Study of Energy Eigenvalues of Three Dimensional. Quantum Wires with Variable Cross Section

Transcription

1 Adv. Studies Ther. Phys. Vl Study f Eergy Eigevalues f Three Dimesial Quatum Wires with Variale Crss Secti M.. Sltai Erde Msa Departmet f physics Islamic Aad Uiversity Share-ey rach Ira alrevahidi@yah.cm Astract I this paper the effect f radius-sie wire the umer f eergy eigevalues f three dimesial quatum wires are studied. It is shw that the umer f the uded state eergy eigevalues f three dimesial quatum wires with peridic structure relates t the peridictiy ad the sie f the radius. eywrds: ud state eergy eigevalue quatum wires -Itrducti: ecetly the quatum wires have ee the suect f etesive studies. Mst csidered tpics are quatum cectis with a cmperssi effect i quatum wires ad umer f the ud states. Direct sluti f the Schrdiger equati applied t the peridic quatum wires [345]. I this apprach first the trasfrmati matrices fr a uit f the wire cmputes the y crssig matrices fr differet uits the ttal trasfrmati matri will e fud. Therefre the trasfrmati cefficiets ca e cmputed y usig Laduar equati. O the ther had Martrell et.al [6] studied wires i D structures ad fud the electr waves

2 4 M.. Sltai Erde Msa fr a L shape wire y slvig the time-idepedet Schrdiger equati csiderig symmetric ad atisymmetric slutis i differet chaels. I a iteractig wrk aavi [7] studied the uded state fr a 3D quatum wire. He cmputed the eergy eigevalues y usig the epasi methd ad shw that the eergy eigevalues f the physical states f a system relates t the eigevalues f uded states which cause the fiite mti f the electrs i a 3D quatum wire. I this paper we ctiue the study f a 3D quatum wire. I particular we apply the epadig methd t study the radius-sie ad peridicity effect the umer f eergy eigevalues f uded states. - Three dimesial quatum wire with peridic structure I this secti we csider a 3D quatum wire with variale crss secti π where is the radius f the wire ad chage as a peridic fucti f. The regi f is. Let e a si q with udary cditi ± : a si q The parameters a q are cstats that defie the radius f the crss secti f the wire. As a eample quatum wire fr a 0 / 8 q 4 is pltted i Fig.. The dyamics f the system will e taied y slvig the Schrdiger equati f free electrs ψ r ϕ 3 where * m E h 4 * The parameters E ad m dete the eergy ad effective mass respectively. The udary cditi 3 ca e writte as Diricleh cdisi ψ r ϕ We csider a sluti as [8]: 5

3 Study f eergy eigevalues f three dimesial quatum wires 5 si cs r m m A r r m m m m m m ϕ ϕ ϕ ψ 6 I the limit ± [7] the eq. 3 ecmes 0 // 7 Oe shuld te that ly sme f the participate i the icidet ad utcmig waves. T have a uded state the value f i 7 must fulfill the iequality This meas that there is a uded state ly fr the grud state eigefuctis. 3-Cuplig f eigefuctis As 0 is small relative t ad the equati 3 ca e writte as / // 9 / / / C I which the dimesiless matri elemets ad C ca e defie as [8] 3 d C d 0 It ca e ccluded that the effect f the ther eigestates the grud state is t small that the effect f 0 ca e eglected ad 9 takes the frm

4 6 M.. Sltai Erde Msa C The equati ca e writte i the frm G // γ γ G C Ad the aswer f e emmitive mdes ca e csidered as [9] G d e γ The values f γ are large fr differet values. Fr eample at fr the quatum wires we have 83 / k γ 6 S we ca e eglected with respect t 0 fr.... Defiig[9] l ep l ep y y 7 replacig i ad eglectig we ca tai y V y 8 V is the effective ptetilal that relates t the shape f quatum wire ad fr the quatum wire V takes the frm si si si 4048 / q a q q q a V 9 The V plt fr 4 0 / 8 q a is preseted i fig..

5 Study f eergy eigevalues f three dimesial quatum wires 7 4- Numerical results I rder t calculate the eigevalues f uded states we ca rewrite the equati 8 usig fiite differece methd [0] y y y y [ V h ] 0 h h h The relati 0 itrduces a three diagal matri. Calculatig the eigevalues f this matri ad csiderig the terms that fullfils V ± We ca tai the eigevalues related t uded states. Step: We first csider the effect f perid f wire radius f cress secti i the umer f eergy eigevalues. The results fr a are I the quatum wire haset uded state. I the quatum wire has a uded state. 3 I 0.7. the quatum wire has tw uded states. 4 I the quatum wire has a uded state. 5 Ii the quatum wire haset uded state. Step : I this step we chse a fr the quatum wire t study the effect f perid f wires radius f crss secti ad its magitude f scillative part simultausy i the eigevalues f uded state f eergy. The results are give i tale. Tale shws that: Fr 0.0. i there is t ay uded state. Fr i there is always a uded state. 3 Fr t ust fr sme q values there is uded state. I such a way that icreasig decrease related q iterval related. 4 Fr 0.9 i there is uded state. Cclusi The cmputatis shw that the eistece ad umer f uded states relates t the shape ad physical structure amely the radius ad the perid f quatum wires.

6 8 M.. Sltai Erde Msa Ackwledgmet. This wrk was supprted y Islamic Aad Uiversity shahr-e-ey rach. Tale q iterval uded states umer e e always e always e always e e always e e always e e always e e e e efereces D.W. Sprug et.al. Appl. phys D.W. Sprug et.al. Phys.ev D.W. Sprug et.al. Phys.ev

7 Study f eergy eigevalues f three dimesial quatum wires 9 4 D.W. Sprug et.al. Phys.Slid.stat.Cmmu D.W. Sprug et.al. Phys.ev M.Martrel.Accust.s. AM M.aavi Phys.Let.A P.M.Mrse H.Feshach Methds f Theritical physics MacGraw-Hill NewYrk953 9 M. Hr M. aavi Iter..Quat.Chem XXIV L.W. hs.d.iess Numerical Aalysis Spriger-Verlag NewYrk98 eceived: Decemer 008 Fig.: quatum wire with redius crss secti

8 0 M.. Sltai Erde Msa Fig: plt f V fr a 0 / 8 q 4