Role of the vacuum fluctuation forces in microscopic systems

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Information about Role of the vacuum fluctuation forces in microscopic systems

Published on January 5, 2017

Author: AndreaBenassi3

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1. Role of the vacuum fluctuation forces in microscopic systems Colloquia Doctoralia XXI ciclo, Modena 9/2/09 Candidate Andrea Benassi Supervisor Prof. Carlo Calandra Buonaura

2. § Introduction to the vacuum fluctuation forces i) Macroscopic and microscopic forces ii) The sign of the force § New Developments i) Quantum size effects ii) Surface effects § Applications i) Stability of thin films ii) A vacuum force based device Outline

3. Macroscopic and Microscopic Forces Van  der  Waals-­‐London  force   MICRO  MACRO   NON  RELATIVISTIC  (small  d)   RELATIVISTIC  (large  d)   Casimir-­‐Polder  force   F(d) = − 18 πd7 ∞ 0 α1(ω)α2(ω)dω α1 α2 d α1 α2 d Casimir  force   F(d) = − cπ2 240d4 L2 Hamaker  force   d L F(d) = − H(1(ω), 2(ω)) d3 L2 1 2 d L (0) → ∞ H.G.B. Casimir Proc.Ned.Akad.Wet. (1948) S.K. Lamoreaux Rep.Prog.Phys. (1997)

4. Macroscopic and Microscopic Forces Van  der  Waals-­‐London  force   MICRO  MACRO   NON  RELATIVISTIC  (small  d)   RELATIVISTIC  (large  d)   Casimir-­‐Polder  force   F(d) = − 18 πd7 ∞ 0 α1(ω)α2(ω)dω α1 α2 d α1 α2 d Lifshitz  theory   F = F(d, 1(ω), 2(ω), 3(ω)) •   Extension  to  other  materials   •   Extension  to  other  geometries   •   Extendion  to  finite  temerature   E.M. Lifshitz Sov.Phys.JEPT (1956) S.K. Lamoreaux Physics Today (2007)

5. The sign of the force d d AIracJve   BETWEEN  ONTO   ISOLATED   INTERACTING   Repulsive  if   Stretching  if    Squeezing   dd 3 1 2 1 2 33 1 1 3 2 1 3 J.N. Munday and F. Capasso Nature (2009) Benassi and Calandra J.Phys.A. 40,13453

6. Electron Confinement (EC) When  the  dimension  of  the  interacJng  objects  is  comparable  to  the  electron  wavelength   the  quantum  nature  of  the  electrons  cannot  be  neglected,  this  is  the  case  of  nanometric   thickness  films:   The  film  is  considered  as  a  quantum  well:   •   The  electron  spill  out  is  simulated  arJficially        in  the  case  of  an  infinite  well,  it  comes        out  naturally  in  the  case  of  a  finite  well   •   The  discreJzaJon  of  the  energy  levels      gives  rise  to  kinks  in  the  electron  density      or  in  the  Fermi  energy     Wood and Ashcroft Phys.Rev.B (1982)

7. The  RPA  dielectric  tensor  of  the  nanometric  film  can  be  calculated…     •   The  system  is  anisotropic   •   The  dielectric  tensor  components  depends  on  the  film  (well)  thickness   •   The  zz  component  becomes  semiconducJng  while  the  parallel  ones  remains  metallic   •   The  dielectric  funcJon  depends  on  the  electron  density  so  kinks  are  also  present  in  the   dielectric  funcJon   Quantum Models for the film Benassi and Calandra Europhys.Lett. 82, 61002

8. The interaction between films Some  plot  of  the  relaJve  percent  difference  of  the  force  with  and  without  the  EC:   •   Including  the  EC  the  force  strength  decreases,  along  the  z  direcJon  the  metallic  film   becomes  transparent,  trapping  less  modes  inside  the  cavity     •   Kinks  appear  each  Jme  that  a  new  level  fall  below  the  Fermi  energy     •   For  large  film  thickness  and  large  electron  density  the  EC  is  negligible     and  the  relaJve  difference  goes  to  zero     δP = Fbulk − FEC Fbulk d d Ωp = 1014 rad/s = 10nm Ωp = 1014 rad/s = 50nm Ωp = 5 · 1014 rad/s = 50nm Benassi and Calandra J.Phys.Conf.Series (2009)

9. Conclusions I     The  inclusion  of  EC  gives  correcJons  depending  on  the  film  density,  the  film  separaJon   and  the  film  thickness,  these  correcJon  can  vary  between  few  percent  up  to  50%     The  EC  correc*ons  can  be  improved  in  order  to:     •   include  atomisJc  descripJon  of  the  film  la]ce   •   include  surface  effects   We  believe  that  for  all  these  purposes  an  ab-­‐iniJo  approach  can  be  suitable  !     The  final  goal:     •   treat  arbitrary  shape  objects  (designing  MEMS)        

10. A Silicon Surface: (111)2x1 Surface  properJes  start  to  be  relevant  when  the  system  of  interest  is  strongly  confined.     A  semiconducJng  surface  introduces  some  new  features:     •   Presence  of  surface  states  inside  the  bulk  gap   •   Surface  reconstrucJon   Benassi and Calandra in preparation

11. Modification of Dielectric Properties An  RPA  dielectric  funcJon  can  be  calculated  both  for  the  bulk  and  the  slab:   •   The  off-­‐diagonal  components  are  negligible     •    A   strong   anisotropy   is   present   in   the   slab:   the  Lifshitz  theory  must  be  extended  to  treat   anisotropy   •    For   symmetry   reasons   the   surface   states   affect  only  the  yy  component     Slab  real   slab   imaginary  yy   xx   zz   bulk   imaginary  RPA   exp   Benassi and Calandra in preparation

12. Modification of Vacuum Interaction Using  the  Lifshitz  formalism  the  force  between  films  can  be  calculated  using  RPA  bulk  and   the  RPA  slab  dielectric  funcJons.  Their  difference  tells  us  how  important  can  be  the   surface  effects:   d = 2nm •   enlarging  the  film  thickness  the  surface  effects  become  negligible   •   the  relaJve  percent  difference  is  large  for  large  separaJons,  where  only  the  staJc  value   is  important   Benassi and Calandra in preparation

13. Conclusions II   The  calculaJon  of  the  vacuum  force  between  thin  films  gives  very  different  results  when   performed  using  an  ab-­‐iniJo  calculated  dielectric  funcJon  instead  of  the  ordinary  bulk   one.  This  is  mainly  due  to:     •   the  appearance  of  surface  states  inside  the  bulk  band  gap  that  gives  rise  to  a  strong   absorpJon  at  low  frequency   •   the  presence  of  the  surface  reconstrucJon  that  brings  to  a  highly  anisotropic  dielectric   response     To  improve  the  calculaJons  one  has  to  go  beyond  the  RPA  approximaJon  of  the  dielectric   tensor  including  many-­‐body  effects     A   separaJon   of   the   confinement   contribuJon   from   the   surface   one   must   be   done:   hydrogen  passivaJon  prevent  surface  reconstrucJon.    

14. A Model for the Film Stability The  vacuum  force  can  cause  a  change  in  the  surface  morphology  of  a  thin  film.    In  an   epitaxial  film  3  main  contribuJons  can  be  considered…   La]ce  mismatch  stress   Surface  stress   Vacuum   stress   The  first  two  contribuJon  are  several  order  of  magnitude  larger  that  the  Vacuum  force   however,  close  to  the  equilibrium  they  cancel  out  and  the  system  becomes  sensible  to   the  vacuum  force     R. Asaro and W. Tiller. Met.Trans (1971)

15. A Model for the Film Stability With  a  slight  (                                                                  )  sinusoidal  corrugaJon  we  have:  q d λ The  change  in  energy  is:   CriJcal  thickness  and  wavelength  exist  if  the  vacuum  force  is  repulsive:   ∆E = − 1 − ν2 E σ2 πq2 + γ π2 q2 λ − q2 λH 8πd4 Benassi and Calandra J.Phys.A 41,175401

16. Film Stability Diagrams (ω) = 1 − Ω2 1 ω2 (ω) = 1 − Ω2 3 ω2 Corrugated  interface   Plasma  model  for  simple  metals:  the   plasma  frequency  is  proporJonal  to   the  electron  density  of  the  metal   Benassi and Calandra J.Phys.A 41,175401

17. Conclusions I We   have   shown   how   the   morphology   of   a   thin   deposited   film   can   be   affected   by   the   presence  of  vacuum  forces,  its  behaviour  depending  on  the  dielectric  properJes  of  both   the  film  and  the  substrate.     This  phenomenon  can  be  used:     •   to  modify  the  film  properJes  during/acer  their  growth       •   to  measure  the  Casimir  force  on  a  single  object  (not  yet  measured)     However,  to  compare  our  result  with  realisJc  situaJons  some  improvements  are  needed:     •   void  defects  and  dislocaJon  must  be  included        in  the  elasJc  model  of  the  film   •   the  relaxaJon  of  the  substrate  la]ce  must  be  included  

18. A Vacuum Force Based MEMS Understanding   vacuum   forces   properJes   is   crucial   in   the   world   of   micro   and   nano   mechanics:     •   to  prevent  micro  and  nano-­‐devices        from  sJcJon,  adhesion  and  breaking       •   to  actuate  micro  and  nano-­‐devices        using  vacuum  forces     Vacuum  forces  depends  on  the  dielectric  properJes  of  the  interacJng  media,  tuning  the   dielectric  properJes  we  can  control  the  force  tailoring  the  mechanical  moJon.     •   The  force  depends  on  the  integral  over  frequencies  of  the  dielectric  funcJon,  a  drasJc   change  in  the  dielectric  properJes  is  needed  to  modify  a  rather  insensible  force   •   This  change  must  be  reversible  in  order  to  be  able  both  to  increase  and  decrease  the   vacuum  force   H.B.Chen et al.Science (2001) Buks and Rouckes Europhys.Lett. (2001)

19. The candidate: GeTe A   good   candidate   is   the   Germanium   Telluride,   which   undergoes   a   fast   and   reversible   crystalline-­‐amorphous  transiJon…     In  the  transiJon  the  dielectric  properJes  change  strongly  moving  from  a  metal  to  an   insulator  with  1  eV  gap.       CalculaJng  the  vacuum  force  we  found  we  found  large  differences  in  the  interacJon   between  two  amorphous  plates  FAA  and  two  crystalline  plates  FCC     Benassi and Calandra Europhys.Lett. 84,11002

20. with  the  equilibrium  condiJon:         the  soluJon  is  a  bi-­‐stable  potenJal:       •    the   distance   between   the   maximum   and   the   minimum   gives  the  mechanical  excursion  of  the  device     •   The  height  of  the  barrier  gives  an  idea  of  the  stability       •   The  posiJon  of  the  barrier  gives  the  s*c*on  point     The Model Device We  can  model  a  mechanical  device  by  a  fixed  plate  interacJng  with  a  mobile  plate  with  a   given  elasJcity  represented  by  a  spring.   The  two  plates  are  covered  with  a  thick  GeTe  film  whose  phase  can  be  switched  using  a   laser  pulse  or  a  current  pulse.     F(x) = Fres. − Fdisp. = k(x − x0) − ΣF(x − x0) (x − x0) − Σ k F(x − x0) = 0  Σ k Batra et al. Europhys.Lett. (2007)

21. Tailoring the Device motion What  happen  if  we  change  the  GeTe  phase?   •   The  mechanical  excursion  can  be  modified  by  10  %     •   The  stability  of  the  device  can  be  changed  up  to  80%     •   The  sJcJon  point  can  be  moved  by  10%       x/x0 CC   AC   AA   Benassi and Calandra Europhys.Lett. 84,11002

22. Conclusions IV We   have   shown   the   feasibility   of   a   vacuum   forces   based   device   that   exploit   a   metal-­‐ insulator  phase  transiJon  to  modify  the  mechanical  properJes  of  a  micro  oscillator.     The  device  model  is  sJll  rough  and  can  be  improved  in  a  number  of  ways  before  it  can  be   used  to  design  a  real  device:     •   Surface  roughness  effects  are  known  to  be  relevant  in  the  vacuum  interacJon  between   plates,  they  must  be  included  in  our  calculaJons     •    The   finiteness   of   the   plates   introduces   edge   effects   whose   importance   increase   decreasing  the  device  dimension       •   In  our  model  we  considered  a  global  phase  change  of  the  plates  media,  but  each  phase   change  process  has  its  own  penetraJon  depth  inside  the  GeTe  film     •   Finite  Jme  required  by  the  phase  change  

23. Acknowledgments I  would  like  to  express  my  special  thanks  to  professor  Carlo  Calandra  Buonaura  for  his  in-­‐   valuable  guidance,  help  and  support  over  these  years.       Another  special  thanks  to  professor  Elisa  Molinari  who  hosted  me  in  S3  na*onal  research   center  giving  me  the  opportunity  to  aIend  schools,  workshop  and  seminars  all  over  the  world.       I  want  to  acknowledge  CINECA  consorzio  interuniversitario  for  funding  my  Ph.D.  fellowship.

24. Field and charge quantization •   QuanJzed  charges  (perturbaJon  theory)   •   ConJnuous  fields   MICRO  MACRO   •   ConJnuos  media  (linear  response  theory)   •   QuanJzed  fields   α1 d d 1 2 Two  approximaJons  of  the  same  theory,   the   QED,   in   which   both   the   fields   and   the  charges  are  quanJzed     The  field-­‐charge  interacJon  is  hidden   inside  the  response  funcJon  

25. Temperature: a way to eliminate the kinks The  kinks  issue  can  be  solved  introducing  a  finite  value  for  the  temperature,  the  energy   levels   becomes   conJnuously   populated,   the   kinks   disappear   from   the   energy   and   the   force:   T = 0◦ K T = 1◦ K T = 2◦ K T = 30◦ K

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