Appendix A Time Evolution of the Wave Packet

Transcription

1 Appendix A Time Evolution of the Wave Packet This appendix presents how to calculate the evolution of the wave packet Û(t) ϕ RP and [ ˆX, Û(t)] ϕ RP which are used in the application of the real space method to calculate the transport properties. In order to do that, we divide the time t into small time steps T = t/n and approximate Û(T ) with the series of orthogonal Chebyshev polynomials Q n (Ĥ) i ĤT Û(T ) = e = c n (T )Q n (Ĥ) (A.) The original Chebyshev polynomials T n which satisfy the recurrent relations T 0 (x) = T (x) = x T 2 (x) = 2x 2. T n+ (x) = 2x T n (x) T n (x) (A.2) (A.3) (A.4) (A.5) and act on the interval [ ; ] are rescaled to the rescaled Chebyshev polynomials Q n which cover the bandwidth of system Hamiltonian E [a : a + ], with the band center and bandwidth are a and 4b, respectively. These rescaled Chebyshev polynomials Q n satisfy Q n (E) = ( ) E a 2T n ( n ) (A.6) Q 0 (E) = (A.7) Q (E) = 2 E a (A.8) Springer International Publishing Switzerland 206 D.V. Tuan, Charge and Spin Transport in Disordered Graphene-Based Materials, Springer Theses, DOI 0.007/

2 44 Appendix A: Time Evolution of the Wave Packet Q 2 (E) = 2 ( ) E a (A.9). ( ) E a Q n+ (E) = 2 Q n (E) Q n (E) (A.0) With above definition, we have the orthonormal relations for Q n (E) Q n (E)Q m (E)p Q (E)dE = δ mn (A.) with respect to the weight p Q (E) = 2πb ( ) (A.2) E a 2 Once the Q n polynomials are well defined, one can compute the related c n (T ) coefficients c n (T ) = de p Q (E)Q n (E)e i E T (A.3) ( E a ) 2Tn = de 2πb ( ) e i E T (A.4) E a 2 = 2 dx T n (x) x 2 (x+a) e i T (A.5) π = ( 2i n e i a T J n ) T, n (A.6) and the first coefficients c 0 (T ) = i n e i a T J 0 ( T ) with J n (x) is the Bessel function of the first kind and order n We can now calculate ϕ RP (T ) ϕ RP (T ) =Û(T ) ϕ RP (A.7) N N ϕ RP (T ) c n (T )Q n (Ĥ) ϕ RP = c n (T ) α n (A.8) where α n =Q n (Ĥ) ϕ RP. With the definitions introduced in Eqs. (A.7, A.8 and A.9) and the recurrence relation Eq. (A.0), we obtain

3 Appendix A: Time Evolution of the Wave Packet 45 α 0 = ϕ RP (A.9) ( ) Ĥ a α = α 0 (A.20) ( ) Ĥ a α 2 = α 2 α 0 (A.2) b ( ) Ĥ a α n+ = α n α n ( n 2) (A.22) b Following the same reasoning as for ϕ RP (T ), ϕ RP (T ) can be evaluated first writting ϕ RP (T ) =[ˆX, Û(T )] ϕ RP (A.23) N ϕ RP (T ) N c n (T )[ ˆX, Q n (Ĥ)] ϕ RP = c n (T ) β n (A.24) with β n =[ˆX, Q n (Ĥ)] ϕ RP. Using the Eqs. (A.0) and (A.9 A.22), we obtain the recurrence relation for β n β 0 =0 (A.25) [ ˆX, Ĥ] β = ϕ RP (A.26) ( ) Ĥ a β n+ = β n β n + b b [ ˆX, Ĥ] α n ( n ) (A.27) which contain α n and the commutator [ ˆX, Ĥ] determined by the hopings and the distances between neighbours 0... Hij X ij [ ˆX, Ĥ] =.... H ji X.. ji 0 (A.28) where X ij = (X i X j ) is the distance between orbitals ϕ i and ϕ j.

4 Appendix B Lanczos Method In this appendix the Lanczos method is introduced. Instead of diagonalizing the Hamiltonian the Lanczos method is a useful method to transform the Hamiltonian into tridiagonal matrix which is more convenient to compute the density of state or spin polarization. The general idea of this method is building from the initial state ϕ RP a new basis in which the Hamiltonian is tridiagonal. Here are the basic steps: The first step starts with the first vector in the new basis ψ = ϕ RP and builds the second one ψ 2 which is orthonormal to the first one a = ψ Ĥ ψ (B.) ψ 2 =Ĥ ψ a ψ (B.2) b = ψ 2 = ψ 2 ψ 2 (B.3) ψ 2 = b ψ 2 (B.4) All other recursion steps ( n ) are identical, we build the (n + ) th vector which is orthonormal to the previous ones and given by a n = ψ n Ĥ ψ n (B.5) ψ n+ =Ĥ ψ n a n ψ n b n ψ n (B.6) b n = ψ n+ ψ n+ (B.7) ψ n+ = b n ψ n+ (B.8) The coefficients a n and b n are named recursion coefficients which are respectively the diagonal and off-diagonal of the matrix representation of Ĥ in the Lanczos basis (that we write Ĥ). Springer International Publishing Switzerland 206 D.V. Tuan, Charge and Spin Transport in Disordered Graphene-Based Materials, Springer Theses, DOI 0.007/

5 48 Appendix B: Lanczos Method while a b b a 2 b 2 Ĥ =. b bn b N a N With simple linear algebra, one shows that ϕ RP δ(e Ĥ) ϕ RP = ψ δ(e Ĥ) ψ = lim ( ) η 0 π I m ψ E + iη Ĥ ψ (B.9) ψ E + iη Ĥ ψ = E + iη a E + iη a 2 E + iη a 3 b (B.0) which is referred as a continued fraction G with the definition of G n as, b 2 b 2 2 G n = E + iη a n E + iη a n+ b 2 n b 2 n+ E + iη a n+2 b2 n+2... (B.) G = G n = E + iη a b 2G 2 E + iη a n bn 2G n+ (B.2) (B.3) Since we compute a finite number of recursion coefficients, the subspace of Lanczos if of finite dimension (N), so it is crucial to terminate the continued fraction by an

6 Appendix B: Lanczos Method 49 appropriate choice of the last {a n=n, b n=n } elements. Let us rewrite the continued fraction as G = E + iη a E + iη a 2 b 2 E + iη a 3 b 2 2 b E + iη a N b 2 N G N+ (B.4) where G N+ denotes such termination. The simplest case is when all the spectrum is contained in a finite bandwidth [a ; a + ], a the spectrum center and 4b its bandwidth. Recursion coefficients a n and b n oscillate around their average value a et b, and the damping is usually fast after a few hundreds of recursion steps. The termination then satisfies G N+ = E + iη a b 2 G N+2 = E + iη a b 2 G N+ (B.5) from which a polynomial of second degree is found (b 2 )G 2 N+ + (E + iη a)g N+ = 0 (B.6) and straightforwardly solved = (E + iη a) 2 () 2 G N+ = (E + iη a) i 2 G N+ = (E + iη a) i () 2 (E + iη a) 2 2 (B.7) (B.8) (B.9)

7 Curriculum Vitae Dinh Van Tuan Contact information Postdoctoral Researcher Theoretical and Computational Nanoscience Group, Catalan Institute of Nanoscience and Nanotechnology Research Interests Quantum Condensed Matter Theory: Charge and spin transport, quantum Hall effect, spin Hall effect, topological electronic phases and disordered electronic systems. Professional Preparation Ph.D., Materials Science, Department of Physics, Autonomous University of Barcelona, Spain Thesis Topic: Charge and Spin Transport in Disordered Graphene-Based Materials Supervisor: Prof. Stephan Roche Springer International Publishing Switzerland 206 D.V. Tuan, Charge and Spin Transport in Disordered Graphene-Based Materials, Springer Theses, DOI 0.007/

8 52 Curriculum Vitae M. Sc., Theoretical and Mathematical Physics, Department of Theoretical Physics, Ho Chi Minh city University of Science, Vietnam Thesis Topic: The Graphene Polarizability and Applications Supervisor: Associate Prof. Nguyen Quoc Khanh Professional Appointments 9/204 Postdoctoral Researcher, Catalan Institute of Nanoscience and Nanotechnology 9/20 9/204 Ph.D. student, Catalan Institute of Nanoscience and Nanotechnology 9/2007 9/20 Research Assistant, Ho Chi Minh city University of Science Publications. Dinh Van Tuan, Frank Ortmann, David Soriano, Sergio O. Valenzuela, and Stephan Roche. Pseudospin-driven spin relaxation mechanism in graphene. Nature Physics, 0, (204) 2. Alessandro Cresti, David Soriano, Dinh Van Tuan, Aron W. Cummings, and Stephan Roche. Multiple Quantum Phases in Graphene with Enhanced Spin- Orbit Coupling: from Quantum Spin Hall Regime to Spin Hall Effect and Robust Metallic State. Physical Review Letter, 3, (204) 3. Aron W. Cummings, Dinh Loc Duong, Van Luan Nguyen, Dinh Van Tuan, Jani Kotakoski, Jose Eduardo Barrios Vargas, Young Hee Lee and Stephan Roche. Charge Transport in Polycrystalline Graphene: Challenges and Opportunities. Advanced Materials, 26, Issue 30, (204) 4. David Jiménez, Aron W. Cummings, Ferney Chaves, Dinh Van Tuan, Jani Kotakoski, and Stephan Roche. Impact of graphene polycrystallinity on the performance of graphene field-effect transistors. Appl. Phys. Lett., 04, (204) 5. Dinh Van Tuan, Jani Kotakoski, Thibaud Louvet, Frank Ortmann, Jannik C. Meyer, and Stephan Roche. Scaling Properties of Charge Transport in Polycrystalline Graphene. Nano Letters, 3 (4), (203) 6. Alessandro Cresti, Frank Ortmann, Thibaud Louvet, Dinh Van Tuan, and Stephan Roche. Broken Symmetries, Zero-Energy Modes, and Quantum Transport in Disordered Graphene. Phys. Rev. Lett., 0, 9660 (203). 7. Alessandro Cresti, Thibaud Louvet, Frank Ortmann, Dinh Van Tuan, Paweł Lenarczyk, Georg Huhs and Stephan Roche. Impact of Vacancies on Diffusive and Pseudodiffusive Electronic Transport in Graphene. Crystals, 3, (203). 8. Dinh Van Tuan and Nguyen Quoc Khanh. Plasmon modes of double-layer graphene at finite temperature. Physica E: Low-dimensional Systems and Nanostructures, 54, 267 (203)

9 Curriculum Vitae Dinh Van Tuan, Avishek Kumar, Stephan Roche, Frank Ortmann, M. F. Thorpe, and Pablo Ordejon. Insulating behavior of an amorphous graphene membrane. Phys. Rev. B, 86, 2408 (Rapid Communications) (202) 0. Dinh Van Tuan and Nguyen Quoc Khanh. Temperature effects on Plasmon modes of double-layer graphene. Communications in Physics, 22, 45 (202). David Soriano, Dinh Van Tuan, Simon M.-M. Dubois, Martin Gmitra, Aron W. Cummings, Denis Kochan, Frank Ortmann, Jean-Christophe Charlier, Jaroslav Fabian, and Stephan Roche. Spin Transport in Hydrogenated Graphene. accepted for publication in 2D Materials as a Topical Review, (205) Honors and Awards Award for the best graduate student, 200 Vietnamese Ministry of Education award, 2005 Sliver Medal at the National Physics Olympiad for the students of national universities, 2005 Bronze Medal at the National Physics Olympiad, 2003