ 11 2 The real-space Kubo formalism 13 2.1 Derivation of the real-space Kubo formula . 13 2.1.1 Preliminary notations . 13 2.1.2 Rewriting the Kubo-Greenwood formula . 15 2.1.3 Time evolution . 17 viii CONTENTS 2.2 Three regimes of transport . 17 2.2.1 Ballistic regime . 18 2.2.2 Diffusiveregime ....................... 18 2.2.3 Localization.......................... 19 2.2.4 Simple model for diffusion and localization . 20 3 Numerical methods 22 3.1 Exact solution to the time-dependent Schr¨odinger equation . 22 3.2 The Chebyshev method . 23 3.2.1 Basic properties of Chebyshev polynomials . 23 3.2.2 Expansion of Uˆ(t) in Chebyshev polynomials . 23 3.2.3 Convergence properties . 24 3.2.4 Convergence of the coefficients . 25 3.2.5 Comparison of Chebyshev and Taylor . 26 3.3 Continued fraction technique . 26 3.4 Overview of the computer codes . 28 3.4.1 Important parameters . 28 3.4.2 Dataflow ........................... 29 4 The one dimensional chain: Testing of the method 31 4.1 Local Density of States . 31 4.1.1 The infinite chain . 32 4.1.2 Comparison of numerical and analytical LDOS . 32 4.1.3 Comparing the velocities . 34 4.2 Disorderedchain ........................... 35 4.2.1 Random initial states . 36 4.2.2 Time dependent diffusion coefficient (E,t)........ 36 D 4.2.3 Meanfreepath ........................ 39 4.2.4 Conductance of disordered wire . 40 4.3 Carbon nanotubes: Comparison with published results . 43 4.4 ParallelChains ............................ 46 CONTENTS ix 4.4.1 Calculation of the conductance . 47 5 The Landauer formula and recursive Green’s function method 50 5.1 Recursive Green’s function method . 50 5.1.1 Introduction to the Landauer formalism . 50 5.1.2 Recursive (N)growthprocess............... 52 O 6 Building a Tight-binding model 55 6.1 Introduction and ideas behind the model . 55 6.1.1 Finding the tight-binding parameters . 56 6.2 From atomic basis set to orthogonal tight-binding . 57 6.2.1 Hamiltonian for the full wire . 58 6.2.2 L¨owdin transformation . 59 6.2.3 Approximate orthogonalization . 60 6.2.4 Example - The one-dimensional chain . 61 7 Modelling of silicon nanowires 64 7.1 Metallic silicon nanowires . 64 7.1.1 Truncating the Hamiltonian . 65 7.1.2 Conductance of a pristine wire . 66 7.1.3 Anderson disorder . 67 7.1.4 Sub-conclusion . 69 7.2 Hydrogen passivated Si-wire . 71 7.2.1 Setting up the model . 71 7.2.2 Length and energy dependent conductance . 73 7.2.3 Distribution of conductances . 74 7.2.4 Resistance vs. length . 76 7.2.5 Mean free path and peaks of σG(L)............. 77 7.2.6 Comparison with analytical models . 78 7.2.7 Scaling of le and ξ ...................... 80 7.2.8 Sub-conclusion . 81 x CONTENTS 8 Summary and outlook 82 8.1 Evaluation of the real-space Kubo method . 82 8.2 Modelling of silicon nanowires . 85 8.3 Outlook ................................ 86 Appendix 87 A L¨owdin transformation 88 A.1 The generalized eigenvalue problem . 88 A.2 Formal orthogonalization procedure . 90 A.3 How to do the orthogonalization . 91 B Numerical Methods 93 B.1 The Continued Fraction Technique . 93 B.1.1 Truncation of the continued fraction - self-energy . 94 B.2 Tridiagonalization procedure . 96 B.3 Exact solution to the time-dependent Schr¨odinger equation .