Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I

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Far from equilibrium steady states of

1D-Schrödinger-Poisson systems with quantum wells I

Virginie Bonnaillie-Noël, Francis Nier, Mamodyasine Patel

To cite this version:

Virginie Bonnaillie-Noël, Francis Nier, Mamodyasine Patel. Far from equilibrium steady states of

1D-Schrödinger-Poisson systems with quantum wells I. 2007. �hal-00124683�

1D-Shrödinger-Poisson systems with quantum wells I

V.Bonnaillie-Noël

∗

,F. Nier

†

,Y. Patel

Abstrat

We desribe the asymptoti of the steady states of the out-of equilibrium Shrödinger-

Poissonsystem, intheregime ofquantumwells ina semilassial island. After establishing

uniformestimatesonthenonlinearity,weshowthatthenonlinearsteadystateslieasymptoti-

allyinanite-dimensionalsubspaeoffuntionsandthattheinvolvedspetralquantitiesare

reduedto anitenumberof so-alled asymptotiresonant energies. Theasymptotinite

dimensionalnonlinearsystemiswritteninageneralsettingwithonlyapartialinformationon

itsoeients. Afterthisrstpart,aompletederivationoftheasymptotinonlinearsystem

willbedoneforsomespeiasesinaforthomingartile[BNP2 ℄.

MSC (2000): 34L25;34L30;34L40;65L10;65Z05;81Q20;82D37.

Keywords: Shrödinger-Poissonsystem;Asymptotianalysis;Multisaleproblems.

Contents

1 Introdution 2

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Quantumframework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Shrödinger-Poissonsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 A prioriEstimates 10 3 Resultson the DirihletProblem 13 3.1 Somenotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Deayestimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Spetrumforonesinglewell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Spetruminthemultiple wellsase. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Resolventestimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Complexdeformation 21 4.1 AreduedStone'sformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Resonanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Analysisoftheresolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

∗

IRMAR, UMR-CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, Frane,

Virginie.Noel-Bonnaillieuniv-rennes1.fr

†

IRMAR, UMR-CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, Frane,

Franis.Nieruniv-rennes1.fr

6 Loal densityof states 28

6.1 Eliminating thenonresonantenergies . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2 Contributionof resonantstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A Agmon identity 33

B Monotony Priniple 34

C Spetral approximation 34

D Satteringstates for the barrier 34

E Pointwiseestimatefor the resolvent 35

1 Introdution

1.1 Motivation

This analysis is motivated by the study of quantum eletroni transport in semiondutor het-

erostrutures, likeresonanttunneling diodes. It ismodelledon thebasisofamean eld Hartree

type desription of the eletrostati interation of partiles, known as the Shrödinger-Poisson

system. Themodellingofresonanttunneling diodesinludesthefollowingharateristifeatures:

1. Steady eletroniurrentsareobserved. This anbe ahievedonlywithin themodellingof

out-of-equilibriumquantum systems.

2. TheI−V ^{urvesofsuhdeviespresentnegativedierentialresistane.Weareinafarfrom}

equilibriumregime,forwhihthelinearresponsetheoryisquestionnable.

3. Averyrihnonlinearphenomenology anbeobservedin suh devies, withhysteresisphe-

nomena(see[JLPS℄,[PrSj℄)andevensteadily osillatingurrents(see[KKetal℄).

4. Thegeneralwisdomaboutthesesystemssaysthatthenonlineareetsaregovernedbylittle

numberofresonantstates.

This artile is a part of a larger program, namely the understanding of the nonlinear dynam-

is of these out-of-equilibriumquantum systems. One issueis to proverigorously that a simple

Shrödinger-Poissonsysteminafarfromequilibriumregime,thatiswhenthesteadystatesshowa

stronganisotropyinthemomentumvariableatthequantumsale,anleadtomultiplesolutionsto

thenonlinearstationaryproblemwithnontrivialbifurationdiagrams.Arsthekwasprovided

by Jona-Lasinio, Presilla and Sjöstrand in [JLPS℄, [PrSj℄. A seond issue whih goes denitely

further thanthosepreviousworksis theexplanationoftheprodutionofomplexbifurationdi-

agrams in terms of the geometry of the potential, whih requires an aurate analysis of tunnel

eets.

The present work was ahieved on the basis of former works by the seond author and of the

ph-D thesisofthethird author. This analysisleadthethree authorsto theintrodution ofsome

reduedmodelwhih happens tobeveryeientin thenumerialsimulationof realistidevies

(see [BNP℄). Onlythe rstpartof themathematial analysisis provided hereand omplements

will bepresentedinaforthomingartile[BNP2℄.

Landauer-Büttikerapproah[BuLa℄,[Lan℄,[ChVi℄and[BDM℄whihinvolvesthesatteringstates.

Thismodellingallowsastronganisotropyoftheoupationnumberwithrespettothemomentum

anditdenitelydiersfromalltheapproahwherethedensitymatrixlookslikeafuntionofthe

Hamiltonian [BKNR1℄, [BKNR2℄. This latter modelling (and probably the entropy maximizing

approahof[DMR ℄aswell)bettersuitsthesituationoflittlevariationsfromthethermodynamial

equilibrium, endswithorreteddrift-diusionmodelsandannotproduemultiple solutionsdue

to monotoniityproperties. It shouldbenotedthatallthese modelling onsiderthereservoirsas

xedobjetswhihonlyprovidesomekindofinhomogeneousboundaryonditions,inomparison

withthetheoretialanalysisofnonequilibriumsteadystateswidelystudiedwithintheframework

ofthevonNeumannalgebraiapproahofstatistial physisandwhihonernstheevolutionof

thefullsystem,smallsystemplusreservoirs(seeforexample[JaPi℄).

Forourmodel,aompletegeneralfuntionalframeworkwhihathesthepropernonlinearsteady

statesand providesawelldened nonlineardynamis wasprovidedin [Ni3℄, afterusing aphase-

spaeapproahwithsomespeitoolsofthetimedependentapproahinsatteringtheory.

Besidesthebuilding ofaproperfuntionalframework,those modelsbeameevenmoreinter-

esting after the artiles of Jona-Lasinio, Presilla and Sjöstrand [JLPS℄, [PrSj ℄ where onvining

heuristiargumentsandalulationsonthosesimplenonlinearsystemswereprovidedasanexpla-

nation forobserved hysteresisphenomena, in agreementwith point 3). Then the questionarose

whether a ompleteexplanation from an asymptotianalysis on theShrödinger-Poisson system

orwhethernewnonlinearphenomenaouldbepreditedinsomemoreomplexgeometrisetting

like amultiple wells problem. Forinstane, noreal explanation isprovided in [JLPS ℄,[PrSj ℄ for

thepreseneortheabseneofhysteresisphenomenaaordingtothegeometryofthebarrierpo-

tentials. Ourreduedmodel(see[NiPa℄,[BNP ℄andforthomingartile[BNP2℄)providessuhan

explanation,withadditionalresults.

Finallypoint4)providestherelevantasymptoti. Resonantstatesareeetivewhentheimag-

inarypartofresonanesaresmall. Suhabehavioranbeahievedwhenthepotentialbarrierare

high orlargeanditiswellformulatedwithinasemilassialasymptoti(small parameterh→0^,

imaginarypartofresonanesoforderO(e^−c/h)^). Neverthelessafullsemilassialasymptotiwith

O(1) ^{largewells would lead to alarge numberof resonant stateswithin axed energy interval.}

Point4)anbefullled byonsideringquantum wellsin asemilassialisland. Theintrodution

ofthesmallparameterh >0^asaresaledFermi-lengthaswellasafulljustiationofthisasymp- totiregimewithin thepresentationofrealistidevieshasbeendonein [BNP℄.

Fromamathematialpointofview,thisproblempresentstwospeidiulties.

• ^{A non usual multiple wells problem has to be onsidered: it is not exatlya} semilassial problemanditisnonlinear.

• ^Theintrodution ofresonanes requires the implementation ofaomplex deformationand thestudy ofnonself-adjointoperators.

Fortunately, the one-dimensional framework provides some simpliations oraurate estimates

whihallowaomplete analysis. Firstauniformontrolonthenonlinearpotentialwiththehelp

of some monotony priniples an be obtained in W^{1,∞. Hene the nonlinear potential an be}

replaed byanh^{-dependentpotential,withuniform boundsin} W^{1,∞. Some standardarguments}

of thesemilassialanalysisforresonanes (see[HeSj1℄), formultiple wells(see [HeSj2℄,[HeSj3 ℄),

or for theBreit-Wigner formula (see [GeMa℄) haveto be adapted. Againthe weak regularityis

partlyompensatedbythefatthatweworkona1Dproblem. Thisartileisalmostself-ontained

in thesense that theproofswhih areexatlythesameasin theusualsemilassialsetting were

havetobehekedinordertoensurethatthesetehniquesanbeadaptedwiththequantumwells

and the limitedregularityof thenonlinear semilassial potential. The 1D Shrödinger-Poisson

systemstudiedhereadmitsnaturalaprioriregularityestimates,uniformwithrespettothesmall

parameter h→ 0^{. This leads}asymptotially to aperfet splitting of the quantum and lassial sales.

1.2 Quantum framework

In the whole study, the framework is the following: h > 0 ^{denotes the} semilassial parameter obtainedinrealistiasesasaresaledFermilength(see[BNP℄)andI:= [a, b]^{isagivenompat}

intervaloftherealline. LetP_B^{h theShrödingeroperatorontherealline:}

P_B^h :=−h² d²

dx² +B, B ≡ BI+B∞, ^(1.1)

where

BI(x) :=−Bx−a

b−a1[a,b](x), B∞(x) :=−B·1[b,+∞)(x), ^(1.2)

andB^{isanonnegativeonstant. Thepotential}B^{simplydesribestheappliedbias. Thereferene}

Hamiltonianistheself-adjointrealizationintheHilbert spaeL²(R)^ofP_B^h:

D(H_B^h) =H²(R), ∀u∈D(H_B^h), H_B^hu:=P_B^hu. ^(1.3)

Sine several self-adjoint (or non self-adjoint) losure of the same dierential operator will be

onsidered,thenotationP ^{referstothedierentialoperatorsatingon}C₀^∞,while H ^{will beused}

foritsrealizationasanunbounded operatoronL^2.

Werestritouranalysis inthisworktooperatorsin theform

P^h[V] :=PB^h+V, V ∈L^∞(I), ^(1.4)

anddenote byH^h[V]^theself-adjointrealizationinL²(R)^of P^h[V]^:

D(H^h[V]) =H²(R), ∀u∈D(H^h[V]), H^h[V]u:=P^h[V]u, ^(1.5)

after identifyingV ∈L^∞(I)^withV(x)1I(x)∈L^∞(R)^.

Ofpartiularinterestistheasewhere thepotentialV =V^{hdependsonthesmallparameter} h^{anddesribesquantum wellsinanislandwithlis. Itsplitsinto}

V^h:=V0+V_{N L}^h , V0:= ˜V0−W^h, V˜0, V_{N L}^h ∈W^1,∞(I). ^(1.6)

The funtion V˜0, ^{whih models the island potential, an be any non negativeLipshitz funtion}

independentofh^{. Pratiallyitissimplyaonstantpotentialon}I^,V˜0(x) =V01I(x)^withV0∈R₊. Thefuntion W^{h,whih desribedthequantumwells,isdenedby}

W^h(x) :=

N

X

i=1

wi

x−ci

h

. ^(1.7)

InthisdenitionofW^{h,thepositions}(ci)^N_i=1 ^areN ^{givenpointsin}(a, b)^andwi ^{arenonnegative}

L^{∞-funtions supportedin the interval}[−κ, κ]^{, with} κ > 0 ^{xed. Wedenote by}U^{h thesupport}

of thefuntion W^{h and} U :=∪^N_i=1{ci} ^{theregionwhere thequantum wells onentrate,and set} c0:=a, cN+1:=b^{(seeFigure1).}

V0

V

x b −B

c1 c2

a Λ0

Λ^∗

Λ∗

Figure1: TotalpotentialB+V^h−W^h.

Assumption 1 Supposethat

Λ0:= inf

x∈I

V˜0(x) +B(x)>0, ^(1.8)

andx the parametersΛ∗ ^andΛ^{∗ so that}0<Λ∗<Λ^∗<Λ0^.

Wewillfousontheenergyrangeλ∈[Λ∗,Λ^∗]^.

FinallythefuntionV_{N L}^{h desribesthemeaneldnonlinearpotentialwhihtakesintoaount}

the repulsive eletrostati interation. It will be givenas asolution to the Poisson equation on

I= [a, b]^{andwill satisfy}

∀h >0, V_{N L}^h ∈W^1,∞(I), V_{N L}^h ≥0. ^(1.9)

Suh Hamiltonians are used in the modelling of quantum eletroni transport in mesosopi

strutureslikeresonanttunellingdiodes(RTD)orsuper-latties. Thenonlinearsteadystatesan

bestudiedwithinaLandauer-Büttikerapproah: see[BuLa℄,[Lan℄,[ChVi ℄and[BDM℄or[Ni3℄for

possible funtional frameworks onerned with the extension to the nonlinearanalysis inluding

the nonlinear dynamis. This approah involves the sattering wave funtions and requires the

analysis of the ontinuous spetrum of H^h[V]^{. Sine for any potential} V ∈ L^∞(I), H^h[V] ^{is a}

ompatlysupportedL^∞-perturbation oftherefereneHamiltonianH_B^{h orthe}Hamiltonianwith steppotential−h²∆ +B∞^{,thelimitingabsorptionprinipleholds.Bystandardarguments([Ya2℄,}

[Pat℄) oneevengetstheabseneofimbeddedeigenvalues

∀h >0, σ^ess(H^h[V]) =σ^a(H^h[V]) = [−B;∞), ^(1.10)

andthesatteringstatesofH^h[V] ^{areindeed welldenedforany}V ∈L^∞(I)^.

Remark 1 Underthe non neessaryadditional assumption

∀i∈ {1, . . . , N}, V˜0(ci) + infσ(−∆−wi)>0, ^(1.11)

oneanevenheklike inTheorem3.4orTheorem3.6thatthereisnoeigenvalueatallforh >0

smallenough (andV_{N L}^h ≥0^);

σ(H^h[V]) =σac(H^h[V]) = [−B,+∞).

Wefousontheenergiesλ∈[Λ∗,Λ^∗]^.

Weonsider theinoming sattering statesψ^h−(k,·)^{of the}Hamiltonian H^h[V]parameterizedby the wavevetork ^{(we omitto write thedependene with respet to the potential forsattering}

states). TheyprovideadiagonalizationofH^h[V]^{overtheontinuousspetrum(seeformula(1.19)).}

Preisely,introduerstthedispersionrelationassoiatedwiththerefereneHamiltonianH_B^h

Denition 1.1 Set for k∈R^∗

λk :=

k^{2 if}k >0,

k²−B ^ifk <0. ^(1.12)

This dispersionrelation (1.12) gives,for thewavevetork^{, the energy}λk ^{ofthe inomingplane}

waverepresentedbyψ^h₋(k,·)^{. Again,wearemostlyinterestedin the}k^'ssuhthat λk ∈[Λ∗,Λ^∗]^.

By denition, the inoming generalized eigenfuntion ψ^h₋(k,·) ^{dened for} k ∈ R solves the dierentialequation:

P^hψ₋^h(k,·) =λkψ^h₋(k,·), ^(1.13)

withthenormalization(ofinomingplanewaves)

fork >0 ψ−(k, x) =







e^ikxh +rke^{−ikxh for} x < a tke^i(λk+B)1

/2x h

for x > b ,

(1.14)

fork <0 ψ−(k, x) =







tke^−i(λk)1

/2x h

for x < a e^ikxh +rke^{−ikxh for} x > b .

(1.15)

Thesquarerootz^{1/2ishosenwiththeramiationalongthehalf-line}iR₋inordertoensurethat

e^{−i(λk)1/2x deays}exponentiallyasx→ −∞^whenλk∈(−B,0)^.

These oeientsdeterminethe satteringmatrix(rk, tk)^{forpositiveenergies}λk >0^{. Theyare}

linkedforλk>0 ^{bytherelation}

|rk|²+ r λk

λk+B|tk|²= 1, λk >0. ^(1.16)

Sinethewavevetork^isalog-derivative,thisnormalizationofthewavefuntionsanbewritten in termsof boundaryonditionsat x=a^and x=b^{, in this spei} one-dimensionalasetting withrealistiproblems:

hh∂x+iλ^1/2_k i

|x=au= 2ike^ikah, hh∂x−i(λk+B)^1/2i

|x=bu= 0, ^fork >0 ^(1.17)

and

hh∂x+iλ^1/2_k i

|x=au= 0, hh∂x−i(λk+B)^1/2i

|x=bu= 2ike^ikbh, ^fork <0. ^(1.18)

ThustheproblemoverthereallineisreduedtoaboundaryproblemonI^{withboundaryonditions}

dependingonthespetralparameter(1.17)-(1.18). Theseboundaryonditionsareexattranspar-

entboundaryonditions. Thissettingmakesrathereasytheomplexdeformationargumentused

in theanalysisofresonanes(see[BaCo℄,[HeSj1℄ or[HiSi℄foramoregeneralintrodution). Here

onsideringaomplexλk ^{aroundanypositivevalueiseasily}implementedbeausetheoeients ontheboundaryonditionsatx=a^andx=b^dependholomorphiallyonλk ^(ork^).

Weendthissetionwiththree elementaryproperties:

1. With this normalization, it appears that for any non negative ontinuous funtion θ ^on [Λ∗,Λ^∗]^,theoperator1Iθ(H^h[V])1I ^{isanintegraloperator. Moreoverthekernelisgivenby}

1Iθ(H^h[V])1I[x, y] = Z

k

θ(λk)ψ^h₋(k, x)ψ₋^h(k, y) dk

2πh, (x, y)∈I×I. ^(1.19)

2. Notethatbeauseoftheregularityofψ₋^h,^{itfollowsbyMerer'stheorem(see[Si,Thm3.5℄)}

thatthisoperator istrae-lass,withatraeequaltothediagonalintegral.

3. Notealso thatbeausethesolutionsto theODE(1.13)in theintervalI ^{is a}2-dimensional linearsubspae, say Sλk ⊂H²(a, b)^{, onditions}(1.17)-(1.18)form anane systemin Sλk^.

Resonanesaroundpositiveenergiesorrespondtotheexeptionalomplexvaluesofλk =z

forwhihtheontinuouslinearfuntionalsdening thissystemareproportional.

1.3 Shrödinger-Poisson system

Here weareinterestedin thestudyofthestationaryase. Werstxtheproleoftheinoming

beamofeletronsoverthestruturebetweena^and b^.

Notation 1 Fixaontinuousnonnegativefuntionk7→g(k)^suhthatg(k) = 0^ifλk∈/(Λ∗,Λ^∗),

see(1.12).

A beamof eletrons orresponds to asuperposition of sattering stateswith density g^{. The}

eletroni densityisthendesribedbythemeasuredng[V]^{dened by} dng[V](x) :=

Z

R

g(k)|ψ−^h(k, x)|² dk

2πh. ^(1.20)

It isonvenientto introdue the funtion g(K₋^h) ^{of theasymptoti momentum operator dened}

(see[DeGe ℄,[Ni3℄foramoregeneralpresentation)aordingto:

g(K₋^h)[x, y] = Z

R

g(k)ψ^h₋(k, x)ψ^h₋(k, y) dk 2πh.

Itsloalizedversion1Ig(K₋^h)1I ^{hastheintegralkernel}

1Ig(K₋^h)1I[x, y] = Z

R

g(k)1I(x)ψ^h₋(k, x)ψ^h₋(k, y)1I(y) dk

2πh. ^(1.21)

Theoperatorg(K₋^h)^{isadensitymatrixandthedensityfullls theweak}formulation

∀ϕ∈ C⁰(I), Z

I

ϕ(x)dng[V](x) =^Tr[1Ig(K₋^h)1Iϕ]. ^(1.22)

Notethatinthepartiularasewhereg(k)^{isafuntion oftheenergy,i.e.} g(k)≡θ(λk)^,g(K₋^h)^is

afuntionoftheHamiltonian

g(K₋^h) =θ(H^h). ^(1.23)

Funtions of the Hamiltonian an be viewed as equilibrium states (and even thermodynamial

equilibrium stateswhen θ ^isdereasing). Forsuh states,theurrentthroughthedevie is null.

Hene out-of-equilibrium steady states with a non vanishing urrent have to be desribed with

a funtion g(k) ^{whih is nota funtion of the energy. Inorder to makethis situation lear, we}

assume the next possibly extendible assumption (see [BNP ℄ for an easy generalization towards

morerealistiproblems).