Non-equilibrium Green function method: theory and application in simulation of nanometer electronic devices

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1 Advnes in Ntul Sienes: Nnosiene nd Nnotehnology REVIEW OPEN ACCESS Non-equilibium een funtion method: theoy nd pplition in simultion of nnomete eletoni devies To ite this tile: Vn-Nm Do 4 Adv. Nt. Si: Nnosi. Nnotehnol Relted ontent - Atomisti theoy of tnspot Alessndo Pehi nd Aldo Di Clo - A eview of seleted topis in physis bsed modeling fo tunnel field-effet tnsistos Dvid Esseni Mo Pl Piepolo Plesti et l. - Self-exited uent osilltions in esonnt tunneling diode desibed by model bsed on the Cldei Leggett Hmiltonin Atsunoi Sui nd Yoshit Tnimu View the tile online fo updtes nd enhnements. This ontent ws downloded fom IP ddess on 5//8 t 3:6

2 Vietnm Ademy of Siene nd Tehnology Advnes in Ntul Sienes: Nnosiene nd Nnotehnology Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 (pp doi:.88/43-66/5/3/33 Review Non-equilibium een funtion method: theoy nd pplition in simultion of nnomete eletoni devies Vn-Nm Do Deptment of NnoSiene Advned Institute fo Siene nd Tehnology (AIST Hnoi Univesity of Siene nd Tehnology (HUST Di Co Viet Rod Hnoi Vietnm E-mil: vnnm.do@gmil.om Reeived 6 Jnuy 4 Aepted fo publition 9 My 4 Published June 4 Abstt We eview fundmentl spets of the non-equilibium een funtion method in the simultion of nnomete eletoni devies. The method is implemented into ou eently developed ompute pge OPEDEVS to investigte tnspot popeties of eletons in nno-sle devies nd low-dimensionl mteils. Conetely we pesent the definition of the fou eltime een funtions the etded dvned lesse nd gete funtions. Bsi eltions mong these funtions nd thei equtions of motion e lso pesented in detil s the bsis fo the pefomne of nlytil nd numeil lultions. In ptiul we eview in detil two eusive lgoithms whih e implemented in OPEDEVS to solve the een funtions defined in finite-size opened systems nd in the sufe lye of semi-infinite homogeneous ones. Opetion of the pge is then illustted though the simultion of the tnspot hteistis of typil semionduto devie stutue the esonnt tunneling diodes. Keywods: non-equilibium nno-eletonis nno-sle simultion tnspot Clssifition numbes: Intodution Fo five dedes of development the non-equilibium een funtion (NEF fomlism hs beome useful nd poweful lultion/omputtion method to the study of dynmil poesses in non-equilibium mny-body systems. Histoilly the NEF theoy stems fom effots in pplying ides estblished in the quntum field theoy to tet poblems in sttistil physis. Aodingly the expettion vlues of obsevbles nd the oeltion funtions between obsevbles n be systemtilly detemined if nowing een funtions whih e ppopitely defined. Fo instne the oeltion funtion between the two obsevbles nd b mesued in system wheein gound stte Φ o theml-equilibium stte e estblished n be lulted though the two-time een funtion defined by [ 5] i = ˆ ˆ ˆ b ( t t T( t b ( t i Φ ˆ ˆ ˆ T( t b Φ β ( t = βh ˆ ˆ ˆ i { e T( t b( t ( } β βh { e } whee â nd ˆb e the opetos defining the two quntities nd b ˆT denotes the ngement of the opetos â nd ˆb oding to the ode of thei time vibles t nd on the el-time xis { } the te opetion s onvention the edued Pln onstnt nd β = T B the invesion of the theml enegy /4/33+$33. 4 Vietnm Ademy of Siene & Tehnology Oiginl ontent fom this wo my be used unde the tems of the Cetive Commons Attibution 3. liene. Any futhe distibution of this wo must mintin ttibution to the utho(s nd the title of the wo jounl ittion nd DOI.

3 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Fo the pogess of the field one should mention to the ey ontibutions of Mtin nd Shwinge in the fomultion of the genel fomlism of the N-ptile een funtion [6]. Howeve in the ptil point of view in ode to lulte suh N-ptile een funtions one needs to intodue ppopite ppoximtions whih usully led to the ppene of mny novel oeltion funtions. In 96 Kdnoff nd Bym disussed some fundmentl spets in the implementtion of the Mtin-Shwinge theoy in the view point of mosopi onsevtion lws the ontinuity eqution fo instne [7]. The development of the quntum field theoy fo theml equilibium mny-body systems led to the fomultion of the tempetue Mtsub een funtion theoy. This theoy is bsed on the onept of imginy time whih ws intodued bsed on the nlogy between the time evolution opeto nd the equilibium sttistil one [ 4]. Howeve diffeent fom the equilibium sitution non-equilibium systems do not hve neithe gound stte no unique sttistil opeto. Tetment of suh systems theefoe equies the intodution of novel een funtions to quntify typil sttistil popeties suh s the distibution of ptiles on the spetum of miosopi sttes. In 965 Keldysh intodued wy to unify lss of fou bsi two time-vibles een funtions using the onept of timeontou [8] defined on whih is the time-ontou een funtion o the Keldysh een funtion. Invesely by ming the ontinuum limit whih is expessed though the so-lled Lngeth theoem [9] the time-ontou een funtion leds to set of fou el-time two-point een funtions whih simultneously desibe the two fundmentl spets of mny-body systems: the spetum of quntum sttes of ptiles nd the stte popultions o the vege numbe of ptiles oupying eh stte. These el-time two-point een funtions e thus ble to desibe genelly mnybody systems without needing the distintion of thei sttistil stte i.e. equilibium o non-equilibium. Sine seies of onfeenes hd been held to epot pogesses nd to seth pespetives of the NEF theoy. Colletions of ppes published in seies of poeedings nmed Pogess in Non-equilibium een funtions epot inteesting ey esults nd bethoughs in pplying this method to vious fields suh s plsm physis solid stte theoy semionduto optis nd tnspot nnostutues nule mtte nd even high enegy physis []. In the lst dede the NEF method hs been developed to investigte nd pedit tnspot popeties of nnosle mteils s well s the opetion nd the pefomne of nnosle devies. The NEF method hs been implemented into mny ompute pges with diffeent sophistited levels fo instne ombined with density-funtionl lultions [ 3] o bsed on effetive physil models [4 7]. Howeve mny tehnil nd physil issues still need to be solved to impove the effetiveness nd the pefomne of suh omputtionl tools in vious pplitions. Ou ompute pge nmed OPEDEVS stnding fo Opto nd Eletoni Devie Simultion [8] hs been eently developed on the bsis of the NEF method to the study of the eletoni nd tnspot popeties of nnosle mteils nd devies with diffeent geometil stutues. The pupose of this ppe is theefoe to pesent shot eview of fundmentl spets in the implementtion of the NEF method into the OPEDEVS pge. Nevetheless ll bsi onepts nd ey equtions of the theoy e pesented in detil in moe suitble fom fo the omputtionl puposes the thn fo full desiption of the theoy. The lyout of the ppe is pesented s follows. In setion we eview pogesses in the lst dedes of the NEF method in nno-eletonis. In setion 3 we pesent the definition of the fou el-time een funtions viz the etded dvned lesse nd gete ones nd thei equtions of motion nd bsi eltions mong them. In ptiul we disuss the issue of how to lulte impotnt physil quntities fom the een funtions in the quntifition of the tnspot popeties of nno-eletoni systems. In setion 4 two effiient eusive lgoithms e pesented whih e implemented in the OPEDEVS pge to solve the een funtions defined in finite-size opened systems nd in the sufe lye of semi infinite-size homogeneous systems. Setion 5 is dedited to the illusttion of the pplition of OPEDEVS to study typil semionduto devie stutue the esonnt tunneling diode (RTD. Setion 6 onludes the ppe.. Pogess of NEFs method in nno-eletonis Sine the invention in 96 the tehnology bsed on the omplementy metl-oxide-semionduto (CMOS iuit nmed the CMOS tehnology hs been pidly nd stongly developed in the tend of minimizing the size of the building blos suh s the p- nd n-type metl-oxide-semionduto field-effet tnsistos (MOSFETs. This tend howeve fo the lst dedes eveled sevee obstles genelly nown s the shot-hnnel effets when the devie gte length ppohes the nnomete sles. Though the 3 nm tehnology (implying tht the typil length of the gte eletode of the MOSFET stutue is bout 3 nm hd been stted sine 7 the Intentionl Tehnology Rodmp fo Semiondutos [9] pedits tht this tend will be soon stopped beuse of the non-pesevtion of the tnsfe hteistis s those of onventionl devie stutues. Fundmentlly suh poblems oiginted fom the emegene of quntum effets tht limit the devie sling tend. Theefoe it hs been uil to study quntum behvios nd tnspot mehnisms of hge ies whih goven the hteistis of nno-sle devies. On the one hnd tht eseh helps lifying the lowe limit of the sling poess. On nothe hnd it my shed light on the finding of ltentive fbition methods nd even on the invention of novel devie genetions. In suh ontext it is neessy to develop effiient theoetil ppohes nd poweful omputtionl tools to study igoously the ole of quntum effets involving in the tnspot of hge ies in nnosle devie stutues nd hene thei dominnt influene on the devie pefomne.

4 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Figue. Shem of typil two teminl devie stutues ( nd of the meshing long the tnspot dietion (b. Figue ( illusttes typil devie stutue of two teminls. The entl egion is lled the tive one tht govens the behvios of the whole devie while the two end egions lled the leds ply the ole of the soue nd the sin of hge ies. The tive egion is thus onsideed s n opened system whih ommunites with its envionment vi the leds. At the sles of onventionl devie stutues with the gte length in the ode of μmvious lssil nd semi-lssil ppohes fo instne the dift-diffusion model nd/o the ones bsed on the semi-lssil Boltzmnn eqution hve been developed []. Suh ppohes howeve beome invlid in ptuing the quntum effets suh s the quntum intefeene onfinement nd tunneling whih beome emeging when the size of the devies is ompble to o even smlle thn the phse being length Lϕ of hge ies. When the wve funtion of ies n extend lgely ove the entl egion to the leds ies n move oheently though the whole devie. The tnspot in this sense is sid to be in the oheent egime o in the bllisti tnspot egime. The uent of hge ies flowing though the stutue theefoe n be detemined by the simple nd intuitive fomul intodued by Lndue in 947 whih ws then vlidted nd genelized fo multipobe systems by Buttie using the fmewo of the stteing theoy [ 4] whee e e de I = π ( E nl( E nr( E ( is the bsolute vlue of the eleton hge nd ( E is lled the tnsmission oeffiient whih is tully the summtion of tnsmission pobbilities (the tnsmittnes of hge ies ove ll possible single-ptile modes. The tnsmission oeffiient is theefoe physil quntity quntifying the tnspeny of the opened system whih is viewed s the gent stteing the ie plne wves inident fom the teminl egions. In the bove eqution the ole of the ie soue (the left-l nd sin (the ight-r egions is expliitly efleted though the pe- n E n E whih define the sene of the two funtions L nd R vege ouption numbe of ies on sttes with enegy E. Sine these egions e usully muh lge thn the devie tive one they e usully ssumed to be in theml equilibium stte nd thus hteized by tempetue T ν nd hemil potentil μ ν ν = L R. In ptie the funtions nlr( E e usully ppoximted by the Femi - Di funtion with some speil noties involving in the speifi vlue of the tempetue nd the hemil potentil [5]. The lultion of the uent of hge ies by eqution ( theefoe evolves ound the detemintion of the tnsmission oeffiient. Diffeent methods my be used to this im inluding the use of the stteing method with the help of the tnsfe mtix tehnique [6] o the use of the el-spe een funtion method ssoited with the Fishe Lee eltion [7]. Though eqution ( is useful it is bsed on the singleptile pitue. In ptie even when the bllisti nd oheent tnspot egime dominte in the nnosle stutues the poblem of ies stteing off impuities lttie vibtions nd the sufes/intefes in/of mteil lyes is in genel inevitble. These stteing poesses my even goven typil behvios of devies fo instne the effet of phonon-ssistne tnspot nd the supe-poissonin noise in double-bie esonnt tunneling stutues [8]. Suh effets ppently nnot be miosopilly ounted fo in the fmewo of the Lndue Buttie fomlism (the stteing theoy. A genel theoy tht enbles ntully onneting the two tnspot egimes i.e. the bllisti nd the diffusion ones is theefoe needed. To the simultion of eletoni devie stutues the tnspot equtions of hge ies in piniple must be pesented nd solved in the el time-spe. In 99 Coli pointed out tht some physil quntities quntifying the tnspot hteistis of n opened system n be expessed in tems of the een funtions whih e defined only in the tive egion [9]. In 99 Mei nd Wingeen bsed on the NEF fomlism to fomulte n expession fo the hge uent flowing though finite stteing egion nd eoveed the well-nown Lndue fomul in the limit of no-stteing i.e. in the oheent tnspot egime [3]. In the next setions we will eview these ey esults sine they ply the bsi ole in the development of the NEF fomlism fo the devie simultion. In 995 Dtt intodued boo wheein mny ptil spets of implementing the NEF method e disussed in the pedgogil wy using the mtix epesenttion of the een funtions of the two teminl stutues [3]. Ptilly the most diffiult pt in the use of the NEF method fo investigting tnspot popeties of system lies in the tetment of mny-ptile intetion effets. Digm tehniques developed fo the mny-body een funtions in genel n be used in the NEF fomlism. It howeve is 3

5 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 fomidble ts [4]. Fo the ses in whih intetion poesses ply the ole of ming the tnsition of ptile fom one stte to nothe but do not use signifint hnges in the dispesion eltion the NEF fomlism n be used in the quntum-vesion fom of the Boltzmnn eqution fo the so-lled Wigne funtion [3]. Mthemtilly the Wigne funtion plys the ole simil to the Boltzmnn distibution funtion in the detemintion of the ineti popeties of quntum system but it is not positively defined. In ddition sine the tnsition poesses n be teted using the fmewo of the Femi golden ule mny useful methods nd tehniques ledy developed in the lssil sttistis suh s the Monte Clo method n be used to solve the quntum Boltzmnn eqution [33 34]. Regding the solution of the NEFs it is woth noting thei eusive stutue in tems of the so-lled Dyson eqution. The ft is tht the eliztion of mthemtil stutue in system of equtions will stongly filittes omputtionl poesses. This point will be detiled in setion 4 but infomtively it should mention some impotnt ontibutions in the field of numeil lultions. In 997 Le et l. suessfully used the eusive method to lulte ll digonl elements of the etded een funtion mtix defined in the devie tive egion [4]. This tehnique enbles ombining diffeent pts of whole devie using the onept of oupling self-enegy. In Svizheno et l. stted fom the mtix stutue of the tnspot equtions to intodue the genel lgoithm to ompute not only impotnt elements of the etded/dvned een funtions but lso of the othe oeltion funtions i.e. the lesse nd gete een funtions (see next setion [5]. This lgoithm is vey useful nd now ommonly used to find in systemti wy ll of the een funtions defined in the devie tive egion given the self-enegies hteizing the teminl (soue/sin egions. Fo some simple tnspot models suh oupling self-enegies n be nlytilly found [6 3]. Howeve when deling with omplited models suh s multi-bnds nd/o tomi tight-binding desiption it is not esy to hndle the nlytil lultions. The lgoithm poposed by Snho nd Rubio in 984 to tet semiinfinite systems ws usully evisited. It is now nown s the genel stble nd effiient method to lulte the oupling self-enegies [35]. All these tehniques e the ey ingedients fo the implementtion of the NEF method to the poblem of quntum devie simultion. In the next setions we will eview these points whih hve been suessfully implemented in the OPEDEVS pge. 3. Fomlism of NEFs 3.. Definitions iven n eleton system whih is desibed by Hmiltonin Ĥ epesented in tems of the field opetos ˆα ˆβ whee α nd β denote fo shot the sets of quntum indies lbeling quntum sttes of the system. The time-ontou Keldysh Figue. Shem of the time-ontou C. It n be seen s ompised by: ( two bnhes nd fo the se of zeo tempetue o (b thee bnhes nd 3 fo the se of finite tempetues. The ontinuum limittion implies tht depending the time vibles τ nd τ on whih bnhes the el-time een funtions e defined. een funtion is defined by (see ppendix A: i ( τ; τ = T ˆ ˆ ( τ ˆ ( τ (3 αβ α β wheein denotes the time-ontou suh tht τ τ (see figues ( nd (b ˆT is the time-odeing opeto on the time-ontou nd the ngle bet denotes the ensemble vege s speified in eqution (. The time-ontou n be seen s omposed of thee bnhes = + iη + + iη = + iη t iη nd ( t i t i 3 = η β whee η> nd β = T B. The omplex time vibles τ τ in eqution (3 theefoe n be speified s τ = t + iη if τ C τ = t iη if τ C nd τ = t it if τ C 3. The definition of the NEF given by eqution (3 tully is the genel one fo the fou el-time een funtions: the lesse ( gete ( > usl (time-odeed ( nd nti-usl (nti time-odeed ( een funtions when ting the limit η [ 4]: αβ τ τ = i = ˆ αβ ( t t Tˆα ( t ˆ β τ τ ( t & i αβ ( t = + ˆ β ˆα τ τ ( t ( t & (4 > i αβ = ˆα ˆ β τ τ ( t t ( t ( t & i = ˆ αβ ˆ α ˆ ( t t T ( t β τ τ ( t & By the definition the usl nd nti-usl een funtions t nd αβ t e elted to the funtions αβ 4

6 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 αβ t nd αβ t by the expessions: > > αβ ( t = θ( t αβ ( t + θ( t αβ ( t > αβ ( t = θ( t αβ ( t + θ( t αβ ( t (5 (6 whee θ is the step funtion s onvention. On the bnh 3 we obtin the Mtsub een funtion [ 3]. Ptilly one usully does not wo with the usl nd nti-usl een funtions (due to thei nlytil popeties [ 3]. The two othe el-time een funtions the etded αβ nd the dvned αβ ones e used insted: i t = ˆ ( t ˆ θ t { } αβ α β αβ αβ > = θ t t t (7 i ( t =+ ˆ ( t ˆ θ ( t ( t t { } αβ α β αβ αβ > = θ t t t (8 whee the uly bet denotes the nti-ommutto. Fom equtions (5 (8 we lely elize the genel eltionships mong the fou el-time funtions: > + = + (9 αβ αβ αβ αβ > = ( αβ αβ αβ αβ > = + = + ( αβ αβ αβ αβ αβ > = = ( αβ αβ αβ αβ αβ In most ses we e usully inteested in the stedy stte of system. The two-time oeltion/een funtions beome dependent on only the diffeene between the two time vibles t = t t. Theefoe it is moe onvenient to ν wo with thei Fouie tnsfoms { αβ ( E } fo instne E i ( t t ν = ν E d( t t e ( t t. (3 αβ αβ Invesely we hve ν de E ν αβ( t t = αβ π ( E e i ( t t (4 In the bove equtions nd so lte the supesipt ν is used to denote the ones { } (exept some ses whih will be noted nd E is the enegy vible. Ptilly it is useful to em the dimension of the defined een funtions in equtions (4 nd (3. Sine the field opetos e usully hosen to be dimensionless the el-time een funtions ν αβ ( t t nd thei Fouie tnsfoms ν αβ ( E in the Intentionl System of units theefoe hve the dimension of (J s nd J espetively. Fo eleton systems it is woth to mention to the spetl density mtix A αβ ( E whih is defined s follows = αβ αβ αβ A E i E E (5 > = i E E (6 αβ αβ As genelly pesented in [ 3] the digonl elements i.e. when α = β of this quntity e positively definite nd povide infomtion of the enegy spetum of elementy exittions in the systems. The spetl density mtix theefoe hs the dimension of the enegy invesion. 3.. Equtions of motion In ppendix B we pesent the Dyson equtions fo the timeontou een funtion ( τ τ (In the following the bold htes e used to denote the mtix nd veto nottion i.e. to omit quntum indies. Mthemtilly it is n integl eqution obtined fom the petubtion expnsion of eqution (A.. In this setion we will pply the Lngeth theoem (poved in ppendix B to the Dyson equtions (B.5 nd (B.6 to speify the equtions of motion in the genel fom fo the fou el-time een funtions i.e. the etded dvned lesse nd gete. Indeed ming use of the ules given in tble B the ontinuum limit of eqution (B.5 beomes [3]: ˆ ( t t = δ t + d t Σ ( t t t (7 ˆ ( t t = d t Σ ( t t ( t + Σ ( t t ( t (8 Similly the ontinuum limit of eqution (B.6 yields ( t ( = δ t + d t ( t t Σ t (9 ( t = d t ( t t Σ ( t + ( t t Σ ( t ( Beuse of the time-onvolution integls in the bove equtions it is useful to wite down the equtions fo the Fouie tnsfoms of the een funtions. Stightfowdly it esults in nd E E = I + Σ ( E ( E ( E E = Σ E E + Σ E E ( ( E ( E = I + ( E Σ ( E (3 ( E ( E = ( E Σ ( E + ( E Σ ( E. (4 whee E = E = E + iη I H. Clely the obtined equtions fo the etded/dvned lesse nd gete een funtions e not independent. In ft they fom set of dependent intego-diffeentil equtions in the el-time epesenttion o set of mtix equtions in the enegy epesenttion. In setion 4 we will pesent some useful lgoithms to solve numeilly these equtions. To end this setion we would 5

7 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 lie to emphsize tht ll the bove equtions fo the fou eltime een funtions e ext nd genel in the mening of the petubtion expnsion i.e. when one n define the self-enegies. iven n expession fo the full Hmiltonin Hˆ = Hˆ + H ˆ the hedhe poblem lwys is how to speify the self-enegies. Tehnilly the self-enegies n be onstuted using the Feynmn digm tehnique [ 3] o the method of eqution of motion. These two e the most genel ommon nd poweful tehniques. Fo the ltte edes ould onsult [36] snexmple Some bsi popeties nd eltions In this setion some genel nlytil popeties of the een funtions e summized. Fom the definition of the el-time een funtions it is stightfowd to he tht ( τ τ = ( τ τ (5 ( τ τ = ( τ τ. (6 Aodingly we n lso esily veify the following expessions: ( E = ( E (7 = E E (8 A E = A E. (9 Whee the dgge symbol denotes the Hemitin onjugtion of the involved opetos/mties. Fom these eltions we see tht though thee e fou defined een funtions only thee of them e independent. Ptilly the etded lesse nd gete funtions e usully hosen to me set of independent funtions. Eqution (9 lely expesses the Hemitin popety of the spetl density mtix [3]. Sine the non-inteting ppoximtion is usully used s the stting point fo the tetment of inteting systems it is thus useful to pesent the expessions fo some typil een funtions. The field opetos fo non-inteting eleton system n be witten in the bsis of plne wves s follows: E i i t ˆ α( t = ˆα( e e (3 V E i i ˆ β( = ˆ β( e e (3 V whee E( expesses the dispesion lw of eleton nd V the volume of the system. Substitute these expessions into the definition of the lesse een funtion we obtin αβ ( t; E E i i i i i t = e e e e ˆ β ˆ V α( (3 Sine ˆ ˆ = δ δ β α αβ n E whee n E ( is the vege numbe of ptiles oupying the E then stte with enegy αβ ( t; E iδαβ i = e ( i ( t t e n E ( (33 V Clely we see tht αβ ( t; = αβ ( t. The Fouie tnsfom of this funtion is thus esily detemined to be: E = πin E δ E E δ (34 αβ In the sme wy the expessions fo the gete etded nd dvned een funtions e deived s follows E = πi n E δ E E δ (35 > αβ αβ δαβ E = (36 E E ± iη Next fom eqution (7 we n show tht the etded/ e nlyti in the uppe/ dvned een funtions ( E lowe one-hlf omplex plne espetively (see eqution (36 s n exmple. This popety llows expessing / (E in the fom: ( E de A E = (37 π E E i η whee η is positive infinitesiml numbe. Eqution (37 is well nown s the Kme Konig eltion of nlyti funtions [ 3]. In genel the self-enegies e omplex funtionls of the een funtions. Howeve in some ses fo instne the oupling of n open system with its leds (see eqution (57 nd/o the phonon eleton intetion in the self-onsistent Bon ppoximtion [4] the self-enegies simply te the line fom s follows Σ ν de ν ( E = D( E ( E E π αβ αβ (38 whee D E ould be the devie-led o eleton phonon oupling ftos. In these ses we n wite down n eqution simil to eqution (37 fo the etded/dvned self-enegies: Σ ( E = de Γ E (39 π E E iη whee Γ ( E is defined s the imginy pt of the etded 6

8 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 self-enegy > Γ E = i Σ E Σ E = i Σ E Σ E. (4 Simil to the spetl density mtix A( E the digonl elements of Γ ( E lso hve physil mening. Speifilly they e detemined s the one-hlf widths of spetl pes [3]. In opened systems we distinguish two inds of soues whih indue the hnges in the spetl nd ineti/tnsition pitues of hge ies: one involves in the onnetion of the opened system to its envionment (the devie led oupling nd the othe the intinsi intetion/stteing poesses of hge ies inside the system. The self-enegies Σ ν ( E in equtions (7 (4 theefoe e the summtion of the two ontibutions Σ ν ( E fo the fome soue nd Σ ν ( E ν ν ν s fo the ltte one i.e. Σ E = Σ E + Σs E. It n be shown (see setion 3.5 tht when ssuming the leds s semi-infinite homogeneous systems the lesse nd gete self-enegies Σ expliitly te the following fom [3 3]: Σ Σ E μ E = iγ E f (4 T B E μ E = iγ E f (4 T > whee in f ( x = + e x is the Femi funtion detemining the popultion of hge ies in sttes with enegy E in the led egions nd Γ ( E is defined by equtions (4 nd ( Eleton nd hole densities The im of this setion nd of the next one is to me lin between the een funtions defined in the pevious setions nd some physil quntities whih ply n impotnt ole in the tnspot poblem. Suh physil quntities inlude the distibution densities of hge ies nd the hge uent density. We will show tht nowing the een funtions ν { ( E } is suffiient to lulte these physil quntities. In ode to fomulte in detil ll the following expessions in ou tnspot poblems we need to speify the quntum indies α nd β lbeling the opetos nd een funtions. In ou onvention these quntum indies efe to the sets of the thee ones { q λ } in whih the fist one denoted by Ltin htes suh s p o q is used to lbel the unit ells o the meshing nodes long the tnspot dietion see figue (b. The seond index λ lbels the ondution eletoni bnds o the tomi vlene obitls ontibuting to the tnspot in unit ell. The lst one is n option whih my be used depending on the speifi geometil stutue of the systems unde onsidetion to hteize the tnsvese motion of ptiles. When the oss-setion of devie B stutue is suffiiently lge so tht the effet of its boundies is not impotnt seves s wve-numbe veto. The illusttion of using these thee indies is pesented in the sethed devie stutue in figue (b. Denote η s the fto ounting ny degeney of some quntum sttes (spin nd/o vlleys of enegy bnds fo instne the eleton density distibuted in unit ell o in mesh-sping is detemined s the expettion vlue of the eleton numbe opeto ˆ Nq λ = ˆq λ ˆq λ η e s n = ˆ q Nq λ x S λ ηs = ˆq λ ˆq λ (43 S x λ whee in S is the oss-setion of the system nd x is the width of the unit ell o the mesh-sping. Using the definition of the lesse een funtion given in setion 3. we n expess n e q s follows: η = e s nq lim q λ; q λ( τ i S τ x λ η = E E s d i τ lim e q λ; q λ( E i S τ x π λ ηs de = T i ( E qq (44 S π x with the symbol T [ ] denoting the te of the mties ove the index λ. By the sme mnne but noting to the nti-ommuttion eltion between the etion nd nnihiltion opetos { ˆq λ ˆ q λ } = we n obtin simil expession fo the hole density in tems of the gete een funtion η h s n = ˆ q Nq λ x S λ ηs = ˆq λ ˆq λ x S λ ηs de = T > i ( E qq. (45 S π x Knowing n e q nd n h q the distibution of hge density in system is detemined h e D A ρ = e n n + n n q q q q q wheein D nq n A q e the dono nd epto densities s the gound hges. The hge density is neessy to desibe the eletostti effets whih ffet on the fomtion of eleti uent in the system. This point will be lified in setion Chge uent density We show tht the hge uent density n lso be lulted by mens of the een funtions. To do so it is neessy to find n ppopite expession fo the eleti uent density in tems of the bsi opetos. Beuse the fomtion of hge uent is the esult of the esponse of system to tnsvese eletomgneti field whih is 7

9 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 hteized by veto potentil A( [37]. Remembe tht the Hmiltonin of system in the seond quntiztion epesenttion n be witten s the summtion of singleptile tem nd othe mny-ptile ones [4 5] fo instne [ A] [ A] ˆ H = H ˆ ˆ pq pq p q + U ˆ ˆ ˆ ˆ + pqp q pqp q p q p q (46 wheein the hopping integls H pq is funtionl of the veto potentil oding to the Peiels substitution nd is given by (see ppendix C: [ ] H H A = H pq pq pq e exp i d ξa( Rq + ξ. Rpq Rpq (47 whee Rpq = Rp R q. The veloity nd thus uent opeto is thus detemined s follows [4 38]: δhˆ ( A J = η = η q ev( R (48 s q s δa( R q A η e s = i R R H ˆ ˆ (49 pq q p p q p q Applying this esult to the onsideed se of the hge tnspot long one dietion sys OX we hve x η e ˆ s Jq = i ( Xq Xp S pq λλ ; H ˆ ˆ. (5 p λ; q λ p λ q λ In the ppoximtion of neest-neighbo oupling this fomul beomes x η e ˆ s x = Jq i Hq λλ ; q+ ( ˆq λ ˆq+ λ S λλ H ˆ ˆ (5 q λλ ; q q λ q λ whee x = Xq+ Xq = Xq X q. By ming the ensemble veging of this opeto with the notie of the definition of the een funtions we obtin the expession fo the density of eletil uent flowing though the ell q s follows [4 5]: I q = x Jˆ q η e s de = x S π { Re H ( ( E qq q q } (5 This fomul is genel nd one n use it to lulte dietly the uent density. Howeve sine the uent density long the tnspot dietion is unifom in the stedy stte i.e. Ip = Iq = I it is inteesting to efomulte eqution (5 in the onvenient fom. Indeed by onsideing the ell q = the mtix blos nd of the lesse een funtion mtix e speified s follows: = H g L + H g L (53 = g L H + g L H (54 whee g ν L ν = e the mtix blo of the left-ontted een funtions oesponding to the sufe ell q = of the left led (see next setion. Beuse { } Re H = + H H nd by noting to the = H H popeties of the te opetion we obtin { L L Re H = H g g H + L Hg H (55 It is helpful when noting to the ft tht the leds ontting to the devie tive egion e usully lge omped to the tive egion. They theefoe ould be ssumed to be in (qusi- theml equilibium stte i.e. being hteized by funtion nη ( x detemining the vege ouption numbe of hge ies on quntum stte (η = L R lbels the left nd ight led egions. Aoding to equtions (34 (36 the lesse een funtion mtix L g ( E is witten in the fom L = L L g E g E g ( E n ( E L (56 Set Σ E = g g E H (57 L the etded/dvned (left led devie oupling self-enegies nd use eqution (4 it yields { L ( Re{ H E = Γ E E + n E E E (58 L Substitute this esult into eqution (5 the uent density is thus ewitten in the fom η e s de I = i Γ { L ( E ( ( E S π + n E E E (59 L } } This fomul fo the uent density is extly the esult deived by Mei nd Wingeen in efeene [3]. We n go futhe by notiing tht = = Σ Σ = iγ iγ (6 η= LR ( Σ Σs η = + [ ] [ ] η = Γ i n + Σ (6 η= LR whee Γ = i Σ Σ s s η s s s nd Σ η = iγ ηnη E (see 8

10 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 eqution (4. Substitute these expessions into eqution (58 we e ble to septe the uent density I into the two ontibutions I nd I s in whih η e s de I = nd S π R { } ΓL E E ΓR E E n E n E (6 L η e s de Is = Γ { L ( E ( E S π Γ ( En ( E + iσ ( E s L s ( E } (63 Eqution (6 is vey notieble beuse if set = { L R } E Γ E E Γ E E (64 then eqution (6 beomes extly identil to eqution ( the Lndue Buttie fomul. The quntity ( E defined by eqution (64 is theefoe intepeted s the pobbility of ptile oupying the tnsvese mode tnsmitting though the devie tive egion. In the se of equilibium sttes both uent density omponents I nd I s beome vnished utomtilly sine nl( E = nr( E = n( E nd Σs = iγsn E. Fo the ltte omponent i.e. I s it is lely indued by the unblne of intetion/stteing poesses of hge ies. This tem lely does not ontibute to the totl uent density if intetion/stteing effets e not ten into ount. 4. Numeil methods nd tehniques In the pevious setion the ey onepts nd equtions of the NEF method hve been pesented s the bsis fo employing the theoy s lultion method. iven Hmiltonin nd funtionl expessions fo the self-enegies it loos stightfowd to just pefom some mtix lgebs suh s the mtix invesion nd the mtix multiplition to obtin the een funtion mties see equtions (9 nd (. It is howeve in ptie fomidble ts beuse ll involved mties e usully vey lge. Theefoe the numeil omputtion is vey expensive in the mening of onsuming huge ompute esoues. Fotuntely s shown in the pevious setion only smll numbe of elements of suh mties is needed to lulte physil quntities fo instne the hge ie densities nd the hge uent density see equtions (44 (45 nd (5 o(6 nd (63. It is theefoe vey useful to develop numeil methods suh tht only inteested mtix elements e lulted. In this setion we will pesent two suh lgoithms one fo solving the een funtions defined in the finite tive egion nd the othe fo the een funtions defined only on the sufe of semi-infinite homogeneous egions. 4.. Reusive lgoithm fo the etded/dvned een funtions We fist pesent the lgoithm fo solving eqution ( i.e. lulting some impotnt elements fo the etded/ dvned een funtion mties. Sine the equtions fo the etded nd dvned een funtions hve the sme stutue the following pesenttion is fo the etded one. iven Hmiltonin H nd self-enegy Σ whih e epesented s ti-digonl nd digonl blo mties espetively to hteize the dynmis of hge ies in the devie egion. Cll N ell the numbe of mtix ows/olumns. Fom eqution ( the etded een funtion mtix ( E Σ is detemined s the invesion of the mtix E E E = E Σ E = E + iη I H Σ E (65 To invet the ight-hnd side mtix we septe it s follows E + iη I H Σ = H (66 whee = E + iη I H d Σ E H d nd H e the digonl nd off-digonl blo mties espetively. The im of this seption is to speify the mtix blos whih onnet to two digonl blos. Conetely the blos Hqq ± will onnet the two digonl blos nd q ± q ± qq. By this seption the etded een funtion mtix ( E is ewitten in the fom of the Dyson eqution s follows = + H (67 The eusive stutue of this eqution suggests the following view: the whole devie tive egion n be seen s the onstution of N ell ells sted sequentilly stting fom the fist ell q = (left-ontted o fom the lst one q = N ell (ight-ontted. Aodingly let us onside system of ( q + ells nd define L g q + q the left-ontted + een funtion in the mening tht it n be divided into fou L mtix blos { g L g L g L g qq qq + q+ q q+ q+ } whee g L is lso qq the left-ontted een funtion of the system of only q L ells. In the mtix fom g q + q is witten s follows + g L q+ q+ g = gq g L L qq qq + L L g + q q+ q+. (68 9

11 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Now pplying the Dyson eqution (67to L + q+ with H speified by H = Hq+ q we dedue these two equtions: whee H qq + g q (69 g L = g + g H g L (7 q+ q+ q+ q+ q+ q+ q+ q qq + L L L g = g H g (7 qq + qq qq + q+ q+ g = E + iη H (7 q+ q+ q+ q+ is nothing the thn the etded een funtion of the isolted ( q + th L ell. Substitute g qq fom eqution (7 + into eqution (7 nd fte some simple lgeb it esults in g = E + i η H H g H L L (73 q q q q q q qq qq This eqution is emble beuse of its eusive stutue. It implies tht L g q + q n be lulted if ledy nowing + L L g qq mtix blo simil to g q + q but of the left-ontted + L een funtion g qq of the system with only q ells. Eqution (73 theefoe suggests poedue of lultion s follows: stting fom the een funtion g defined by eqution (7 fo the fist ell we onstut the funtion g L using eqution (73. We then stt fom g L L to onstut g 33 L nd so on. Fo q = ( Nell we obtin g N ell N whih is ell nothing the thn the lowe ight one blo of the een funtion mtix of the devie tive egion L = g. (74 Nell Nell Nell Nell To te into ount the the oupling effet of the tive egion with the left nd ight leds one should modify only two een funtions g nd g N ell N by dding tem of ell LR oupling self-enegy σ qq into the definition eqution (7 i.e. + LR g = E + i H σ (75 q+ q+ q+ q+ q q Equtions (73 nd (74 with the notie of eqution (75 onstitute the so-lled fowd lultion poedue fo solving the etded een funtion of the opened system. The bwd poedue is theefoe neessy to omplete this ts. To do so we should speify the mening of the bed funtion. Assume tht we ledy now the blo q + q +. The etded een funtion of system of (q + ells but the (q + th ell does not ouple to the blo of q ells theefoe must be L g = qq gq + q+ (76 Aodingly ming use of the Dyson eqution (67 we dedue these two equtions: L L qq = g + g Hqq+ q+ q (77 qq qq L = H g (78 q+ q q+ q+ q+ q qq + Combine these we obtin the following eusive eqution: L L L qq= g + g Hqq+ q+ q+ H + g (79 qq qq q q qq Simil to eqution (73 this esult expesses the bwd lultion poedue s follows: stting fom N ell N ell ll the digonl blos of the ext etded een funtion n be suessively lulted. In genel not only the digonl blos q q but lso ny off-digonl blo n be lulted using the fowd nd bwd lultion poedue. It is tully elized by ombining with both the g L qq nd the ight-ontted fun- left-ontted funtions { } tions { g R qq} whih e defined nd onstuted simil to the fome ones but stting fom q = N ell. The eusive equtions fo the he off-digonl blos e thus given s follows: pq L g H + + p q pp pp p q = R g H p q + pp pp p q (8 The fowd nd bwd lultion poedues s desibed n be dietly pplied to lulte the dvned een funtion. Howeve in ptie this funtion is detemined by invoing the eltion = the thn pe- foming gin the eusive lgoithm. 4.. Reusive lgoithm fo the lesse/gete een funtions A eusive lgoithm n be lso developed to lulte the lesse/gete een funtions s shown by Svizheno et l in [5]. In this setion stting fom the Dyson eqution fo these funtions we will e-estblish tht lgoithm with two typil steps of fowd nd bwd. Remembe tht the Dyson eqution fo the lesse/gete een funtions is not the sme s tht fo the etded/dvned een funtions. In ppendix B we pesent the ules fo ming the ontinuum limit of the time-ontou Dyson eqution to deive the eltime Dyson equtions fo the lesse/gete een funtions [ 9 7 3]. Sine the equtions fo the lesse nd gete funtions hve the sme mthemtil stutue following we only pesent the lgoithm fo the fome one. Fom eqution ( the lesse een funtion mtix ( E is

12 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 fomlly detemined s follows Σ = Σ (8 d On the bsis of the seption H = H + H we thus ewite the bove eqution in the fom ( H = Σ (8 In ode to find in eusive mnne we lso onstut the left-ontted lesse een funtions L { g q + q + } fo the system of ( q + ells. The eqution fo this funtion tes the sme fom of eqution (8 of ouse. Aodingly we n wite down these two equtions: L + L L H g g g = Σ g (83 q+ q qq + q+ q+ q+ q+ q+ q+ q+ q+ L L L L g g H g = Σ g ( qq qq qq q+ q+ qq qq + L g qq Fom eqution (84 we dedue nd then substitute it + into (83 it yields L L ( g Hq+ qg H g q q qq qq q+ q+ L L L = H g Σ g + Σ g (85 q+ q qq qq qq + q+ q+ q+ q+ L By noting to eqution (73 fo g q + q nd eqution (7 fo + L g qq+ (emembe tht the etded nd dvned funtions hve the sme mthemtil stutue we deive the following eqution fo g : q + q + g L L = g L ( H g L H q q q q q+ q qq qq + + Σ L g (86 q+ q+ q+ q+ L L L L whee g qq is the lowe ight one blo of g = g Σ g qq qq qq qq the left-ontted lesse een funtion of the system with only q ells. Eqution (86 obviously hs the ole simil to tht of eqution (73 in the onstution of the fowd lultion poedue with the sme notie tht the ells with q = nd q = N ell ouple to the left nd ight leds (it mens tht the ontt-indued lesse self-enegies should be ppopitely integted into the genel self-enegy Σ q + q +. Applying eqution (86 suessively fo q unning fom upto q = Nell we obtin the lowe ight one (digonl blo of the ext lesse een funtion L = g. (87 Nell Nell Nell Nell Now in ode to onstut the bwd step to lulte ll digonl nd some off-digonl blos of fom the blo we ewite eqution (8 in the fom N ell N ell = H + Σ (88 We futhe use eqution (67 to expnd nd thus obtin whee = + H + H (89 = Σ (9 Aodingly we deive the following eusive equtions: = L + L L g g H + g H (9 qq qq qq qq + q+ q qq qq + q+ q L L = H g + H g (9 q+ q q+ q+ q+ q qq q+ q+ q+ q qq These esults obviously onstitute the bwd poedue to ompletely detemine ll digonl nd some impotnt off-digonl blos of the lesse een funtion mtix. Fo the gete een funtion sine thee is no simple lgeb eltion to the lesse nd/o etded/dvned funtions it must be independently detemined. Fotuntely sine both hve the sme mthemtil stutue the lgoithm desibed by equtions (86 (9 n be dietly > used to lulte Snho Rubio lgoithm As shown in the pevious setions to find the een funtions of n opened system one hs to ledy now the self-enegies hteizing the oupling between this system nd its envionment. iven H nd H. the Hmiltonin mtix blo defining the oupling between the ell q = of the devie tive egion nd the sufe ell of the left led. Aoding to equtions (57 (4 nd (4 we see tht it is mndtoy to detemine the blo g L of the left-led etded een funtion. In the se tht the led egions e finite in size the eusive lgoithm pesented in setion 4. n be dietly used. Howeve beuse the led egions e usully muh lge thn the tive egion they ould be seen s infinite-size systems. This ssumption tully n simplify the lultion fo g L L. Indeed the blo g n be fomlly witten in the fom L g = E + iη H H M E (93 wheein M( E is lled the tnsfe mtix nd is eusively given by M( E = E + iη H H M( E H (94 Ptilly itetion shemes n be used to solve dietly eqution (94. Howeve the onvegene is usully vey slow. In 984 Snho nd Rubio pointed out tht the onvegent te n be stongly impoved by ting speil e of the eusive stutue of the Dyson eqution. The sheme pesented below is fo suh e nd now nown s the Rnho Rubio lgoithm [35]. Denote g L the left-ontted etded een funtion of semi-infinite system. By invoing Dyson eqution (67 we

13 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 L obtin the following esults fo the blos g nd fo wheein with q L = + η g E i H I + H g (95 q L = + η L L g E i H H g + H g q L = tg + g L q q+ q q+ (96 t = E + i η H H (97 t = E + i η H H (98 Applying eqution (96 gin to L g nd q g q we hve L + L L L g = t g + g (99 q q q+ [ ] t = I t t t ( [ ] = I t t t ( Sine eqution (99 is isomophi to eqution (96 the poess n be epeted itetively fte n itetions it yields nd L L L g = t g n + g n ( q n q n q+ n [ n n n n ] n t = I t t t (3 n [ n n n n ] n = I t t t (4 Now letting q = n in eqution ( the following hin of equtions is obtined Aodingly we hve ( L L L g = t g + g L L L g = t g + g 4 L L L g = t g + g + n n n n (5 L g = t + t + t + t 3 L L + + t t g + g n (6 n n n + The poess is epeted until t n + nd + s smll s L needed then g n+. Fom eqution (6 by setting M = t + t + t + t3 + + t n t n. (7 nd substitute it into (95 fte some lgeb we lely elize T the tnsfe mtix needed in eqution (93. t n 5. OPEDEVS pge nd the tnspot hteistis of RTDs In this setion we pesent some lultion esults obtined fom numeil investigtion of the tnspot hteistis of typil eletoni devie the RTDs. The lultion ws pefomed using module HETS integted in the OPEDEVS pge [8] whose onstution is bsed on the NEF method desibed in the pevious setions. The module HETS (ppeition of Heteo-semionduto stutues simultion ws designed to solve the eletoni tnspot poblem of heteo-semionduto stutues. Speifilly HETS is olletion of ompute pogms to numeilly solve ouple of two diffeentil equtions the effetive Shodinge eqution nd the Poisson one whih e defined in finitesize tive egion of length L nd suffiiently lge osssetion. These two equtions espetively ed nd d d m + ε δ + E ( x + U( x dx m ( x d x m ( x eff eff ψ( x = E ψ( x d κ ( x d dx d x U x = e ε ρ ( x (8 wheein meff ( x : the effetive mss of ondution eletons. It is genelly ssumed to vy fom lye-to-lye of mteils ε = : the ineti enegy of one eleton involving m its motion long the tnsvese dietions δe ( x : the pofile of the enegy offset of the ondution bnd fo instne U ( x: the eletostti potentil indued by the distibution of ondution eletons κ ( x: the pofile of the stti dieleti onstnt defined in eh mteil lye nd ρ( x: the pofile of the hge density Figue 3( is the piniple shem of the RTD. Aodingly this devie n be seen s mde up of five piniple semionduto lyes. The ente lye plys the ole of quntum well (W. It is sndwihed between two othe semionduto lyes s the potentil bies (B B. The system of these thee mteil lyes is then onneted to two othe thi semionduto blos whih e doped to me two good ontt egions the emitte (E nd the olleto (C. The doping is not neessy unifom. In ptie the dopnt onenttion in the outside-end egions of these two lyes is high while it is vey low in the egions lose to the ente one. Regding this ft one n distinguish the buffe egions (BF BF fom these ontt egions. In figue 3(b we dw the pofile of the ondution bnd edge in the flt fom to illustte the fomtion of the potentil bies due to the finite offset enegy. As shown below the ente lye nd

14 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Figue 3. Shem of the esonnt tunneling diodes ( nd of the ondution bnd edge long the tnspot dietion (b. the two djent ones essentilly goven the tnspot hteistis of the whole stutue. It is thus lled the devie tive egion. The opetion of the HETS module is govened by the pmetes defined in thee input files. The fist nd seond files nmed EOMETRY_STRUCTURE nd MATERI- AL_STRUCTURE e designed to speify the vlues of the pmetes whih define the geomety nd the mteils in use of the devie. The thid input file nmed OPER- ATION_PARAMETERS speifies the initil vlues fo some physil quntities s well s othe pmetes to ontol the numeil lultion the uy of the self-onsistent eletostti potentil fo instne. One the thee input files e estblished (tully the templtes e povided in the pge the fist step of the lultion poedue is to pefom the self-onsistent solution of the Shodinge nd Poisson equtions to obtin the oet vlues of the eletostti potentil nd the ie density distibutions. This step is elized with n initil guess fo the eletostti potentil. It n be mde mnully o by hoosing the defult option. The self-onsistent lultion is pefomed by invoing the pogm ModelClultion s line-ommnd s follows ModelClultion -s suffix input_file o ModelClultion -s suffix -d In the fist fom the option -s is speified to onfim the sting suffix s the suffix ofoutputfiles tht the pogm genetes to sve the vlues of the eletostti potentil nd of the ie densities. Fo instne if suffix is speified by Vp.5 the output-files nmes will be Pots_Vp.5 nd Dens_Vp.5. In the se tht the option -s is used but suffix is fogotten to povide the sving mode of the pogm will set the output-file nmes in the defult fom s Pots_ nd Dens_. Running the pogm by this ommnd it equies to speify input_file s the file nme inluding the pth of ouse ontining n ppopite vlue fo the eletostti potentil U(x s the initil guess. This unning mode n be used to estt the lultion fom pevious step in the ses tht the self-onsistent poess is stopped (by eleti utting fo instne befoe ehing the onvegene. The seond fom of the ommnd is onvenient to stt the lultion poess fom the beginning with the option -d dded to imply tht the defult vlue of U(x is used s the initil guess. One the self-onsistent solution fo the eletostti potentil nd hge ie densities is hieved we n pefom the lultion fo inteested physil quntities. Fo instne the hge uent density t given voltge n be lulted by invoing the pogm ChgeCuent vi the line ommnd: ChgeCuent input_file > output_file whee input_file is sting speifying the file (inluding the pth of ouse sving the eletostti potentil vlue (i.e. the Dt/Pots_suffix file nd output_file speifies the tget fo sving the uent vlue s output. In the theml equilibium stte the module HETS lso povides pogm nmed Condutne to lulte the eletil ondutne t two egimes the bsolute zeo tempetue nd the finite tempetue ones using the ommnd: Condutne input_file > output_file By vying the vlues of some pmetes in the input files one n hnge the opetion ondition of the simulted devie. One thus n me n investigtion of some tnspot popeties of the devie fo instne the uent voltge (I V hteistis o the ondutne. Note tht when invoing the pogm ChgeCuent it esults in only one vlue fo the uent density oesponding to the vlue of the bis voltge speified in the file OPER- ATION_PARAMETERS. Fo the pogm Condutne it lso esults in one vlue fo the ondutne fo povided tempetue exept fo the se of zeo tempetue file sving the ondutne vlue s funtion of Femi enegy. In OPEDEVS we howeve ommonly use the shell-sipt tehnique to utomte some lultions fo instne utomtilly lulting ll desied points of the I V uve. Apt fom the desibed funtionlities module HETS povides some othe utilities to lulte vious fundmentl physil quntities whih my help to nlyze the physil pitue behind the tnspot hteistis. Fo instne if invoing these thee ommnds: CheContteen > output_file CheTnsmission input_file > output_file 3

15 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Figue 4. Cuent voltge hteisti of esonnt tunneling diode. Figue 5. Density distibution of ondution eletons fo thee vlues of bis voltges V =..6 V nd.34 V. SpetlFuntions input_file > output_file one n obtin espetively dt fo the ontt-sufe een funtion o the tnsmission oeffiient o the lol density of sttes (LDOS nd the ie distibution funtions in the el-enegy spe. Module HETS is n open pge in the mening tht uses n feely develop diffeent pogms intefing with the pogm ModelClultion to lulte inteested physil quntities. We now pesent the simultion esults fo onete RTD. In figue 4 we disply the I V hteisti of RTD smple with ll pmetes given in tble. The figue obviously shows the well-nown N-shpe of the I V uve [6 39]. In the smll bis voltge nge the uent ineses when ising up the vlue of bis voltge. Howeve when the voltge psses one itil vlue lled V (whih detemines the uent pe I P the uent suddenly flls to muh lowe vlue (the uent vlley I V nd then slowly ineses if ontinuing inesing the voltge. This typil non-line fom of the I V uve hteized by vey lge negtive diffeentil esistne hd been topi ttted n intensive onsidetion of both fundmentl nd tehnologil esehes in the s peiod [4 6 4]. To shed the light on the opetion of the RTDs we pesent in figues 5 nd 6 the pofiles of the ondution eleton density nd the ondution-bnd bottom E( x = δe( x + U( x fo sevel vlues of the bis voltge. In the se of no bis voltge ( V = the devie is in the theml equilibium stte. Sine the stutue of the onsideed devie smple is symmetil the pofiles of both quntities e symmetil though the ente egion. Comping the pofile of the ondution bnd bottom whih ws self-onsistent lulted (figue 6 to tht of the enegy offset plotted in figue 3(b we obviously see the effet of the non-unifom doping in the emitte nd olleto s the ise of the potentil in the buffe nd ente egions. When inesing the bis voltge we see the down-shift of the ondution bnd Figue 6. Pofiles of the ondution bnd edge fo thee vlues of bis voltges V =..6 V nd.34 V. Tble. Pmetes defining esonnt tunneling diode. Quntity Vlue Unit Desiption L W 5 nm Width of the As quntum well L B/ 3 nm Width of the Al x As x potentil bies L BF/ nm Width of the As buffe lyes L EC nm Width of the As emitte/ olleto lyes x. nm Mesh-sping m eff.67 m Effetive mss in As lyes m eff.69 m Effetive mss in Al x As x lyes δe.3 ev Enegy-offset between As nd Al x As x T 3 K Tempetue bottom in the olleto egion (ight. The pofile of the eleton density is thus hnged into the symmetil fom. Initilly when the voltge ineses we see the inese of the eleton density in the quntum well. Howeve when the voltge psses 4

16 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Figue 7. Pitue of lol density of sttes plotted fo V =. V. Figue 8. Pitue of stte popultion plotted fo V =.6 V. the itil vlue V the eleton density in this egion suddenly beomes exhusted. This pitue is onsistent with the behvio of the uent. The high uent density mens the lge numbe of eletons flowing though the quntum well. Howeve wht is elly the mehnism govening the tnspot of eletons nd thus the I V hteisti of the devie. In othe wods why the uent suddenly deeses when the bis voltge is ove the vlue V.Tofind the nswe we need to ompute nd nlyze othe physil quntities. In figue 7 we show the LDOS of the simulted devie fo V =. The figue lely shows the existene of LDOS pes in the quntum well egion. These pes obviously eflets the quntiztion of the eleton sttes in the quntum well due to the onfinement used by two finite potentil bies. We found tht the position of suh quntized levels eltive to the ondution bnd bottom e vey onsistent with the vlues estimted fom the eqution fo finite bie quntum wells [39] whee ( L tn. = β β W = meff. E ε (9 nd β = meff. δe + ε E. Fo instne the lowest quntized level detemined fom eqution (9 is.89 ev while it is bout.85 ev fom the figue. In figue 8 we show the pitue of the distibution of eletons n e (x E on eh quntum sttes. The dt ws obtined togethe with the LDOS when invoing the pogm SpetlFuntions fo V =.6 V. Fom the figue we ppently elize tht ondution eletons e stongly distibuted on the sttes of low enegies in the emitte nd olleto egions. Fo the vlue of tempetue of 3 K ne ( x E is detemined to boden ove n enegy nge fom to.5 ev. Additionlly bsed on the olo mp of the figue we n lso elize the lely the ouption of eletons in some quntized levels in the quntum well. Aoding to the Lndue Buttie fomul (eqution ( the tnsmission oeffiient is quntity Figue 9. Pobbility fo eletons tnsfeing though the double potentil bie (tnsmittne plotted fo the tnsvese mode = nd t diffeent bis voltges V =. V (theml equilibium stte.6 V (stte of highest tunneling uent nd.34 V (stte of vlley tunneling uent. hteizing the tnspot of eletons. In figue 9 we pesent the esults fo this quntity obtined by invoing the pogm CheTnsmission. The figue ws plotted fo thee uves oesponding to V =. V = V =.6 nd V =.34 V. The tnsmission oeffiient fo eh vlue of the tnsvese veto is nothing the thn the tnsmittne of eleton though the system. The figue plotted fo = ppently shows shp pe in the enegy nge lowe thn the potentil bies nd n osilltion behvio in the enegy nge bove. The pe ws found to point extly the position of the fist quntized level in the quntum well. It theefoe implies tht the eleton in the emitte egion n esily tunnel though the double potentil bies when its enegy ligns with the quntized level in the quntum well. When inesing the voltge this tnsmission pe shifts down due to the bnd bending nd then disppes when V supsses V. The disppene of the tnsmittne pe is simply due to the position of the quntized level in the 5

17 Adv. Nt. Si.: Nnosi. Nnotehnol. 5 (4 33 Figue. Right pnnel shows the spetum of uent density flowing though the devie t V =.6 V whees the left one displys the eleton popultion on diffeent tnspot sttes to illustte the fomtion of the spetum pes. quntum well does not mth with ny ondution sttes in the emitte egion. The quntized level thus beomes lost its ole of oheent onduting hnnel. To nlyze the fomtion of the hge uent we dditionlly pesent in figue the spetum of the uent density I(E. This quntity is defined s the funtion of enegy unde the integl in eqution (. The lultion ws pefomed by invoing the utility CuentSpetum. As expeted the uent spetum shows the signifint pes oesponding to the tnsmittne pes with diffeent tnsvese modes. These behvios ppently e the evidenes onfiming tht the tnspot of eletons in the RTD devie is govened by the quntiztion of eleton sttes in the quntum well. An eleton in the emitte egion n be tnsmitted though the double-bie system if it oupies stte whose enegy ligns with the quntized level in the quntum well. By this wy the tnspot of eletons is in the esonnt egime. Tht is why the tem esonnt tunneling is ommonly dded to the devie nme. 6. Conlusion We hve pesented shot eview of fundmentl spets in the implementtion of the NEF method into the ompute pge OPEDEVS tht we hve eently developed to study tnspot popeties of nno-sle devie nd low-dimensionl mteil stutues. The definition of the fou el-time een funtions the etded dvned lesse nd gete een funtions s well s thei equtions of motion nd bsi eltions mong them e pesented in the mtix fom whih is suitble fo the numeil omputtionl pupose. Ptiully by ewiting the equtions of motion of the een funtions in the fom of the Dyson equtions we hve eonstuted two effiient omputtion shemes one fo solving the fou een funtions defined in finite-size opened systems nd the othe fo finding the een funtions defined only in the sufe lye of semi-infinite homogeneous systems. These two dvned omputtionl tehniques hve been suessfully implemented in the pge OPEDEVS. As n illusttion fo the opetion of OPEDEVS we hve pesented the omputtionl poedue of numeil investigtion of the eletoni tnspot hteistis of typil semionduto devie stutue the RTD. The RTD simultion ws elized by the module HETS integted in the pge OPEDEVS whih ws genelly designed to solve the tnspot poblem of semionduto heteo-stutues. The tnspot hteistis of the devie inluding the I V uve the hge ie density distibution nd othe physil quntities hve been shown nd nlyzed to lify the physil pitue behind the opetion of this devie stutue. Anowledgements This wo is finnilly suppoted by the Ministy of Edution nd Tining (MOET Vietnm though the pojet B--9. The utho would lie to thn Phm Nm Phong fo eding the mnusipt nd hving fuitful disussions. Appendix A. Definition of the time-ontou Conside system whih is expeiened fom n extenl tion. The Hmiltonin of suh system n be fomlly witten s the sum of two tems Ĥ nd Ĥ ˆ = ˆ + ˆ H H H t ( (A. wheein Ĥ is the solvble pt nd Ĥ the petubtion pt inludes the tem desibing the extenl tion s well s the ones fo ll intinsi mny-body spets of the system. Assuming tht the extenl tion is tuned on t the time t. It mens tht befoe the time t the system is in n equilibium stte whih is detemined eithe by gound stte Φ if the tempetue is equl to zeo o by theml equilibium density mtix ρ if the tempetue is finite. Fo simpliity in the following we will pesent the fomultion fo the fome se. Redes ould onsult efs. [] nd [3] fo the ltte one. Afte the time t the system is pushed out of the equilibium stte. Assuming tht we me n obsevtion t the time t > t. It is thus esonble to define el-time usl een funtion s the expettion vlue of the T-podut of the field opetos in n bity stte Φ tht the system n sty in [] Φ ˆ i Tˆα( t ˆ β αβ ( t; = ΦΦ Φ (A. whee ˆα( t ˆ β t e the field opetos in the Heisenbeg epesenttion. 6