Magnetism in. nanostructured graphene

Transcription

1 Magnetism in L. Brey nanostructured graphene Juan José Palacios Funding MEC Spain FIS , MAT , and CONSOLIDER CSD Generalitat Valenciana Accomp Dresden, May 29th, 2008 J. Fernández Rossier

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4 2 dimensional systems (top down) graphene Novoselov, K.S. et al. "Electric Field Effect in Atomically Thin Carbon Films", Science, Vol 306 (5696), 666, 2004

5 2 dimensional systems (bottom up) graphene Ethylene decomposition Vazquez de Parga et al., PRL (2008)

6 1 dimensional systems (top down) graphene Melinda Y. Han, Barbaros Oezyilmaz, Yuanbo Zhang, Philip Kim, Phys. Rev. Lett, 98, (2007)

7 1 dimensional systems (bottom up) graphene Chemically Derived, Ultrasmooth Graphene Nanoribbon Semiconductors Xiaolin Li, Xinran Wang, Li Zhang, Sangwon Lee, and Hongjie Dai* Science Express (2008)

8 0 dimensional systems (top down) graphene L. A. Ponomarenko, F. Schedin, M. I. Katsnelson, R. Yang, E. H. Hill, K. S. Novoselov, A. K.Geim, Science (2008)

9 0 dimensional systems (bottom up) graphene Vazquez de Parga et al., PRL (2008)

10 0 dimensional systems (bottom up) polycyclic aromatic hydrocarbons (PAH s) Naphthacene Coronene Chrysene Benz[d]ovalene Corannulene Ovalene Pentacene Triphenylene Anthracene Pyrene Benz[a]ovalene Benzo[a]pyrene

11 0 dimensional systems (bottom up) disk like polycyclic aromatic hydrocarbons (PAH s) J. Wu, M. D. Watson, K. Müllen, Angew. Chem. Int. Ed. 42, 5329 (2003)

12 Outline Nanographenes J. Fernández Rossier and J. J. Palacios, Phys. Rev. Lett. 99, (2007) Vacancies and voids in graphene and graphene ribbons J. J. Palacios, J. Fernández Rossier, and L. Brey, Phys. Rev. B 77, (2008)

13 Outline Nanographenes J. Fernández Rossier and J. J. Palacios, Phys. Rev. Lett. 99, (2007) Vacancies and voids in graphene and graphene ribbons J. J. Palacios, J. Fernández Rossier, and L. Brey, Phys. Rev. B 77, (2008)

14 Theoretical description of graphene A Hˆ = ε i nˆi + tij cˆi+ cˆ j i B t t ij t tij = t = 2.5 ev if first neighbors usually ε i = 0 B B

15 Electronic structure of graphene DOS Energy Band structure

16 Nanographenes

17 Nanographenes

18 Nanographenes

19 Nanographenes

20 Nanographenes

21 Understanding the spectra PˆA = i i ( i A PˆB = j j ) j B ( ) Hˆ PˆA + PˆB φ = E PˆA + PˆB φ Hˆ PˆA φ = EPˆB φ ( ) and Hˆ PˆB φ = EPˆA φ ( ) ( ) H PˆA PˆB φ = E PˆB PˆA φ = E PˆA PˆB φ What if N odd, e.g., N A N B = 1? At least 1 state with E = 0 Let us consider a state with weight only on sites A : Hˆ φ =Eφ A Hˆ PˆA PˆB φ A ( A ) ( ) = E PˆA PˆB φ A Hˆ φ A E =0 = E φ A What if N A N B > 1? At least N A N B states with E = 0 and weight only on sites A M. Inui, S. A. Trugman, and E. Abrahams Phys. Rev. B 49, 3190 (1994)

22 Understanding the spectra PˆA = i i ( i A PˆB = j j ) j B ( ) Hˆ PˆA + PˆB φ = E PˆA + PˆB φ Hˆ PˆA φ = EPˆB φ ( ) and Hˆ PˆB φ = EPˆA φ ( ) ( ) H PˆA PˆB φ = E PˆB PˆA φ = E PˆA PˆB φ What if N odd, e.g., N A N B = 1? At least 1 state with E = 0 Let us consider a state with weight only on sites A : Hˆ φ =Eφ A Hˆ PˆA PˆB φ A ( A ) ( ) = E PˆA PˆB φ A Hˆ φ A E =0 = E φ A What if N A N B > 1? At least N A N B states with E = 0 and weight only on sites A M. Inui, S. A. Trugman, and E. Abrahams Phys. Rev. B 49, 3190 (1994)

23 Understanding the spectra PˆA = i i ( i A PˆB = j j ) j B ( ) Hˆ PˆA + PˆB φ = E PˆA + PˆB φ Hˆ PˆA φ = EPˆB φ ( ) and Hˆ PˆB φ = EPˆA φ ( ) ( ) H PˆA PˆB φ = E PˆB PˆA φ = E PˆA PˆB φ What if N odd, e.g., N A N B = 1? At least 1 state with E = 0 Let us consider a state with weight only on sites A : Hˆ φ =Eφ A Hˆ PˆA PˆB φ A ( A ) ( ) = E PˆA PˆB φ A Hˆ φ A M. Inui, S. A. Trugman, and E. Abrahams Phys. Rev. B 49, 3190 (1994) E =0 = E φ A What if N A N B > 1? At least N A N B states with E = 0 and weight only on sites A

24 Triangles vs. hexagons

25 Triangles vs. hexagons

26 Electron electron interactions

27 Electron electron interactions Superatomic Hunds s rule

28 Electron electron interactions Hubbard model H = ε iσ nˆiσ + tij cˆi+σ cˆ jσ + U nˆi nˆi iσ ijσ i Mean field self consistent approximation : ( ) H = ε iσ nˆiσ + tij cˆi+σ cˆ jσ + U nˆi nˆi + nˆi nˆi + cte iσ ijσ i Density functional theory (GAUSSIAN03) Kohn Sham equations (unrestricted) GGA (BLYP) approximation to the functional (avoid hybrids, e.g., B3LYP) Saturation of dangling bonds with H

29 DFT vs mean field Hubbard Hubbard model (U=3.85 ev, t=2.5 ev) DFT

30 Ferromagnetic order

31 Lieb s theorem/ Longuet Higgins conjecture E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989) Longuet Higgins, J. Chem. Phys. 18, 265 (1950) (1) The number of unpaired electrons present in the ground state is at least as great as the number of carbon atoms having a deficiency of valence bonds in any principal resonance structure. (2) With a few special exceptions, these odd electrons are distributed over just those atoms which have a deficiency of valence bonds in one or more of the principal resonance structures. (3) In singly charged hydrocarbon anions or cations the ionic charge is located on just those atoms which bear charges in the various principal resonance structures.

32 Ferrimagnetic order

33 Ferrimagnetic order

34 Outline Nanographenes J. Fernández Rossier and J. J. Palacios, Phys. Rev. Lett. 99, (2007) Vacancies and voids in graphene and graphene ribbons J. J. Palacios, J. Fernández Rossier, and L. Brey, Phys. Rev. B 77, (2008)

35 Vacancies = H adsorption Jannik C. Meyer, C. O. Girit, M. F. Crommie, A. Zettl, arxiv:

36 A few words about ribbons (armchair) Son et al., Phys. Rev. Lett. 97, (2006)

37 Basic definitions AN A BN B A3 B1 B A4 B4 NI = 2 N I = 1 NI = 0 Local imbalance charge : N A N B = N I N min Z = N α + I (α ) + N ( β ) β I N Zmax = N I+ (α ) + N I ( β ) α β

38 Basic scenarios 1. N Zmin = N Z = N Zmax : All the voids are of the same sign. The coupling between them is always ferromagnetic and the spin of the ground state is 2S = N Z. The splitting with smaller spin states will depend on the inter void coupling. 2. N Zmin = N Z < N Zmax : All the voids of different sign are in proximity and interact, yielding a 2 S = N Zmin state. Calculations will be necessary to ascertain the spin texture in these situations. 3. N Zmin < N Z = N Zmax : All the voids of different type are separated and uncoupled. The ground state has 2S = N Zmin, but the spin flip gap is negligible since there are uncoupled magnetic moments. 4. N Zmin < N Z < N Zmax : There are voids of different sign, but some of them are uncoupled and some not. This is the most general case. The ground state has 2S = N Zmin, but, as in the previous case, the spin flip gap is negligible since there are uncoupled magnetic moments. Calculations will be necessary to ascertain the spin texture in these situations as well.

39 Basic scenarios 1. N Zmin = N Z = N Zmax : All the voids are of the same sign. The coupling between them is always ferromagnetic and the spin of the ground state is 2S = N Z. The splitting with smaller spin states will depend on the inter void coupling. 2. N Zmin = N Z < N Zmax : All the voids of different sign are in proximity and interact, yielding a 2 S = N Zmin state. Calculations will be necessary to ascertain the spin texture in these situations. 3. N Zmin < N Z = N Zmax : All the voids of different type are separated and uncoupled. The ground state has 2S = N Zmin, but the spin flip gap is negligible since there are uncoupled magnetic moments. 4. N Zmin < N Z < N Zmax : There are voids of different sign, but some of them are uncoupled and some not. This is the most general case. The ground state has 2S = N Zmin, but, as in the previous case, the spin flip gap is negligible since there are uncoupled magnetic moments. Calculations will be necessary to ascertain the spin texture in these situations as well.

40 Basic scenarios 1. N Zmin = N Z = N Zmax : All the voids are of the same sign. The coupling between them is always ferromagnetic and the spin of the ground state is 2S = N Z. The splitting with smaller spin states will depend on the inter void coupling. 2. N Zmin = N Z < N Zmax : All the voids of different sign are in proximity and interact, yielding a 2 S = N Zmin state. Calculations will be necessary to ascertain the spin texture in these situations. 3. N Zmin < N Z = N Zmax : All the voids of different type are separated and uncoupled. The ground state has 2S = N Zmin, but the spin flip gap is negligible since there are uncoupled magnetic moments. 4. N Zmin < N Z < N Zmax : There are voids of different sign, but some of them are uncoupled and some not. This is the most general case. The ground state has 2S = N Zmin, but, as in the previous case, the spin flip gap is negligible since there are uncoupled magnetic moments. Calculations will be necessary to ascertain the spin texture in these situations as well.

41 Basic scenarios 1. N Zmin = N Z = N Zmax : All the voids are of the same sign. The coupling between them is always ferromagnetic and the spin of the ground state is 2S = N Z. The splitting with smaller spin states will depend on the inter void coupling. 2. N Zmin = N Z < N Zmax : All the voids of different sign are in proximity and interact, yielding a 2 S = N Zmin state. Calculations will be necessary to ascertain the spin texture in these situations. 3. N Zmin < N Z = N Zmax : All the voids of different type are separated and uncoupled. The ground state has 2S = N Zmin, but the spin flip gap is negligible since there are uncoupled magnetic moments. 4. N Zmin < N Z < N Zmax : There are voids of different sign, but some of them are uncoupled and some not. This is the most general case. The ground state has 2S = N Zmin, but, as in the previous case, the spin flip gap is negligible since there are uncoupled magnetic moments. Calculations will be necessary to ascertain the spin texture in these situations as well.

42 Single vacancy: A

43 Single vacancy: A Σ= mi 2 i η = φv (i ) i 4

44 Spin charge separation Charge Q=0 Q =1 Spin

45 A and A2

46 Two vacancies A+A A+B

47 Two vacancies: A and B (no interactions)

48 Two voids: A2 and B2 (no interactions)

49 Large voids with NI=0

50 Two notches

51 Single vacancy in bulk graphene 2 R = ri r0 φv (i ) i M A( B ) d = ri r0 mi i A ( B ) Md = M + M A d B d

52 Single vacancy in bulk graphene

53 Finite density of vacancies: Compensated graphene (NI=0)

54 Future work Magnetic structure of more generic PAH s or nanographenes Thermal fluctuations: Superparamagnetism Stability of open shell structures (radicals) Change of properties when deposited on surfaces Devices F AF