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1、基于格林函数的纳米器件模型分析—.TheGreen’sfunctioni.定义AGreen，sfunction,s),ofalineardifferentialoperatorL-Z(x)actingondistributionsoverasubsetoftheEuclideanspaceR；atapoints、isanysolutionofLG(xs=5(x—s}⑴where(5istheDiracdeltafunction.ThispropertyofaGreen’sfunctioncanbeexploitedtosolvedifferenti
2、alequationsoftheform=/(x)(2)Asasidenote,theGreen’sfunctionasusedinphysicsisusuallydefinedwiththeoppositesign;thatis，LG(x,s)=-5(x-s).(3)Iftheoperatoristranslationinvariant，thatiswhenLhasconstantcoefficientswithrespecttox、thentheGreen’sfunctioncanbetakentobeaconvolutionoperator，
3、thatis，GrlX,=G(x—s).(4)Inthiscase,theGreen’sfunctionisthesameastheimpulseresponseoflineartime-invariantsystemtheory.2.推算Looselyspeaking，ifsuchafunctionGcanbefoundfortheoperator人，thenifwemultiplytheequation(1)fortheGreen’sfunctionbyf{s),andthenperformanintegrationinthesvariable
4、,weobtain;/邮，s)/(s)ds=Js{x-S)/(5)(fe=f(x)(5)Therighthandsideisnowgivenbytheequation(2)tobeequaltoLu(x),thus:Lit⑻=fLG(j,3f(6)BecausetheoperatorL-人(义)is1inearandactsonthevariablexalone(notonthevariableofintegrations),wecantaketheoperatorLoutsideoftheintegrationontherighthandside
5、,obtaining;=LG{xss)f(s)dsj(7)Andthissuggests;=JG(x3s)/(s)d#3.格林函数在纳米器件屮作用[pk]=fodf^L+(^k-M),])[p]=y^Pk=Fq([Hl-"/])(8)-U0C[Pk}=IdEf0(E+sk-m)•oc-{*3Cp}=/dEF^E—A(9)对比方程⑴，可知：=+带入方程(9),可知：1广+oc[pk]=2n-,/—DCdEfo(E4—■M)M(£)]1r+oc[p]=2tt,/—DCdEFq(E■-M)[A(E)]其中，[A(E)]=z([G(£)]-[G(£)]+
6、)这就是格林函数表示电荷密度和密度矩阵，其中[A(E)]是谱函数(spectralfunction)，谱函数A的物理意义:Indeedthediagonalelementsof[八(E)]/2氺pi卜intherealspacerepresentationgiveusthelocaldensityofstatesatdifferentpointsinspace(aquantitythatcanbemeasuredwithscanningprobemicroscopy)4.自能矩阵(self-energy)(1)物理含义Theconceptofsel
7、f-energyisusedinmany-bodyphysicstodescribeelectron-electronandelectron-phononinteractions.Tnthepresentcontext,however,weareusingthisconcepttodescribesomethingmuchsimpler,namely,theeffectofasemi-infinitecontact.(2)自能矩阵的推导在考虑了电极时，沟道的总哈密顿量为HThrr是电极对沟道的作用矩阵是电极的哈密顿矩阵总的格林函数为GGDRl=［（
8、E+iQ^I-H一r一1_GRDG、」--（£+/.0，/-/^从上面的公式可得：G=（£+/0^）/-H-L£/-//-
