=-$:,6.$.217$.72:(
3RáF]HQLD PHWDO±SyáSU]HZRGQLN OXE SyáSU]HZRGQLN±SyáSU]H
ZRGQLN Z\WZDU]DM REV]DU\ SU]HMFLRZH ]áF]D JG]LH NRQFHQWUDFMD
QRQLNyZ RGELHJD RG NRQFHQWUDFML Z REM WRFL PDWHULDáX :áDFLZRFL
W\FK REV]DUyZ ]PLHQLDM VL SRQDGWR SR SU]\áR*HQLX QDSL FLD
]HZQ WU]QHJR &KDUDNWHU\VW\NL SUGRZRQDSL FLRZH ]áF]\ V QD RJyá
QLHOLQLRZH 1LHOLQLRZRFL WH SRZRGXM *H ]áF]D V SRGVWDZRZ\P
HOHPHQWHPVNáDGRZ\PZL NV]RFLSU]\U]GyZSyáSU]HZRGQLNRZ\FK
3RM FLHP LVWRWQ\P GOD ]UR]XPLHQLD ]MDZLVN NRQWDNWRZ\FK MHVW
SUDFD Z\MFLD HOHNWURQyZ : SU]\SDGNX PHWDOX HOHNWURQ\ VZRERGQH
]QDMGXMFHVL ZVLHFLNU\VWDOLF]QHMGRGDWQLFKMRQyZSRVLDGDMXMHPQ
HQHUJL SRWHQFMDOQ áDGXQHN HOHNWURQX MHVW XMHPQ\ -H*HOL HQHUJL
HOHNWURQX Z SUy*QL SU]\MPLHP\ ]D ]HURZ WR PHWDO VWDQRZL GOD
HOHNWURQyZVWXGQL SRWHQFMDOQU\V
m
q
-
0
0
T
>
0
=
T
W
n(W)
(b)
(a)
W
F
5\V(OHNWURQ\VZRERGQHZPHWDOXREVDG]DMFHSR]LRP\HQHUJHW\F]QHZVWXGQL
SRWHQFMDOQHMDRUD]NU]\ZHUR]NáDGXHOHNWURQyZZIXQNFMLHQHUJLLQ:GODGZyFK
Uy*Q\FKWHPSHUDWXUE:
)
MHVWHQHUJL)HUPLHJRPDNV\PDOQDHQHUJLDHOHNWURQyZ
ZPHWDOXZWHPSHUDWXU]H7 .T
Φ
P
R]QDF]DWHUPRG\QDPLF]QSUDF Z\MFLD
$E\SU]HQLHüHOHNWURQ]PHWDOXGRSUy*QLWU]HEDZ\NRQDüSUDF
-HVW RQD PLQLPDOQD ZWHG\ JG\ SU]HQRV]RQ\ HOHNWURQ SRFKRG]L
]SR]LRPX )HUPLHJR L QRVL ZyZF]DV QD]Z WHUPRG\QDPLF]QHM SUDF\
Z\MFLDT
Φ
P
:WHPSHUDWXU]H7!LVWQLHMHOHNWURQ\NWyU\FKHQHUJLH
SU]HNUDF]DM SR]LRP ]HURZ\ U\VE ZWHG\ SHZQD LFK F] ü MHVW
]GROQDRSXFLüPHWDO5LFKDUGVRQSRGDá]DOH*QRüRNUHODMFZLHONRü
SUGXWHUPRHOHNWURQRZHJRMNWyU\]ZL]DQ\MHVW]WHPSHUDWXUPHWDOX7
RUD]SUDFZ\MFLDHOHNWURQyZ
M $7
T
N7
P
=
−
H[S
Φ
JG]LH$MHVWW]ZVWDá5LFKDUGVRQD
3RGQLHVLHQLHWHPSHUDWXU\OXE ]PLDQD SUDF\ Z\MFLD QS SRSU]H]
QDáR*HQLHWOHQNXQDSRZLHU]FKQL PHWDOXSRZRGXMHJZDáWRZQ\Z]URVW
SUGX WHUPRHOHNWURQRZHJR JG\* RELH WH ZLHONRFL Z\VW SXM Z
Z\NáDGQLNX
(OHNWURQ\RSXV]F]DMFHSyáSU]HZRGQLNPRJSRFKRG]Lü]DUyZQR
]SDVPDSU]HZRGQLFWZDMDNUyZQLH*]SDVPDZDOHQF\MQHJRF]\WH*]H
VWDQyZGRPLHV]NRZ\FKU\V
7HUPRG\QDPLF]QSUDF Z\MFLDGODSyáSU]HZRGQLNDRNUHODVL
DQDORJLF]QLH MDN GOD PHWDOX MDNR Uy*QLF HQHUJLL PL G]\ SR]LRPHP
]HURZ\P L SR]LRPHP )HUPLHJR ERZLHP W\ONR WDNLH SU]HMFLH QLH
SRZRGXMH]PLDQ\WHPSHUDWXU\SyáSU]HZRGQLND
0
C
W
F
W
d
W
V
W
p
q
-
n(W)
5\V(PLVMDHOHNWURQyZ]SyáSU]HZRGQLND
: ]MDZLVNDFK NRQWDNWRZ\FK V]F]HJyOQLH GOD SyáSU]HZRGQLNyZ
LVWRWQURO PR*HRGJU\ZDüREHFQRüVWDQyZSRZLHU]FKQLRZ\FK6WDQ\
WDNLH ]ZL]DQH ] QLHFLJáRFL VWUXNWXU\ NU\VWDOLF]QHM GHIHNWDPL VLHFL
MDNUyZQLH*FKHPLF]QLH]ZL]DQ\PLREF\PLDWRPDPLF]\F]VWHF]NDPL
VWDQRZLGRGDWNRZHFHQWUDZ\FKZ\WXMFHOXEXZDOQLDMFHHOHNWURQ\7H
GRGDWNRZH FHQWUD áDGXQNRZH Z\ZLHUDM ZSá\Z QD ZDUVWZ
SU]\SRZLHU]FKQLRZ SU]\FLJDMF áDGXQNL SU]HFLZQHJR ]QDNX
LSRZRGXMFMHMZ]ERJDFHQLHOXE]XER*HQLHZQRQLNL:]DOH*QRFLRG
URG]DMX FHQWUyZ áDGXQNRZ\FK L W\SX SyáSU]HZRGQLND PRJ SRZVWDü
Uy*QHURG]DMHREV]DUyZSU]\SRZLHU]FKQLRZ\FK3U]\NáDGRZRMH*HOLQD
SRZLHU]FKQL LVWQLHM VWDQ\ DNFHSWRURSRGREQH áDGXQHN XMHPQ\
DSyáSU]HZRGQLN MHVW W\SX Q WR SRZVWDMH SU]\ SRZLHU]FKQL REV]DU
]XER*RQ\ GRGDWQL áDGXQHN SU]HVWU]HQQ\ ] QLHGRERUHP HOHNWURQyZ
VZRERGQ\FK3U]\GX*HMJ VWRFLVWDQyZSRZLHU]FKQLRZ\FK1
66
NWRUH
SRREVDG]HQLXZ\ND]XMáDGXQHNSRZLHU]FKQLRZ\
4
T1
66
66
= −
ZDUVWZD
SU]\SRZLHU]FKQLRZD PR*H ]PLHQLü W\S SU]HZRGQLFWZD FR QD]\ZDP\
LQZHUVM SyáSU]HZRGQLND U\V =DNU]\ZLHQLH SDVP HQHUJHW\F]Q\FK
NX JyU]H ]ZL]DQH MHVW ]H Z]URVWHP HQHUJLL HOHNWURQyZ
SU]HPLHV]F]DMF\FKVL ]SyáSU]HZRGQLNDNXSRZLHU]FKQL
2%6=$5
,1:(56<-1<
C
W
V
W
F
W
4
66
1
66
G]LXU\
5\V,QZHUVMDW\SXSU]HZRGQLFWZD
ZREV]DU]HSU]\SRZLHU]FKQLRZ\PSRG
ZSá\ZHP áDGXQNXSRZLHU]FKQLRZHJR
4
66
-H*HOL 1
66
Z SyáSU]HZRGQLNX SU]HNUDF]D
FP
WR
RZáDFLZRFLDFK ]áF]D QLH GHF\GXM SUDFH Z\MFLD DOH áDGXQHN
SRZLHU]FKQLRZ\ :VSyáF]HVQH WHFKQRORJLH Z\WZDU]DQLD VWUXNWXU
SyáSU]HZRGQLNRZ\FK QS WHFKQRORJLD SODQDUQD VNXWHF]QLH UR]ZL]Dá\
SUREOHPVWDQyZSRZLHU]FKQLRZ\FKJáyZQLHSRSU]H]VWRVRZDQLHZDUVWZ
SDV\ZXMF\FK W\SRZD ZDUVWZD SDV\ZXMFD 6L2
SRZVWDMH áDWZR
ZZ\QLNXWHUPLF]QHJRXWOHQLDQLDSRZLHU]FKQL6L
.217$.70(7$/3Ïà35=(:2'1,.2 5
'RSURZDG]HQLH GR EH]SRUHGQLHJR NRQWDNWX PHWDOX ]
SyáSU]HZRGQLNLHP SRZRGXMH Z\PLDQ HOHNWURQyZ PL G]\ REX
PDWHULDáDPL D* GR XVWDOHQLD VL UyZQRZDJL WHUPRG\QDPLF]QHM
=DNáDGDMF *H SUDFD Z\MFLD T
Φ
P
HOHNWURQX ] PHWDOX MHVW ZL NV]D QL*
T
Φ
S
GOD SyáSU]HZRGQLND W\SX Q X]\VNXMH VL VWDQ UyZQRZDJL ] W]Z
REV]DUHP]XER*RQ\PSRVWURQLHSyáSU]HZRGQLNDU\V
p
q
-
m
q
-
d
T
'
Φ
5\V6WDQUyZQRZDJLSU]\
EH]SRUHGQLPNRQWDNFLHPHWDOXL
SyáSU]HZRGQLNDW\SXQGMHVWJUXERFL
ZDUVWZ\]XER*RQHMT
Φ
'
R]QDF]D
Z\VRNRüEDULHU\SRWHQFMDáX
3RQLHZD*
Φ
S
Φ
P
ZL F VWUXPLH HOHNWURQyZ ] SyáSU]HZRGQLND GR
PHWDOXSU]HZD*DSRF]WNRZRQDGVWUXPLHQLHPHOHNWURQyZ]PHWDOXGR
SyáSU]HZRGQLND : Z\QLNX WHJR PHWDO ]DF]QLH VL áDGRZDü XMHPQLH
SR]LRP)HUPLHJR:
)P
SRGQRVLVL DSyáSU]HZRGQLNGRGDWQLRSR]LRP
)HUPLHJR:
)S
REQL*DVL 3RMDZLDVL EDULHUDGODHOHNWURQyZSá\QF\FK
] SyáSU]HZRGQLND GR PHWDOX L LFK VWUXPLH PDOHMH *G\ SR]LRP\ :
)
Z\UyZQDMVL QDVW SXMHVWDQUyZQRZDJLDEDULHUDSRWHQFMDáXRVLJQLH
ZDUWRü
U
D
m
n
=
−
Φ
Φ
= WHJR Z]JO GX *H NRQFHQWUDFMD QRQLNyZ VZRERGQ\FK Z
SyáSU]HZRGQLNXMHVW]QDF]QLHPQLHMV]DQL*ZPHWDOXDáDGXQNLSRREX
VWURQDFK ]áF]D PXV] E\ü UyZQH JUXERü áDGXQNX SU]HVWU]HQQHJR Z
SyáSU]HZRGQLNXMHVW]QDF]QLHZL NV]DQL*ZPHWDOXSU]\F]\PJUXERü
ZDUVWZ\ HOHNWURQyZ MHVW U] GX NLONX VWDá\FK VLHFLRZ\FK 'OD V\WXDFML
SRND]DQHM QD U\V SR VWURQLH SyáSU]HZRGQLND SRMDZLD VL ZDUVWZD
QLHUXFKRPHJRáDGXQNX]MRQL]RZDQ\FKGRQRUyZSR]EDZLRQDHOHNWURQyZ
SU]HZRGQLFWZD]XER*RQD*UXERüWHMZDUVWZ\Z\QRVL
G
1 T
'
G
=
ε ε Φ
JG]LH
Φ
'
MHVWRNUHORQHZ]RUHP1
'
MHVWNRQFHQWUDFMGRQRUyZ
ε
ε
R]QDF]DSU]HQLNDOQRüHOHNWU\F]QSyáSU]HZRGQLND
3U]\NáDG 'OD NU]HPX R NRQFHQWUDFML GRQRUyZ 1
G
FP
L
Z\VRNRFL EDULHU\
Φ
'
9
ε
RWU]\PXMH VL ] JUXERü
REV]DUX]XER*RQHJRG
µ
P
2PDZLDQDEDULHUDWZRU]FDVL QDVW\NXPHWDOSyáSU]HZRGQLNQD]\ZDQD
MHVWEDULHU6FKRWWN\CHJR
Φ
'
3U]\áR*HQLH QDSL FLD ]HZQ WU]QHJR GR ]áF]D PS SRZRGXMH
Z]URVW OXE REQL*HQLH EDULHU\ Z ]DOH*QRFL RG NLHUXQNX QDSL FLD 3R
SU]\áR*HQLX QDSL FLD Z NLHUXQNX SU]HZRG]HQLD SRWHQFMDá XMHPQ\ SR
VWURQLH SyáSU]HZRGQLND W\SX Q ZV]\VWNLH SR]LRP\ SyáSU]HZRGQLND
SRGQRV]VL RT8LRWDNVDPZDURüPDOHMHEDULHUDSRWHQFMDáX3R
SU]\áR*HQLXQDSL FLDZNLHUXQNX]DSRURZ\PEDULHUDSRWHQFMDáXURQLHR
T8LURQLHWH*JUXERüZDUVWZ\]DSRURZHM
(
)
G
8
T1
'
G
=
+
ε ε Φ
=PLDQ\ Z\VRNRFL EDULHU\ VSRZRGRZDQH SU]\áR*HQLHP QDSL FLD
]HZQ WU]QHJR PDM ZSá\Z QD ZLHONRü Sá\QFHJR SU]H] ]áF]H SUGX
3U]\EUDNXQDSL FLDSUGM
P
Sá\QF\]PHWDOXGRSyáSU]HZRGQLNDLSUG
j
p
Sá\QF\]SyáSU]HZRGQLNDGRPHWDOXNRPSHQVXMVL (
)
j
j
j
m
p
s
=
=
3U]\áR*HQLH QDSL FLD Z NLHUXQNX SU]HZRG]HQLD L ]PQLHMV]HQLH EDULHU\
RT8 VSRZRGXMH *H ]JRGQLH ]H VWDW\VW\N %ROW]PDQQD M
S
Z]URQLH
H[ST8N7 UD]\ 3UG M
P
QLH ]PLHQL VL L Z ]ZL]NX ] W\P SUG
Z\SDGNRZ\E G]LHZ\QRVLá
M M
M
M
T8 N7
M
M
T8 N7
S
P
V
V
V
=
−
=
− =
−
H[S>
@
H[S>
@
3R SU]\áR*HQLX QDSL FLD Z NLHUXQNX ]DSRURZ\P WU]HED ]PLHQLü ]QDN
SU]\ 8 Z Z\UD*HQLX L ZWHG\ RWU]\PXMH VL MX* GOD VWRVXQNRZR
QLHGX*\FK QDSL ü QDV\FDMF VL ZDUWRü SUGX M M
V
]ZDQ SUGHP
ZVWHF]Q\P3RQLHZD*SUGZNLHUXQNXSU]HZRG]HQLDMHVWGX*RZL NV]\
RG SUGX Z NLHUXQNX ]DSRURZ\P RWU]\PXMH VL NRQWDNW PS
RZáDFLZRFLDFKSURVWXMF\FKU\V
j
U
S
j
5\V3URVWRZDQLHQD]áF]XPHWDO
SyáSU]HZRGQLNGOD]áF]DP6LQDSL FLH
ZNLHUXQNXSU]HZRG]HQLDQLHSU]HNUDF]D
QDRJyáZDUWRFL9
'RNáDGQD WHRULD SU]HSá\ZX SUGX SU]H] EDULHU PS XZ]JO GQLD ZLHOH
F]\QQLNyZ WDNLFK MDN J VWRü VWDQyZ SRZLHU]FKQLRZ\FK UR]PLDU
EDULHU\ URG]DM UXFKX QRQLNyZ Z REV]DU]H EDULHU\ =QDQH WHRULH
GLRGRZD %HWKHJR G\IX]\MQD 6FKRWWN\CHJR GDM Z\UD*HQLD ]EOL*RQH
NV]WDáWHPGRDUy*QLFHSRMDZLDMVL ZZ\UD*HQLDFKQDM
V
=à&=(5 3
=áF]HP SQ QD]\ZDP\ REV]DU SyáSU]HZRGQLND Z NWyU\P
QDVW SXMH]PLDQDSU]HZRGQLFWZD]W\SXSQDW\SQ6DPDSRZLHU]FKQLD
JUDQLF]QD UR]G]LHODMFD SyáSU]HZRGQLNL R RGPLHQQ\FK W\SDFK
SU]HZRGQLFWZDQD]\ZDVL ]áF]HPWHFKQRORJLF]Q\P3RREXVWURQDFK
]áF]D WHFKQRORJLF]QHJR ]QDMGXMH VL ZDUVWZD SU]HMFLRZD E GFD
GLSRORZZDUVWZQLHUXFKRPHJRáDGXQNX
:áDFLZRFL ]áF]D ]DOH* RG URG]DMX X*\WHJR SyáSU]HZRGQLND
KRPR]áF]H OXE URG]DMX REX Uy*Q\FK SyáSU]HZRGQLNyZ WZRU]F\FK
W]ZKHWHUR]áF]HW\SXGRPLHV]HNLVSRVREXLFKUR]NáDGX
5R]NáDG GRPLHV]NL XZDUXQNRZDQ\ MHVW Z ]QDF]QHM PLHU]H
URG]DMHP WHFKQRORJLL X*\WHM GR Z\WZRU]HQLD ]áF]D =áF]D
ZVSyáF]HQLHZ\WZDU]DQHPR*QDSRG]LHOLüQDVW
SXMFR
=áF]HG\IX]\MQH
=áF]D WHJR W\SX Z\WZDU]DQH V PHWRG G\IX]ML SLHUZLDVWND
VSHáQLDMFHJRURO GRPLHV]NLGRQRURZHMGRSyáSU]HZRGQLNDW\SXSOXE
DNFHSWRURZHMGRSyáSU]HZRGQLNDW\SXQ'\IX]M SU]HSURZDG]DVL Z
SRGZ\*V]RQHM WHPSHUDWXU]H U] GX
R
& GOD 6L D GRPLHV]ND PR*H
E\üZSRVWDFLSDU\OXEFLHF]\QDQLHVLRQHMQDSRZLHU]FKQL'\IX]M ERUX
GR NU]HPX SU]H] RNQR Z\WUDZLRQH Z ZDUVWZLH 6L2
VSHáQLDMFHM URO
PDVNLSU]HGVWDZLDU\VD
p-Si
n-Si
2
SiO
B
(a)
p
n
(b)
j
x
d
N
(x)
N
a
)
0
(
N
a
x
5\VD'\IX]MDERUXGRNU]HPXZZ\EUDQ\PREV]DU]HSyáSU]HZRGQLNDRUD]E
SURILOUR]NáDGXWHMGRPLHV]NL
5R]NáDG ZSURZDG]RQHM GRPLHV]NL Z\QLND ] SUDZ G\IX]ML L ]DOH*\
UyZQLH*RGWHJRF]\Z\GDMQRü(UyGáDZWUDNFLHSURFHVXE\áDVWDáDF]\
PDOHMFDZF]DVLH3URILOUR]NáDGXGRPLHV]NLDNFHSWRURZHMGOD(UyGáDR
QLHRJUDQLF]RQHM Z\GDMQRFL ZG\IXQGRZDQHM GR SyáSU]HZRGQLND
]DZLHUDMFHJR GRPLHV]N GRQRURZ SU]HGVWDZLD U\VE .V]WDáW
NU]\ZHM 1
D
[ RNUHORQ\ MHVW W]Z IXQNFM Eá GX :VSyáU] GQD ]áF]D
WHFKQRORJLF]QHJR[
M
Z\]QDF]RQDMHVWSU]H]UyZQDQLH
1 [
1
D
M
G
−
=
:SREOL*X[
M
SURILOUR]NáDGXPR*QDDSURNV\PRZDü]DOH*QRFLOLQLRZ
1 [
1
1
D[
D
G
=
−
≈ −
L]áF]HWDNLHQRVLQD]Z
OLQLRZHJR]áF]DSQ
=áF]HLPSODQWRZDQH
=áF]D WHJR W\SX Z\WZDU]D VL Z Z\QLNX ERPEDUGRZDQLD
NU\V]WDáX MRQDPL GRPLHV]HN UR]S G]RQ\FK GR HQHUJLL U] GX VHWHN NH9
>@ =PLHQLDMF HQHUJL ERPEDUGXMF\FK MRQyZ PR*QD NV]WDáWRZDü
SURILOGRPLHV]NRZDQLDU\V
a
N
N
d
N
p
n
x
j
x
5\V5R]NáDGNRQFHQWUDFML
GRPLHV]NLGRQRURZHMZSURZDG]RQHM
GRSyáSU]HZRGQLNDPHWRG
ZLHORNURWQHMLPSODQWDFMLMRQyZR
Uy*Q\FKHQHUJLDFK
,QQ\PL ]DOHWDPL PHWRG\ LPSODQWDFML V PR*OLZRü Z\ERUX GRZROQHM
GRPLHV]NL PR*OLZRü SURZDG]HQLD SURFHVX Z QLVNLFK WHPSHUDWXUDFK
RUD] áDWZRü MHJR DXWRPDW\]DFML =áF]H SU]HGVWDZLRQH QD U\V MHVW
SU]\NáDGHPW]Z]áF]DVNRNRZHJRDQJDEUXSW
=áF]HHSLWDNVMDOQH
: WUDNFLH QDQRV]HQLD ZDUVWZ\ HSLWDNVMDOQHM SyáSU]HZRGQLND QD
]RULHQWRZDQH SRGáR*H PR*QD ZSURZDG]Lü GRGDWNRZR GR DWPRVIHU\
JD]RZHM SDU\ ]ZL]NX NWyU\ Z\WZRU]\ GRPLHV]NL Z SLHUZLDVWNX
PDFLHU]\VW\PZZ\QLNXUHDNFMLFKHPLF]Q\FK:WHQVSRVyEáDWZRMHVW
Z\WZRU]\ü]áF]HW\SXVNRNRZHJRU\V
a
N
d
N
d
a
-N
N
j
x
x
5\V3U]\NáDG]áF]DVNRNRZHJR
RWU]\PDQHJRPHWRGHSLWDNVML
6WDQUyZQRZDJRZ\]áF]D
%H]SRUHGQLNRQWDNW REV]DUyZ Q RUD] S SURZDG]L GR Z]DMHPQHM
G\IX]MLQRQLNyZHOHNWURQyZGRREV]DUXW\SXSLG]LXUGRREV]DUXW\SX
QQDVNXWHNLVWQLHMF\FKJUDGLHQWyZNRQFHQWUDFML:WHQVSRVyESRREX
VWURQDFK ]áF]D WHFKQRORJLF]QHJR SRZVWDQ ZDUVWZ\ QLHUXFKRPHJR
áDGXQNXSU]HVWU]HQQHJRWZRU]FHZDUVWZ GLSRORZ:SyáSU]HZRGQLNX
W\SX S SRZVWDQLH ZDUVWZD áDGXQNX XMHPQHJR ]MRQL]RZDQ\FK
DNFHSWRUyZ D Z SyáSU]HZRGQLNX W\SX Q ZDUVWZD áDGXQNX GRGDWQLHJR
]MRQL]RZDQ\FKGRQRUyZU\V
a
-qN
d
qN
n
d
q(x)
x
p
d
5\V:DUVWZDGLSRORZDáDGXQNX
SU]HVWU]HQQHJRGOD]áF]DVNRNRZHJRS
QZREV]DU]HG
S
G
Q
EUDNMHVWáDGXQNyZ
VZRERGQ\FKREV]DU]XER*RQ\
&DáNRZLW\áDGXQHNZDUVWZ\GLSRORZHMPXVLE\üMHGQDNUyZQ\]HURFR
GOD]áF]DSU]HGVWDZLRQHJRQDU\VGDMH
qd N
qd N
n
d
p
a
−
=
0
5yZQDQLH RNUHOD Já ERNRü ZQLNDQLD ZDUVWZ\ áDGXQNX
SU]HVWU]HQQHJRZREV]DUSyáSU]HZRGQLND:QLNDQLHWRMHVWW\PZL NV]H
LPPQLHMV]DMHVWNRQFHQWUDFMDGRPLHV]NLZGDQ\PREV]DU]H
àDGXQHN SU]HVWU]HQQ\ ]JRGQLH ] SUDZDPL HOHNWURVWDW\NL MHVW
SU]\F]\Q SRZVWDQLD SROD HOHNWU\F]QHJR ] RGSRZLHGQLP UR]NáDGHP
SRWHQFMDáX FR PR*QD Z\]QDF]\ü ] SUDZD 3RLVVRQD 'OD SU]\SDGNX
MHGQRZ\PLDURZHJRSUDZRWRPDSRVWDü
G 9
G[
= − ρ
ε ε
(
9
= −
JUDG
D
JG]LH
ρ
R]QDF]DJ VWRüáDGXQNXSU]HVWU]HQQHJR
E
MHVWQDW *HQLHPSROD
HOHNWU\F]QHJR D 9SRWHQFMDáHP 'OD ]áF]D VNRNRZHJR UR]NáDG
áDGXQNXMDNQDU\VUR]ZL]DQLHUyZQDLDGDMHSU]HELHJL
SU]HGVWDZLRQHQDU\V
:EXGRZDQH SROH HOHNWU\F]QH Z\WZDU]D SUG\ XQRV]HQLD
HOHNWURQyZ L G]LXU NWyUH V VNLHURZDQH SU]HFLZQLH GR SUGyZ
G\IX]\MQ\FK : VWDQLH UyZQRZDJL SUG\ G\IX]\MQH L XQRV]HQLD
UyZQRZD*VL
= DQDOL]\ SUGyZ REX URG]DMyZ PR*QD Z\OLF]\ü Z\VRNRü EDULHU\
SRWHQFMDáX
ψ
=
=
N7
T
Q
Q
N7
T
S
S
Q
S
Q
S
OQ
OQ
JG]LH ZVND(QLN GROQ\ R]QDF]D NRQFHQWUDFM QRQLNyZ Z PDWHULDOH
RGSRZLHGQLHJR W\SX Z VWDQLH UyZQRZDJL WHUPRG\QDPLF]QHM
=UyZQDQLDZ\QLND*HEDULHUDMHVWW\PZ\*V]DLPZL NV]DMHVW
Uy*QLFD PL G]\ NRQFHQWUDFMDPL QRQLNyZ GDQHJR ]QDNX Z REX
REV]DUDFK
max
E
E(x)
0
p
d
n
d
x
V(x)
(a)
(b)
k
V
p
V
x
x
j
ψ
=
−
9 9
N
S
5\V3ROHHOHNWU\F]QH
ÄZEXGRZDQH´DLSRWHQFMDáE
Z\WZRU]RQHZVNRNRZ\P]áF]X
SQZZ\QLNXLVWQLHQLDáDGXQNX
GLSRORZHJR
ψ
R
MHVWEDULHU
SRWHQFMDáX
3U]\NáDGREOLF]HQLDZ\VRNRFLEDULHU\
0DWHULDáZ\MFLRZ\6L'ODNRQFHQWUDFMLGRQRUyZ1
G
⋅
FP
RWU]\PXMHVL NRQFHQWUDFM HOHNWURQyZQ
Q
RSRGREQHMZDUWRFL
'OD NRQFHQWUDFML DNFHSWRUyZ 1
D
⋅
FP
RWU]\PXMH VL
NRQFHQWUDFM G]LXUS
S
RSRGREQHMZDUWRFL
:WHPSHUDWXU]H.NRQFHQWUDFMDQRQLNyZVDPRLVWQ\FKZ\QRVL
Q
L
⋅
FP
=SUDZDG]LDáDQLDPDVGOD6LW\SXQRWU]\PXMHVL ]DOH*QRüQ
Q
⋅
S
Q
Q
L
] F]HJR PR*QD Z\OLF]\ü NRQFHQWUDFM G]LXU QRQLNL
PQLHMV]RFLRZHZW\PREV]DU]H
S
Q
Q
Q
L
Q
=
=
⋅
⋅
=
⋅
−
FP
$QDORJLF]QLHGOD6LW\SXSNRQFHQWUDFMDQRQLNyZPQLHMV]RFLRZ\FK
Z\QRVL
Q
Q
S
S
L
S
=
=
⋅
−
FP
3RXWZRU]HQLX]áF]DZREV]DU]H]XER*RQ\PPDP\SUDNW\F]QLHGR
F]\QLHQLD]VDPRLVWQNRQFHQWUDFMQRQLNyZa
FP
ZREHF
NRQFHQWUDFML
⋅
FP
ZREV]DUDFKSR]DEDULHURZ\FKQRUD]S
:\VRNRü EDULHU\ GOD GDQHJR ]áF]D Z\OLF]RQD ] UyZQDQLD
Z\QRVL>@
ψ
=
=
⋅
⋅
=
N7
T
Q
Q
Q
S
OQ
OQ
9
5R]PLDU SU]HVWU]HQQ\ EDULHU\ PR*QD Z\OLF]\ü ]H ]PLDQ
SRWHQFMDáX Z ]áF]X U\VE L ZDUXQNX QHXWUDOQRFL HOHNWU\F]QHM
:Z\QLNXRWU]\PXMHVL
G G
G
1
1
T1 1
8
S
Q
D
G
D
G
=
+
=
+
−
ε ε
ψ
JG]LH8MHVWSU]\áR*RQ\PQDSL FLHP]HZQ WU]Q\P=DWHPGODQDSL FLD
]DSRURZHJR XMHPQHJR V]HURNRü EDULHU\ SRWHQFMDáX URQLH 5RQLH
UyZQLH*Z\VRNRüEDULHU\GRZDUWRFL
ψ
+
8
'ODQDSL FLDSRODU\]DFML
Z NLHUXQNX SU]HZRG]HQLD 8! PDOHMH ]DUyZQR Z\VRNRü MDN L
V]HURNRüEDULHU\
3U]HSá\ZSUGXSU]H]]áF]HSQ
-DN MX* ZVSRPQLDQR Z UR]G] Z VWDQLH UyZQRZDJL EUDN
QDSL FLD]HZQ WU]QHJRSUGZ\SDGNRZ\SU]H]]áF]HMHVWUyZQ\]HUR
U\V
C
W
F
W
V
W
D
qU
p
n
p
D
j
n
D
-j
p
U
j
n
U
-j
5\V=áF]HSQZVWDQLHUyZQRZDJL8
3UGG\IX]\MQ\HOHNWURQyZM
'Q
MHVWUyZQRZD*RQ\SU]H]SUGXQRV]HQLD
HOHNWURQyZM
8Q
SUGQRQLNyZPQLHMV]RFLRZ\FKRUD]SUGG\IX]\MQ\
G]LXU M
'S
MHVW UyZQRZD*RQ\ SU]H] SUG XQRV]HQLD G]LXU M
8S
SUG
QRQLNyZPQLHMV]RFLRZ\FK
3U]\áR*HQLH QDSL FLD ]DSRURZHJR GR ]áF]D SQ ELHJXQ XMHPQ\
(UyGáDSRODU\]XMDFHJRGRSDGRGDWQLGRQU\VSRZRGXMHZ]URVW
EDULHU\SRWHQFMDáXGODQRQLNyZZL NV]RFLRZ\FKLVLOQ\VSDGHNSUGX
G\IX]\MQHJR
QRQLNyZ
ZL NV]RFLRZ\FK
3UG
QRQLNyZ
PQLHMV]RFLRZ\FK SUDNW\F]QLH QLH ]PLHQLD VL L Z HIHNFLH GHF\GXMH R
W]ZSUG]LH]DSRURZ\P
3U]\ SRODU\]DFML ]áF]D Z NLHUXQNX SU]HZRG]HQLD Z\VRNRü
EDULHU\ GOD QRQLNyZ ZL NV]RFLRZ\FK PDOHMH L SUG W\FK QRQLNyZ
V]\ENRURQLHZUD]]QDSL FLHP0Dá\SUGQRQLNyZPQLHMV]RFLRZ\FK
QLH]PLHQLDVL LQLHRGJU\ZDZW\PSU]\SDGNXZL NV]HMUROL
C
W
p
F
W
V
W
p
U
j
n
U
-j
n
F
W
p
d
n
d
qU
T
8
ψ
+
5\V=áF]HVSRODU\]RZDQHZNLHUXQNX]DSRURZ\P8
-H*HOL]DQLHGEDVL ]MDZLVNDJHQHUDFMLLUHNRPELQDFMLQRQLNyZZ
]áF]X SQ FR MHVW GREU\P SU]\EOL*HQLHP GOD ]áF] FLHQNLFK WR
Z\UD*HQLHQDSUGSá\QF\SU]H]]áF]HSU]\ELHUDNV]WDáW
, ,
8
Q
6
7
=
−
H[S
ϕ
JG]LH8MHVWQDSL FLHPSRODU\]DFML]áF]DDQMHVWW]ZZVSyáF]\QQLNLHP
QLHLGHDOQRFL ]áF]D R ZDUWRFLDFK QLHFR ZL NV]\FK RG MHGQRFL
:VSyáF]\QQLN Q ]RVWDá ZSURZDG]RQ\ DE\ XSRGREQLü FKDUDNWHU\VW\N
LGHDOQHJR ]áF]D SQ GR ]áF] U]HF]\ZLVW\FK 0D WR ]QDF]HQLH SU]\
SURMHNWRZDQLXXNáDGyZJG\Z\PDJDQD MHVW GX*D GRNáDGQRü QS SU]\
SURMHNWRZDQLXXNáDGyZVFDORQ\FK