Non-Equilibrium Thermal Transport in Nanowires with Rough Surfaces

Citation

Permanent Link:: https://ufdc.ufl.edu/UFE0054212/00001

Material Information

Title:: Non-Equilibrium Thermal Transport in Nanowires with Rough Surfaces
Creator:: Abhinav, Siddhartha
Place of Publication:: [Gainesville, Fla.]
Florida
Publisher:: University of Florida
Publication Date:: 2019
Language:: english
Physical Description:: 1 online resource (64 p.)

Thesis/Dissertation Information

Degree:: Doctorate ( Ph.D.)
Degree Grantor:: University of Florida
Degree Disciplines:: Physics
Committee Chair:: Muttalib,Khandker A
Committee Co-Chair:: Hershfield,Selman Philip
Committee Members:: Fry,James N
Hebard,Arthur F
Kopelevich,Dmitry I
Graduation Date:: 5/3/2019

Subjects

Subjects / Keywords:: green -- nanowire -- non-equilibrium -- non-linear -- phonon -- silicon -- thermal
Physics -- Dissertations, Academic -- UF
Genre:: bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:: Heat transport in low dimensional nanostructures is a topic of current interest because of the experimental observation of reduced thermal conductivity in surface-roughness dominated silicon nanowires, and its relevance to thermoelectricity. Here we consider the effects of surface-roughness on the full non-linear thermal current in a nanowire, for arbitrary hot and cold lead temperatures. We use the experimental insights to construct an analytical model of the phonon surface roughness interaction. We show that the surface roughness affects the frequency dependence of the thermal current and leads to a temperature dependent reduction of the net current at high temperatures. We further confirm that for a wire of given diameter and length, the reduction in the heat flux is characterized by a single combination of two different parameters, the mean height of surface corrugation and the correlation length of the disorder. The dependence of net current on this parameter explains the experimentally observed differences in thermal conductivity between smooth vs rough surface nanowires. We argue that in order to best understand optimization of thermoelectric efficiency, we need a systematic study of the thermal current for various strengths of surface disorder as well as the driving temperature gradient it is subjected to. ( en )
General Note:: In the series University of Florida Digital Collections.
General Note:: Includes vita.
Bibliography:: Includes bibliographical references.
Source of Description:: Description based on online resource; title from PDF title page.
Source of Description:: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:: Thesis (Ph.D.)--University of Florida, 2019.
Local:: Adviser: Muttalib,Khandker A.
Local:: Co-adviser: Hershfield,Selman Philip.
Statement of Responsibility:: by Siddhartha Abhinav.

Record Information

Source Institution:: UFRGP
Rights Management:: Applicable rights reserved.
Classification:: LD1780 2019 ( lcc )

UFDC Membership

Aggregations:: University of Florida Theses & Dissertations
INTERNAL UF RETRO ETDS

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Full Text

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PAGE 6

TLTR hlcd J(!)=0!=TL)]TJ /F3 11.9552 Tf 12.015 0 Td[(TRTL=1TL=2 1=!=1:5 J(!)=0!=11:3=TL)]TJ /F3 11.9552 Tf 11.215 0 Td[(TRTL=1TL=2 J(!)=0!TL=1T=1 J(!)=0!TL=2T=1T=2 J(!)=0!TL=1T=1hlc Jnet=(0T)TRT=1T=2 Jnet=(0T)TR=11:3T

PAGE 10

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PAGE 11

kBe2 3ee

PAGE 13

l==2 d15nm

PAGE 14

ZT T Jth=Z10d! 2~!hbL(!))]TJ /F3 11.9552 Tf 11.955 0 Td[(bR(!)i(!):

PAGE 15

bL(R)(!)~!TL(TR)(!)(!)(!)

PAGE 16

~=B=1

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TLTR TLTR s Heff=Xk;n(q+En)cyk;nck;n+Xk;qXm;n(q)Vmncyk+q;nck;m:

PAGE 18

cycVmn(En)]TJ /F3 11.9552 Tf 12.167 0 Td[(Em)[z@=@z+(@=@z)z]mnEnz HCint=XlZdxQ(x)Q(x)(x)]TJ /F3 11.9552 Tf 11.956 0 Td[(xl);Q(x)Qq(x)hirandom D(x;x0))]TJ /F3 11.9552 Tf 21.918 0 Td[(ihT[Q(x)Q(x0)]i;

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hlcd n (n)D(x;x0)=)]TJ /F2 11.9552 Tf 10.494 8.087 Td[(()]TJ /F3 11.9552 Tf 9.298 0 Td[(i)n n!Zdt1:::dtnT)]TJ /F3 11.9552 Tf 5.479 -9.683 Td[(Q(x)Q(x0)Hint(t1):::::Hint(tn): (2)D(x;x0)=1 2Xl;mZdx1dx2hT[Q(x)Q(x0)Q(x1)Q(x1)(x1)]TJ /F3 11.9552 Tf 11.955 0 Td[(xl)Q(x2)Q(x2)(x2)]TJ /F3 11.9552 Tf 11.955 0 Td[(xm)]i: (2)D(x;x0)random=1 2Xl;mZdx1dx2hT[Q(x)Q(x0)Q(x1)Q(x1)Q(x2)Q(x2)]iF(x1;x2);

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F(x1;x2) F(x1;x2)=hhT(x1)]TJ /F3 11.9552 Tf 11.955 0 Td[(xl)(x2)]TJ /F3 11.9552 Tf 11.956 0 Td[(xm)iirandom=Xq1;q2eiq1x1eiq2x2hTq1(t1)q2(t2)i*Xl;me)]TJ /F5 7.9701 Tf 6.587 0 Td[(iq1xle)]TJ /F5 7.9701 Tf 6.587 0 Td[(iq2xm+random: *Xl;me)]TJ /F5 7.9701 Tf 6.586 0 Td[(iq1xle)]TJ /F5 7.9701 Tf 6.587 0 Td[(iq2xm+random=Nimpq1+q2;0+N2impq1;0q2;0;NimpNimpq1=q2=0 F(x1;x2)=NimpXq1eiq1(x1)]TJ /F5 7.9701 Tf 6.586 0 Td[(x2)hTq1(t1))]TJ /F5 7.9701 Tf 6.586 0 Td[(q1(t2i):=a+ay q(t)=uqq(t):uqhuqu)]TJ /F5 7.9701 Tf 6.587 0 Td[(qi F(x1;x2)=iNimpXq1eiq1(x1)]TJ /F5 7.9701 Tf 6.586 0 Td[(x2)uq1u)]TJ /F5 7.9701 Tf 6.587 0 Td[(q1dq1(t1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t2); dq(t1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t2))]TJ /F3 11.9552 Tf 21.918 0 Td[(ihT[q(t1)q(t2)]i:

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Q hT[Q(x)Q(x0)Q(x1)Q(x1)Q(x2)Q(x2)]i=hTQ(x)Q(x1)ihTQ(x1)Q(x2)ihTQ(x2)Q(x0)i+hTQ(x)Q(x2)ihTQ(x2)Q(x1)ihTQ(x1)Q(x0)i+hTQ(x)Q(x1)ihTQ(x2)Q(x2)ihTQ(x1)Q(x0)i+hTQ(x)Q(x2)ihTQ(x1)Q(x1)ihTQ(x2)Q(x0)i+hTQ(x)Q(x0)ihTQ(x1)Q(x1)ihTQ(x2)Q(x2)i+hTQ(x)Q(x0)ihTQ(x1)Q(x2)ihTQ(x2)Q(x1)i: 1=2 Dex(x;x0)=iXx1;x2hTQ(x)Q(x1)ihTQ(x1)Q(x2)ihTQ(x2)Q(x0)i[NimpXq1eiq1(x1)]TJ /F5 7.9701 Tf 6.587 0 Td[(x2)uq1u)]TJ /F5 7.9701 Tf 6.587 0 Td[(q1dq1(t1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t2)]: p(!)=Xq;Wqdq()Dp)]TJ /F5 7.9701 Tf 6.586 0 Td[(q(!)]TJ /F3 11.9552 Tf 11.955 0 Td[(); Wququ)]TJ /F5 7.9701 Tf 6.587 0 Td[(qdq()Dp)]TJ /F5 7.9701 Tf 6.586 0 Td[(q(!)]TJ /F3 11.9552 Tf 12.182 0 Td[()

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D0;7p(!)=)]TJ /F3 11.9552 Tf 9.299 0 Td[(i !p[(1+Np)(!!p)+Np(!!p)] Npp!p=vpvWq S(q) S(q)=2lc 1+q2l2c;=h d;hlcdWq Wq=W02lc 1+q2l2cW0W0=1 hlcd

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hdlc =p lcd h:h=dlchdlc )]TJ /F1 11.9552 Tf 11.211 0 Td[<00420062>-326<0069001c0046>26<0032004d>-325<00690051>]TJ -398.856 -23.908 Td[<0023>-27<0032>-326<001c>-326<0054003f0032004d0051004b0032004d005100480051003b0042002b001c0048>-326<0054001c>26<0060001c004b00320069003200600058>-433<003e0051>26<0072>27<0032007000320060002d>-325<00620042004d002b0032>-326<00420069>-326<001c>26<00600042006200320062>-325<003700600051004b>-326<0069003f0032>-326<004b001c005400540042004d003b>-326<00510037>-326<0069003f0032>-326<0062006d00600037001c002b0032>]TJ 0 -23.907 Td[<00600051006d003b003f004d003200620062002d>-326<00420069>-326<001c004800620051>-326<00600032007e0032002b00690062>-326<0069003f0032>-326<006d004d00420070003200600062001c004800420069>26<0076>-325<0042004d>-326<006900320060004b0062>-326<00510037>-326<002f004200620051>26<0060002f00320060002d>-325<001c0062>-326<0062003f0051>26<0072004d>-325<0042004d>-326<0028>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 370.323 -322.759 Td[<00650079>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 382.039 -322.759 Td[<00290058>-434<0071>26<0032>-325<003f001c00700032>]TJ -382.039 -23.908 Td[<0069003f00320060003200370051>26<00600032>-325<002b003f005100620032004d>-326<0069003f0032>-326<002f004200620051>26<0060002f00320060>-325<002f00320054>-27<0032004d002f0032004d002b0032>-326<00510037>]TJ /F2 11.9552 Tf 218.17 0 Td[()]TJ /F1 11.9552 Tf 11.211 0 Td[<0023>26<0076>-325<006d00620042004d003b>]TJ /F2 11.9552 Tf 43.924 0 Td[()-412(=)]TJ /F6 7.9701 Tf 33.598 -1.794 Td[(0)]TJ /F6 7.9701 Tf 7.315 -1.794 Td[(0)]TJ /F6 7.9701 Tf 7.314 -1.793 Td[(0)]TJ /F1 11.9552 Tf -443.31 -23.908 Td[<0051004d>-326<0062006d00600037001c002b0032>-326<002f004200620051>26<0060002f00320060>-325<00420062>-326<002b003f005100620032004d>-326<00690051>-326<00600032007e0032002b0069>-326<0069003f0032>-326<0037001c002b0069>-326<0069003f001c0069>-326<0048001c>26<0060003b00320060>-325<002f004200620051>26<0060002f00320060>-325<00550062004b001c0048004800320060>]TJ /F3 11.9552 Tf 372.332 0 Td[()]TJ /F1 11.9552 Tf 7.314 0 Td[<0058>]TJ ET BT /F8 11.9552 Tf 170.889 -466.206 Td[<006b0058006a>-1000<00470051>-31<002b001c0048004200780032002f>-367<0053003f0051004d0051004d>]TJ /F1 11.9552 Tf -147.505 -24.439 Td[<001b>-326<00480051>-27<002b001c0048004200780032002f>-326<0054003f0051004d0051004d>-326<00420062>-326<001c>-326<007000420023>26<0060001c006900420051004d>-325<0069003f001c0069>-326<00420062>-326<00480051>-27<002b001c0048004200780032002f>-326<0042004d>-326<00620054001c002b00320058>-434<0068>81<007600540042002b001c004800480076>82<002d>-325<00420069>-326<003f001c0062>-326<001c>-326<007d00740032002f>]TJ -23.384 -23.908 Td[<003700600032005b006d0032004d002b0076>-326<007200420069003f>-326<001c>-326<0062004b001c00480048>-326<00620054>26<00600032001c002f0058>-433<0050006d0060>-326<0054003f007600620042002b001c0048>-326<0042004d0069003200600054>26<006000320069001c006900420051004d>-325<00420062>-326<0069003f001c0069>-326<0069003f0032>-326<0062006d00600037001c002b0032>-326<002b0051>26<00600060006d003b001c006900420051004d>-325<0051004d>]TJ 0 -23.908 Td[<0023>-27<00510069003f>-326<00620042002f00320062>-326<00510037>-326<0069003f0032>-326<0072004200600032>-326<00620051004b003200690042004b00320062>-326<002b0051>26<0060006000320048001c00690032>-325<0042004d>-326<001c>-326<0072>26<001c>27<0076>-325<00620051>-326<001c0062>-326<00690051>-326<0054>26<00600051>-26<002f006d002b0032>-326<001c>-326<00690060001c00540054>-27<0032002f>-326<007000420023>26<0060001c006900420051004d>-325<001c002b0060005100620062>]TJ 0 -23.908 Td[<0069003f0032>-326<00720042006000320058>-434<0050004d0032>-326<002b001c004d>-326<0069003f0042004d0046>-326<00510037>-326<00420069>-326<001c0062>-326<001c00690051004b0062>-326<0051004d>-326<0023>-27<00510069003f>-326<00620042002f00320062>-326<007000420023>26<0060001c00690042004d003b>-325<0042004d>-326<001c>-326<0072>26<001c>27<0076>-325<0069003f001c0069>-326<0054>26<00600051>-26<002f006d002b00320062>-326<001c>-326<00480051>-27<002b001c0048>]TJ 0 -23.907 Td[<004b0051>-27<002f0032>-326<00510037>-326<007000420023>26<0060001c006900420051004d002d>-325<0072003f0042002b003f>-326<002b001c004d>-326<004d00510069>-326<00320062002b001c0054>-27<0032>-326<002f006d0032>-326<00690051>-326<00620051004b0032>-326<00620054>-27<0032002b0042001c0048>-326<00370032001c0069006d00600032>-326<00510037>-326<0069003f0032>-326<002b0051>26<00600060006d003b001c006900420051004d>-325<0069003f001c0069>]TJ 0 -23.908 Td[<004b001c0069002b003f00320062>-326<0051004d>-326<0023>-27<00510069003f>-326<00620042002f003200620058>]TJ ET BT /F1 11.9552 Tf 227.766 -687.422 Td[<006b006a>]TJ ET Q

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H=1 2XkPkPk+!2kQkQk+qq+2qqq+XkVLkqQkq;!kkqQPbyb Qp=s 1 2!p(bp+byp)Pp=)]TJ /F3 11.9552 Tf 9.298 0 Td[(ir !p 2(bp)]TJ /F3 11.9552 Tf 11.955 0 Td[(byp); Pp=_Qp;Pyp=Pp;Qyp=Qp; qqaya q=s 1 2q(aq+ayq)

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q=)]TJ /F3 11.9552 Tf 9.298 0 Td[(ir q 2(aq)]TJ /F3 11.9552 Tf 11.955 0 Td[(ayq); q=_q;yq=q;yq=q: [bp;byp0]=pp0;[bp;bp0]=[byp;byp0]=0;[aq;ayq0]=qq0;[aq;aq0]=[ayq;ayq0]=0: PQ [Qp;Pp0]=ip;p0;[Qp;Qp0]=[Pp;Pp0]=0[q;q0]=iq;q0;[q;q0]=[q;q0]=0: [Qp;q]=[Qp;q]=[q;Pp]=[q;Pp]=0:bby d>qq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0))]TJ /F3 11.9552 Tf 28.56 0 Td[(ihaq(t)ayq(t0)i;dqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0))]TJ /F2 11.9552 Tf 14.021 3.155 Td[(dqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0))]TJ /F3 11.9552 Tf 28.559 0 Td[(ihq(t)q(t0)i=1 2q[d>qq(t0)]TJ /F3 11.9552 Tf 11.955 0 Td[(t)+dqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)+d
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dqq(!)=jdrqq(!)j20>R(!);drqq(!)=1 !)]TJ /F2 11.9552 Tf 11.955 0 Td[(q)]TJ /F2 11.9552 Tf 11.955 0 Td[(0rR(!); R 0R(!)XkjVRkqj2D0>kk(!);0rR(!)XkjVRkqj2D0rkk(!): D0bby D0kk(!)=)]TJ /F3 11.9552 Tf 9.299 0 Td[(i)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(1+Nk;L2(!)]TJ /F3 11.9552 Tf 11.956 0 Td[(!k): 0R(!)=)]TJ /F2 11.9552 Tf 9.298 0 Td[(2i)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R(!))]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(1+NR(!):NR)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(RPkjVLkqj2!kknR(!) )]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R(!)nR(!)jVR(!)j2;NR(!)1 eR!)]TJ /F2 11.9552 Tf 11.955 0 Td[(1: 0aR(!)=XkjVkqj21 !)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k)]TJ /F3 11.9552 Tf 11.955 0 Td[(i=Zd!knR(!k)jVR(!k)j2 !)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k)]TJ /F3 11.9552 Tf 11.955 0 Td[(i=PZd!knR(!k)jVR(!k)j2 !)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k+inR(!k)jVR(!)j2

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R+i)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R: bby dR()]TJ /F3 11.9552 Tf 9.298 0 Td[(!)=)]TJ /F3 11.9552 Tf 15.007 8.088 Td[(i q)]TJ /F5 7.9701 Tf 7.314 -1.794 Td[(RNR(!) (!)]TJ /F2 11.9552 Tf 11.956 0 Td[(q)]TJ /F2 11.9552 Tf 11.956 0 Td[(R)2+)]TJ /F6 7.9701 Tf 19.076 4.118 Td[(2R(!)+)]TJ /F5 7.9701 Tf 7.314 -1.794 Td[(R(1+NR()]TJ /F3 11.9552 Tf 9.299 0 Td[(!)) (!+q+R)2+)]TJ /F6 7.9701 Tf 19.075 4.118 Td[(2R(!)=)]TJ /F3 11.9552 Tf 15.007 8.087 Td[(i q)]TJ /F5 7.9701 Tf 7.315 -1.793 Td[(RNR(!) (!)]TJ /F2 11.9552 Tf 11.956 0 Td[(q)]TJ /F2 11.9552 Tf 11.956 0 Td[(R)2+)]TJ /F6 7.9701 Tf 19.076 4.118 Td[(2R+)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R(1+NR()]TJ /F3 11.9552 Tf 9.299 0 Td[(!)) (!+q+R)2+)]TJ /F6 7.9701 Tf 19.075 4.118 Td[(2R: )]TJ /F5 7.9701 Tf 7.315 -1.794 Td[(R)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R()]TJ /F3 11.9552 Tf 9.298 0 Td[(!)=)]TJ /F5 7.9701 Tf 27.613 -1.793 Td[(R(!)=)]TJ /F5 7.9701 Tf 27.612 -1.793 Td[(R d>qq(!)=)]TJ /F3 11.9552 Tf 10.494 8.088 Td[(i)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(L q1+NL(!) (!)]TJ /F2 11.9552 Tf 11.955 0 Td[(q)]TJ /F2 11.9552 Tf 11.955 0 Td[(L)2+)]TJ /F6 7.9701 Tf 19.075 4.118 Td[(2L+(NL()]TJ /F3 11.9552 Tf 9.298 0 Td[(!) (!+q+L)2+)]TJ /F6 7.9701 Tf 19.075 4.118 Td[(2L:Vkq!0)]TJ /F7 11.9552 Tf 10.635 0 Td[(!0!0 dqq(!)!)]TJ /F3 11.9552 Tf 26.748 8.087 Td[(i q(1+NR(!)(!)]TJ /F2 11.9552 Tf 11.956 0 Td[(q)+NR(!)(!+q); drqq(!)=1 2qdrqq(!)+daqq()]TJ /F3 11.9552 Tf 9.299 0 Td[(!)=1 2q1 !)]TJ /F2 11.9552 Tf 11.955 0 Td[(q)]TJ /F2 11.9552 Tf 11.955 0 Td[(R)]TJ /F3 11.9552 Tf 11.955 0 Td[(i)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R+1 )]TJ /F3 11.9552 Tf 9.299 0 Td[(!)]TJ /F2 11.9552 Tf 11.955 0 Td[(q)]TJ /F2 11.9552 Tf 11.955 0 Td[(R+i)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R=0 q1 (!+i)]TJ /F5 7.9701 Tf 7.314 -1.793 Td[(R)2)]TJ /F2 11.9552 Tf 11.955 0 Td[(02: PkVLkqQkq d
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d>qq(!)=)]TJ /F3 11.9552 Tf 15.006 8.088 Td[(i q)]TJ /F5 7.9701 Tf 7.314 -1.794 Td[(L(1+NL(!))+)]TJ /F5 7.9701 Tf 30.837 -1.794 Td[(R(1+NR(!)) (!)]TJ /F2 11.9552 Tf 11.955 0 Td[(q)]TJ /F2 11.9552 Tf 11.956 0 Td[()2+)]TJ /F6 7.9701 Tf 19.075 3.454 Td[(2+)]TJ /F5 7.9701 Tf 7.314 -1.794 Td[(LNL(!)+)]TJ /F5 7.9701 Tf 26.284 -1.794 Td[(RNR(!) (!+q+)2+)]TJ /F6 7.9701 Tf 19.075 3.454 Td[(2drqq(!)=0 q1 (!+i\2)]TJ /F2 11.9552 Tf 11.955 0 Td[(02 =L+R;)-278(=)]TJ /F5 7.9701 Tf 59.529 -1.793 Td[(L+)]TJ /F5 7.9701 Tf 19.075 -1.793 Td[(R;0=q+L+R1 )]TJ /F1 11.9552 Tf -457.122 -23.908 Td[<00690051>-326<0069003f0032>-326<0054003f0051004d0051004d0058>-434<001b0062>-326<001c>-326<004b0032001c0062006d00600032>-326<00370051>26<0060>-325<00620042004b005400480042007d002b001c006900420051004d002d>-326<0072>26<0032>-325<002b001c004d>-326<001c00620062006d004b0032>-326<0069003f001c0069>-326<0023>-27<00510069003f>-326<00480032001c002f0062>-326<003f001c00700032>-326<0032005b006d001c0048>]TJ 0 -23.908 Td[<002b0051004d0069006000420023006d006900420051004d0062>-326<00690051>-326<0023>26<00600051001c002f0032004d0042004d003b>-325<0069003f0032>-326<0054>-27<0032001c0046>-326<00510037>-326<0069003f0032>-326<00480051>-27<002b001c0048004200780032002f>-326<0054003f0051004d0051004d002d>-326<0042005800320058>]TJ /F2 11.9552 Tf 324.71 0 Td[()]TJ /F5 7.9701 Tf 7.315 -1.793 Td[(L=)]TJ /F5 7.9701 Tf 20.68 -1.793 Td[(R=)]TJ /F3 11.9552 Tf 20.68 0 Td[(=2qq+=0 dqq(!)=)]TJ /F3 11.9552 Tf 11.349 8.088 Td[(i)]TJ ET q .4782 w 168.128 -414.241 m 181.147 -414.241 l S Q BT /F2 11.9552 Tf 168.128 -425.431 Td[(q(1+NLR(!)) (!)]TJ /F2 11.9552 Tf 11.955 0 Td[(q)2+)]TJ /F6 7.9701 Tf 19.075 3.454 Td[(2+NLR(!) (!+q)2+)]TJ /F6 7.9701 Tf 19.076 3.454 Td[(2;drqq(!)=1 (!+i\2)]TJ /F2 11.9552 Tf 11.955 0 Td[(2 NLR(!)=NL(!)+NR(!) 2: TLTR

PAGE 29

HLHRHLCHRCHC1HC2HCint H=HL+HR+HLC+HRC+HC;HL(R)=1 2XkPL(R)kPL(R)k+!2kQL(R)kQL(R)k;HLC(RC)=Xk;pVLC(RC)kpQL(R)kQCp;HC=HC1+HC2+HCint;HC1=1 2XpPCpPCp+!2pQCpQCp;HC2=1 2XqCqCq+2qCqCq;HCintXmnlUmnlQCmQCnCl: !kk!ppq

PAGE 30

t=0Hint=0 H=HL+HC+HLCHLHCkq JL!C=D)]TJ /F2 11.9552 Tf 27.09 3.819 Td[(_H0L(t)E=iD[H0L(t);H(t)]E=iDh1 2leadXk0Pk0(t)Pk0(t);interfaceXkqVkpQk(t)Qp(t)iE=i1 2Xk0kpVkpDPk0(t)[Pk0(t);Qk(t)]Qp(t)+[Pk0(t);Qk(t)]Pk0(t)Qp(t)E; [Qp(t);Pp0(t)]=ip;p0 JL!C=1 2Xk0kpVkpDk;k0Pyk0(t)Qp(t)+k;k0Pk0(t)Qp(t)E=1 2XkpVkpDPk(t)Qp(t)+Pk(t)Qp(t)E=XkpVkpDPk(t)Qp(t)E Pp(t)=_Qp(t) JL!C=XkpVkpDd dtQk(t)Qp(t)E:

PAGE 31

h_Qk(t)Qp(t)hd dtQk(t)Qk(t)!Qk(t0)t0>tD< D(CL)
PAGE 32

C<(t;t0)=Zdt1)]TJ /F3 11.9552 Tf 5.48 -9.684 Td[(Ar(t;t1)B<(t1;t0)+A<(t;t1)Ba(t1;t0): D(CL)
PAGE 33

JL!Cy=Z1d! 2!Trh)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(D(C)r(!)(L)<(!)+D(C)<(!)(L)a(!)yi:D(C)a(!)=D(C)ry(!)(L)<(!)=)]TJ /F2 11.9552 Tf 9.298 0 Td[((L)(L)>)]TJ /F3 11.9552 Tf 11.956 0 Td[(D(C)<(L)>i: JR JR=1 2Z1d! 2!TrhD(C)>(R)>)]TJ /F3 11.9552 Tf 11.955 0 Td[(D(C)<(R)>i: (R)<(R)>J=JL!C=)]TJ /F3 11.9552 Tf 9.299 0 Td[(JR!C J=1 2JL!C+JR!CJ=1 4Z1d! 2!TrhD(C)>((L)<)]TJ /F2 11.9552 Tf 11.955 0 Td[((R)<))]TJ /F3 11.9552 Tf 11.955 0 Td[(D(C)<((L)>)]TJ /F2 11.9552 Tf 11.955 0 Td[((R)>)i: (L)(!)=i(L)r)]TJ /F2 11.9552 Tf 11.955 0 Td[((L)a=i(L)>)]TJ /F2 11.9552 Tf 11.955 0 Td[((L)<;

PAGE 34

(R)(!)=i(R)r)]TJ /F2 11.9552 Tf 11.956 0 Td[((R)a=i(R)>)]TJ /F2 11.9552 Tf 11.955 0 Td[((R)<) <(!)=Neq(!))]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(r(!))]TJ /F2 11.9552 Tf 11.956 0 Td[(a(!)=Neq(!))]TJ /F2 11.9552 Tf 5.48 -9.683 Td[(>(!))]TJ /F2 11.9552 Tf 11.955 0 Td[(<(!);) Neq(!)=1 e!)]TJ /F6 7.9701 Tf 6.586 0 Td[(1 <(!)=Neq(!) 1+Neq(!)>(!): (L)(!)=i(L)>)]TJ /F3 11.9552 Tf 23.286 8.087 Td[(Neq 1+Neq>=i1 1+Neq(L)>: (L)>=)]TJ /F3 11.9552 Tf 9.298 0 Td[(i1+Neq(L)(L)(L)<=Neq(L) 1+Neq(L)(L)>=)]TJ /F3 11.9552 Tf 9.298 0 Td[(iNeq(L)(L): (R)>=)]TJ /F3 11.9552 Tf 9.299 0 Td[(i1+Neq(R)(R);(R)<=)]TJ /F3 11.9552 Tf 9.299 0 Td[(iNeq(R)(R): (!) K(!)=K0h! !Kci2exp2)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(1)]TJ /F3 11.9552 Tf 17.252 8.088 Td[(! !Kc;

PAGE 35

K2L;RK0!KcK (L)>=)]TJ /F3 11.9552 Tf 9.299 0 Td[(i1+Neq(L)(L)=)]TJ /F3 11.9552 Tf 9.298 0 Td[(iL01+Neq(L)h! !Lci2exp2)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(1)]TJ /F3 11.9552 Tf 16.28 8.088 Td[(! !Lc;(L)<=)]TJ /F3 11.9552 Tf 9.299 0 Td[(iNeq(L)(L)=)]TJ /F3 11.9552 Tf 9.299 0 Td[(iL0Neq(L)h! !Lci2exp2)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(1)]TJ /F3 11.9552 Tf 16.279 8.088 Td[(! !Lc; (R)>=)]TJ /F3 11.9552 Tf 9.298 0 Td[(i1+Neq(R)(R)=)]TJ /F3 11.9552 Tf 9.298 0 Td[(iR01+Neq(R)h! !Rci2exp2)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(1)]TJ /F3 11.9552 Tf 16.613 8.088 Td[(! !Rc;(R)<=)]TJ /F3 11.9552 Tf 9.298 0 Td[(iNeq(R)(R)=)]TJ /F3 11.9552 Tf 9.298 0 Td[(iR0Neq(R)h! !Rci2exp2)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(1)]TJ /F3 11.9552 Tf 16.614 8.088 Td[(! !Rc: D7=Dr7Da;LRM =M+L+R:

PAGE 36

~=B=1!D=TD=2p 2645K 2p 2=1:5!Kc !Kc=2 DR=)]TJ /F2 11.9552 Tf 5.48 -9.683 Td[((D0R))]TJ /F6 7.9701 Tf 6.586 0 Td[(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(R)]TJ /F6 7.9701 Tf 6.586 0 Td[(1DA=(DR)y: 7MR(A)M 2Im)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(R(!)=<(w))]TJ /F2 11.9552 Tf 11.956 0 Td[(>(w);

PAGE 37

Re)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(R(!)=1 P Zd!0Im)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(R(!0) !0)]TJ /F3 11.9552 Tf 11.955 0 Td[(!:!DR(A)D<(>)<(>)R(A)

PAGE 38

(T)Jnet(T)=T(T)TRT !J(!)!JnetJ(!)!2=1:5)]TJ /F1 11.9552 Tf 11.211 0 Td[<0051004d>-326<0069003f0032>-326<00510069003f00320060>-326<003f001c004d002f>-326<002f00320054>-27<0032004d002f0062>-326<0051004d>-326<002f004200620051>26<0060002f00320060002d>-325<001c004d002f>-326<0072>26<0032>-325<002b0051004b004b0032004d0069>]TJ -182.14 -23.908 Td[<0051004d>-325<0051>-1<006d0060>-326<002b003f00510042002b0032>-326<001c0069>-326<0069003f0032>-326<0032004d002f>-326<00510037>-326<00620032002b006900420051004d>]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.9552 Tf 174.725 -552.904 Td[<006a00580052>]TJ ET 0 G 0 g BT /F1 11.9552 Tf 189.693 -552.904 Td[<0058>-434<002a003f001c004d003b0042004d003b>-326<0069003f0032>-326<0070001c0048006d00320062>-326<00510037>-326<0069003f003200620032>-326<0054001c>26<0060001c004b00320069003200600062>-325<002f0051>-326<004d00510069>-326<001c007a0032002b0069>]TJ -189.693 -23.908 Td[<0069003f0032>-326<005b006d001c004800420069001c0069004200700032>-326<001c00620054>-27<0032002b00690062>-326<0069003f001c0069>-326<0072>26<0032>-325<003f00510054>-27<0032>-326<00690051>-326<006d004d002f0032006000620069001c004d002f0058>]TJ ET BT /F1 11.9552 Tf 227.766 -687.422 Td[<006a0033>]TJ ET Q

PAGE 39

J(!)0 0=Z10d!(!)d dTb(!)T=Troom;TL=Troom+TTR=TroomT!00limT!0+Jth=TJth (!)=1Troom=300KTroom=0:465!D02:520!J(!)J0=0TT=1TJ0!2Jnet TW0=0 J(!)=0=TL)]TJ /F3 11.9552 Tf 10.439 0 Td[(TRTLJ(!)J(!)/!3

PAGE 40

J(!)=0!=TL)]TJ /F3 11.9552 Tf 11.956 0 Td[(TRTL=1TL=2

PAGE 41

int=0 J(!)=!TrhD(C)>((L)<)]TJ /F2 11.9552 Tf 11.955 0 Td[((R)<))]TJ /F3 11.9552 Tf 11.955 0 Td[(D(C)<((L)>)]TJ /F2 11.9552 Tf 11.955 0 Td[((R)>)i;(L)7(R)7 !TL;TR Neq(L)(!)=1 ew=TL)]TJ /F2 11.9552 Tf 11.955 0 Td[(1TL !Neq(R)(!)=1 ew=TR)]TJ /F2 11.9552 Tf 11.956 0 Td[(1TR !: !!c e2(1)]TJ /F13 5.9776 Tf 9.595 3.259 Td[(! !c)e2: <)]TJ /F3 11.9552 Tf 38.523 0 Td[(i0e2T !2c!>)]TJ /F3 11.9552 Tf 38.523 0 Td[(i0e2!2 !2c)]TJ /F3 11.9552 Tf 6.675 -1.596 Td[(T !+1: D(C)7 D<(>)=Dr<(>)Da;Dr=)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[((D0r))]TJ /F6 7.9701 Tf 6.586 0 Td[(1)]TJ /F2 11.9552 Tf 11.955 0 Td[(r)]TJ /F6 7.9701 Tf 6.587 0 Td[(1;Da=(Dr)y: D0r=1 !2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!2p;

PAGE 42

!pp<(>)R 2Imr=<)]TJ /F2 11.9552 Tf 11.955 0 Td[(>: Rer=H(Imr): D7/!7/! Imr/!2: J(!)/!3 1=(!)(R(!))T=1 TW0=1 =p lcd h;hlcd

PAGE 43

1=!=1:5 h=dlcd=64nm=11:3 h=2:3lc=8:9d=77:5=11:4 TLTL

PAGE 44

J(!)=0!=11:3=TL)]TJ /F3 11.9552 Tf 11.955 0 Td[(TRTL=1TL=2

PAGE 45

T T=1TL=1 T=1T=2TL=2 hlcd d=64nmhlc J(!)=0hlc

PAGE 46

J(!)=0!TL=1T=1 )]TJ /F1 11.9552 Tf 11.211 0 Td[<0023>26<0076>-325<006d00620042004d003b>]TJ /F2 11.9552 Tf 43.924 0 Td[()-321(=)]TJ /F6 7.9701 Tf 31.417 -1.793 Td[(0)]TJ /F6 7.9701 Tf 7.314 -1.793 Td[(0)]TJ /F1 11.9552 Tf -449.044 -23.908 Td[<0051004d>-326<0062006d00600037001c002b0032>-326<002f004200620051>26<0060002f00320060>-325<00420062>-326<002b003f005100620032004d>-326<00690051>-326<00600032007e0032002b0069>-326<0069003f001c0069>-326<0048001c>26<0060003b00320060>-325<002f004200620051>26<0060002f00320060>-325<00550062004b001c0048004800320060>]TJ /F3 11.9552 Tf 330.489 0 Td[()]TJ /F1 11.9552 Tf 7.315 0 Td[<0058>]TJ ET BT /F8 11.9552 Tf 183.299 -527.376 Td[<006a0058006b>-999<0068>91<00510069001c0048>-366<002b006d006000600032004d0069>]TJ /F1 11.9552 Tf -159.915 -24.439 Td[<0068003f0032>-326<004d00320069>-326<0069003f00320060004b001c0048>-326<002b006d006000600032004d0069>-326<00420062>-326<005100230069001c0042004d0032002f>-326<0023>26<0076>-325<0042004d00690032003b0060001c00690042004d003b>]TJ /F3 11.9552 Tf 249.816 0 Td[(J(!)!=0!=!D TRT=1T=2T!0

PAGE 47

J(!)=0!TL=2T=1T=2

PAGE 48

J(!)=0!TL=1T=1hlc Jnet=TT0Jnet=(0T)

PAGE 49

T=1T=2TT (T)Jnet(T)=TTRTTT

PAGE 50

Jnet=(0T)TRT=1T=2

PAGE 51

Jnet=(0T)TR=11:3T

PAGE 54

b;byQ;QQ;Q H=Xq!qbyqbq+1 2 [bq;byq0]=qq0;[byq;byq0]=[bq;bq0]=0 bq(t)=eiHC(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)bq(t0)e)]TJ /F5 7.9701 Tf 6.586 0 Td[(iHC(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0) _bq(t)=eiHC(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)ihHC;bq(t0)ie)]TJ /F5 7.9701 Tf 6.587 0 Td[(iHC(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)=)]TJ /F3 11.9552 Tf 11.955 0 Td[(i!qbq(t) bq(t)=e)]TJ /F5 7.9701 Tf 6.586 0 Td[(i!q(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)bq(t0);byq(t)=ei!q(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)byq(t0) a;ayD>qq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)=)]TJ /F3 11.9552 Tf 11.955 0 Td[(iDbq(t)byq(t0)E=)]TJ /F3 11.9552 Tf 11.955 0 Td[(i1+Neqe)]TJ /F5 7.9701 Tf 6.587 0 Td[(i!q(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0);

PAGE 55

Neq=byq(t)bq(t)Dqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0))]TJ /F3 11.9552 Tf 11.955 0 Td[(Dqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)=)]TJ /F3 11.9552 Tf 11.955 0 Td[(i(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)e)]TJ /F5 7.9701 Tf 6.586 0 Td[(i!q(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0): Q;Q Qq=s 1 2!qbyq+bq:D>qq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)=)]TJ /F3 11.9552 Tf 11.955 0 Td[(iDQq(t)Qq(t0)E=)]TJ /F3 11.9552 Tf 20.008 8.088 Td[(i 2!q)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[(1+Neqe)]TJ /F5 7.9701 Tf 6.586 0 Td[(i!q(t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)+Neqei!q(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0): D
PAGE 56

DRqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)=(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)D>qq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0))]TJ /F3 11.9552 Tf 11.955 0 Td[(Dqq(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)=)]TJ /F3 11.9552 Tf 20.008 8.088 Td[(i 2!q(t0)]TJ /F3 11.9552 Tf 11.955 0 Td[(t)ei!q(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0))]TJ /F3 11.9552 Tf 11.955 0 Td[(e)]TJ /F5 7.9701 Tf 6.587 0 Td[(i!q(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0): a;ayD>qq(!)=Zd(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)ei!(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0)qq>(t)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)=)]TJ /F3 11.9552 Tf 11.956 0 Td[(i1+Neq2(!)]TJ /F3 11.9552 Tf 11.955 0 Td[(!q): D
PAGE 57

Q;Q D>qq(!)=)]TJ /F5 7.9701 Tf 15.865 4.707 Td[(i 2!q)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[(2)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(Neq+1(!)]TJ /F3 11.9552 Tf 11.956 0 Td[(!q)+2Neq(!+!q);D
PAGE 58

HHR264HyHR375HRH264DDDRDRDDR375=264EI)]TJ /F3 11.9552 Tf 11.956 0 Td[(H)]TJ /F3 11.9552 Tf 9.298 0 Td[()]TJ /F3 11.9552 Tf 9.298 0 Td[(yEI)]TJ /F3 11.9552 Tf 11.956 0 Td[(HR375)]TJ /F6 7.9701 Tf 6.587 0 Td[(1:D=EI)]TJ /F3 11.9552 Tf 11.955 0 Td[(H)]TJ /F2 11.9552 Tf 11.955 0 Td[()]TJ /F6 7.9701 Tf 6.587 0 Td[(1=gRygR=EI)]TJ /F3 11.9552 Tf 11.955 0 Td[(H)]TJ /F3 11.9552 Tf 11.955 0 Td[(HR)]TJ /F6 7.9701 Tf 6.586 0 Td[(1:HH+=H++y 2+)]TJ /F2 11.9552 Tf 11.955 0 Td[(y 2=H0)]TJ /F3 11.9552 Tf 11.955 0 Td[(i=2H0H++y 2;i)]TJ /F2 11.9552 Tf 11.955 0 Td[(y:

PAGE 59

HH0H0H0+0)]TJ /F3 11.9552 Tf 11.955 0 Td[(i)]TJ /F1 11.9552 Tf -244.894 -35.863 Td[<0072003f003200600032>]TJ /F2 11.9552 Tf 32.243 0 Td[(0)]TJ /F1 11.9552 Tf 11.212 0 Td[<001c>27<00600032>-325<0069003f0032>-326<002b0051>26<00600060003200620054>-26<0051004d002f0042004d003b>-326<002f>-1<0042001c003b0051004d001c0048>-326<003200480032004b0032004d00690062>-326<00510037>]TJ /F3 11.9552 Tf 212.646 0 Td[(H0H!0)]TJ /F1 11.9552 Tf 11.212 0 Td[<002b001c004d>-325<0023>-28<0032>-326<0070004200320072>26<0032002f>-325<001c0062>-326<0023>26<00600051001c002f0032004d0042004d003b>-325<00510037>]TJ -310.302 -23.908 Td[<0032004d00320060003b0076>-326<00480032007000320048>-326<003700600051004b>-326<001c>-326<002f003200480069001c>-326<0037006d004d002b006900420051004d>]TJ /F3 11.9552 Tf 164.025 0 Td[((!)]TJ /F2 11.9552 Tf 12.027 0 Td[(0))]TJ /F1 11.9552 Tf 11.212 0 Td[<0056>]TJ /F2 11.9552 Tf 203.799 -15.82 Td[()]TJ ET q .4782 w 196.455 -271.955 m 267.541 -271.955 l S Q BT /F2 11.9552 Tf 196.455 -283.144 Td[((!)]TJ /F2 11.9552 Tf 11.956 0 Td[()2+)]TJ /F6 7.9701 Tf 19.076 3.453 Td[(2:1=)]TJ /F1 11.9552 Tf 7.314 0 Td[<0058>]TJ ET BT /F1 11.9552 Tf 227.766 -687.422 Td[<0038004e>]TJ ET Q