Quantumengineeringfor
electronsandspins
RudolfARömer,Physics
Thephysicsofwaves
• Wavesinterfereduetothesuperpositionprinciple
• Waveinterferencecanbedestructiveorconstructive
Interferenceofelectronsfroma
doubleslit
http://www.hitachi.com/rd/
portal/highlight/quantum/in
dex.html#anc04
(Tonomura, 1980’s)
Interferencefromdisorder:
Andersonlocalization
Light,water,sound
(ultra),electrons,
atoms …
J. Billy et al., Direct observation of Anderson localization of matter waves in a
controlled disorder, Nature (2008) 453, 891-894; G. Roati et al., Anderson
localization of a non-interacting Bose-Einstein condensate, Nature (2008) 453, 895898
Segev, M., Silberberg,
Y., & Christodoulides,
D. N. (2013). Anderson
localization of light. Nat
Photon, 7(3), 197–204.
Atransitionin
𝑫 ≥ 𝟑,nonefor
𝑫≤𝟐
Abrahams E.,
Anderson P. W.,
Licciardello D. C.
and Ramakrishnan
T. V. 1979 Phys.
Rev. Lett. 42 673
Howto“cheat”thegangof4!
• Many-bodyphysics
•
Goldsborough, A. M., Römer, R. A., (2014). Selfassembling tensor networks and holography in
disordered spin chains. Physical Review B, 89(21).
Magnetism, superconductivity, …
• Correlatedrandomness
•
•
•
Changesof“universality”
Perfectcorrelation=cleansystem
Short-range (irrelevant inlargesystems)versuslong-range correlation
(relevantwhenever correlationlength comparable/larger thansystem)
de Moura, F. A. B. F., & Lyra, M.
L. (1998). Delocalization in the
1D Anderson Model with LongRange Correlated Disorder.
Physical Review Letters, 81(17),
3735–3738.
“Charge Puddles in Graphene Near the
Dirac Point”, S. Samaddar, I. Yudhistira,
S. Adam, H. Courtois, and C.B.
Winkelmann, arXiv:1512.05304
Ndawana, M. L., Römer, R. A., & Schreiber, M.
(2004). The Anderson metal-insulator transition
in the presence of scale-free disorder. Epl, 68(5),
678–684.
Controlledengineeringofextended
statesindisorderedsystems
Transfer-matrixapproach:
– Rewrite𝐻𝜓 = 𝐸𝜓 as
– isactuallydisordermatrix
Rodriguez, A., Chakrabarti, A., Römer, R. (2012). Controlled engineering of
extended states in disordered systems. Physical Review B, 86(8).
Resonancecondition
𝜖+,- = 𝛼+ 𝛽- foronsitepotentials
𝛾+,- = 𝛼+ 𝜉- fortransversekineticenergy
3NrandomnumbersforN2 terms–
correlation!
Atransformation
• Wecanfactorizematrix𝜖+ = 𝛼+ 𝑃,s.t. 𝑃doesnot
dependon𝑥.
• 𝑝 = 𝑈 67𝑃𝑈 diagonalmatrixofeigenvaluesof𝑃,
𝑈 thevectorofeigenvectors.
• ThenwithΦ+ = 𝑈 67Ψ+ ,wecanwrite
Trafo cont’d:
Explicitly,thisgives
Supposethat
oneofthe
𝑝: ≈ 0,then
thatequation
correspond
toanearperfectly
conducting
channel!
Quantumengineeringwithelectrons
Choosingthedistributionsofrandomnessin{𝛼},
{𝛽} and{𝜉},wecanthenengineerstateswithany
confinementin(x,y),(x)or(y)independentlyor
$log %L&!Ψx,y "2
none!
2
! Χy "
1
1.25
1.5
1.75
2
2.25
#2.5
50
40
Rodriguez, A.,
Chakrabarti, A., Römer,
R. (2012). Controlled
engineering of extended
states in disordered
systems. Physical
Review B, 86(8).
y
30
20
10
!Φx "2
#b$
20
40
x
60
80
100
$log %L&!Ψx,y "2
! Χ y "2
1
1.25
1.5
1.75
2
#2.5
2.25
50
40
y
30
20
10
! Χ y "2
y
!Φx "2
#a$
20
$log %L&!Ψx,y "2
1
1.25
1.5
1.75
2
2.25
40
x
60
80
! Χ y "2
#2.5
50
50
40
40
30
y
20
10
10
20
40
x
60
80
100
1
1.25
$log %L&!Ψx,y "2
1.5
1.75
2
2.25
#2.5
30
20
!Φx "2
#a$
100
!Φx "2
#b$
20
40
x
60
80
100
“Phase”diagramofdelocalization
in2D
Dashedlinesindicateanalyticestimatesfor
“phase”boundaries
Λx !Lx
Λy !Ly
10
8
"m Β "
0
0.5
1
y
x y
0
0.5
y
4
"Ξ"
1
$2
1.5
xy
y
3
x
2
1
2
0
5
y
6
4
Λx !Lx
Λy !Ly
$2
1.5
!5
0
E!t
5
0
!6
!4
!2
0
E!t
2
4
6
Quantumengineeringwithspins
Spintronics (spintransportelectronics),alsoknown
asspinelectronics orfluxtronics,isthestudyofthe
intrinsicspinoftheelectronanditsassociated
magneticmoment,inadditiontoitsfundamental
electroniccharge,insolid-statedevices.
https://en.wikipedia.org/wiki/Spintronics
Applications:metal-based(GMR,TMR),doped
semiconductormaterialswithdiluteferromagnetism
(ZnO,GaMnAs,…),allforlogic/storagedevices
Magneticchainin1D:
Magnetic substrate
B. Pal, A., Chakrabarti, A., Römer, R. (2016). “Spin filter for arbitrary spins by
substrate engineering”, to be submitted to Physical Review B, 86(8).
Themodel
• Additional“Heisenbergterm”addsenergy
ifspinisnotalignedwithmagneticfieldin
substrateandreducesotherwise
• Leadstoalignmentofspinsacrosssystem
Variationsinsubstrate
magnetization
C⁄
D
𝒉@ A 𝑺@
= 𝒉@,+ 𝝈- + 𝒉@,+ 𝝈- + 𝒉@,H 𝝈H
ℎ@ cos𝜃@
ℎ@ sin𝜃@ exp(−𝑖𝜑@ )
=
ℎ@ sin𝜃@ exp(𝑖𝜑@ )
−ℎ@ cos𝜃@
Transportanddensityofstates
• Heisenbergtermsplitsspin-bands
• Spinstransportonlyinappropriateenergies
𝜃@ = 0
𝜃@ = 𝜋/4
Higher-spincases
•
•
•
•
𝑆 = 7⁄\,𝑚^ = − 7⁄\ , 7⁄\ [2]
[2𝑆 + 1]
𝑺 = 𝟏, 𝒎𝑺 = −𝟏, 𝟎, 𝟏 [3]
…
𝑆 = b⁄\ , 𝑚^ = − b⁄\ , − c⁄\ . … , c⁄\ , b⁄\ [7]
Atomicgases:suchhigher-spinatomscanbestudied
𝜃@ = 0
Spiral:flippingspins
𝜃@ = 𝑛𝜋m𝐿
𝜃@ = 𝑛𝜋m(𝐿/2)
S= 1/2
Spin-spiralforS=1
• Perfectlyanaloguous resultsforallhigherS
• Asimplemodelforaspinfilterwithperfect
polarization
Combininglocalizationandspins:
usingdisordertoselectthespin
Silicene:
•
•
•
hexagonallatticeofSiatoms,bucklesandhaslargerspin-orbit(SO)coupling
Suggestionsthatgapopensandcanbecontrolledbyelectricfield
UseSO/substratetoseparatespins,anddisorder/vacanciestolocalizespinsin
differentenergyranges
Liu C C, Feng
W and Yao Y
2011 Physical
Review Letters
107 076802
Transportcalculations
𝑐 = 0.05
• 𝜆SO spin-orbitcoupling
(3.9meV)
• 𝑀 magneticfieldsplitting
(0.1eV,aguess,0.25eVinG)
duetosubstrate(ferroelectric
polymer)
Nano-ribbon:
L=32.22nm(300“sites”in
transportdirection)
W=2.35nm
Results
"Silicene-based spin-filter device: Impact of random
vacancies“, C. Nunez, F. Dominguez-Adame, P. A. Orellana,
L. Rosales, R. A. Römer, accepted for publication in 2D Mat.
(2015)
𝜀Fermi = 0.95𝑒V
𝑐 = 0.05
𝑐 = 0.1
[errors within
symbol size]
Addinganelectricfield
𝑐 = 0.05
• 𝑈 = 0.3𝑒V
𝐴
• 𝜁: = ±1 if𝑖 = { }
𝐵
• spin-dependent
transportregionat
lowenergies
• Butlargedisorder
dependence
Conclusions
• Extendedtransportchannelscanopen
eveninlow-Ddisorderedsystems
• Simplemodelstoshowpossibilities
• Probablyhardtoachieveexperimentally
(butwhoknows!)
• Thanksfortheattention!
DisorderedQuantumSystems
Localization: E.Carnio,AChakrabarti (Calcut),NHine, RLima(Maceio),A
Rodriguez-Gonzales(Freiburg),EProdan (NewYork), DQuigley,HSchulzBaldes (Erlangen)
NanoScience:FDominguez-Adame (Madrid),CNunez(Chile),LRosales
(Chile),POrellana (Chile)
NumericalMethods: M.Bollhoefer(Braunschweig),OSchenk(Basel)
ProteinRigidity: RFreedman,EJimenez,SAWells,JHeal
Many-BodyPhysics:MEPortnoi,AGoldsborough
QuantumHall: JOswald (Leoben)
DNA: CPaez (Brasil), ARodriguez(Madrid),C-TShih(Taichung),SAWells
RogueWaves:ASavojardo,MEberhard (Aston)
Funding: DFG,EPSRC,LeverhulmeTrust,NuffieldFoundation,Royal
Society;HECToR,ARCHER,Hartree Centre