3ROXSURYRGQL PDWHULMDOLmaterijali.elfak.ni.ac.rs/poluprovodnici.pdf · 3ROXSURYRGQL PDWHULMDOL 49 ln(niC*T-3/2) 1/T ~Eg/2k Eg Ec Ev Ed Donorski nivo DNR MH 3 / 2 2 2 2 = h mk C π

�� 3ROXSURYRGQL PDWHULMDOL�� 3ROXSURYRGQL PDWHULMDOL

=D SROXSURYRGQLNH MH NDUDNWHULVWLaQD ]RQVND VWUXNWXUD �6O� ��

(J � HQHUJHWVNL SURFHS

�]DEUDQMHQD ]RQD�

(J (

F� (

Y

6O� �� >HPDWVNL SULND] ]RQVNH VWUXNWXUH NRG SROXSURYRGQLND

8 SROXSURYRGQLFLPD VX PRJX`L PHGMX]RQVNL SUHOD]L HOHNWURQD L] YDOHQWQH XSURYRGQX ]RQX aLPH VH SRYH`DYD NRQFHQWUDFLMD HOHNWURQD X SURYRGQRM L ^XSOMLQD XYDOHQWQRM ]RQL�

.RQFHQWUDFLMD HOHNWURQD X SURYRGQRM ]RQL�

−−=

kT

EENn fcc exp

2/3

2

22

=

h

kTmN nc

π ← HIHNWLYQL EURM VWDQMD VYHGHQ QD GQR SURYRGQH ]RQH

.RQFHQWUDFLMD ^XSOMLQD X YDOHQWQRM ]RQL�

−−=

kT

EENp vfv exp

2/3

2

22

=

h

kTmN pv

π← HIHNWLYQL EURM VWDQMD VYHGHQ QD YUK YDOHQWQH ]RQH

=D SROXSURYRGQLNH YDCL�

2inpn =⋅ � JGH MH QL VRSVWYHQD NRQFHQWUDFLMD QRVLODFD QDHOHNWULVDQMD�

−=

−−

−−=⋅

kT

ENN

kT

EE

kT

EENNpn gvc

vffcvc expexpexp

−=

kT

ENNn gvci

2exp �

Eg

Ec

Ev

Provodna zona

Valentna zona

3ROXSURYRGQL PDWHULMDOL 49

ln(niC*T-3/2)

1/T

~Eg/2k

Eg

Ec

Ev

Ed

Donorski nivo

DNR MH

2/3

2

22

=

h

mkC

πPRCH VH QDSLVDWL GD MH

2/31TCNc = �

2/32TCNv =

−=

kT

ECTn gi 2

exp2/3

−=−

kT

ETCn gi 2

exp* 2/3 � JGH MH & ��&

( )kT

ETCn

gi

2*ln 2/3 −=−

=DYLVQRVW VRSVWYHQH NRQFHQWUDFLMH QRVLODFD QDHOHNWULVDQMD RG WHPSHUDWXUH

GDWD MH QD VO� ��

1D RVQRYX ]DYLVQRVWL QL I�7� PRCH VH

HNVSHULPHQWDOQR RGUHGLWL ^LULQD

HQHUJHWVNRJ SURFHSD (J�

6O� �� 3URPHQD VRSVWYHQH NRQFHQWUDFLMH QRVLODFD QDHOHNWULVDQMD X IXQNFLML RG

WHPSHUDWXUH

=D VRSVWYHQL� EHVSULPHVQL SROXSURYRGQLN YDCL inpn ==

3ULPHVQL SROXSURYRGQLN PRCH ELWL GRQRUVNRJ LOL DNFHSWRUVNRJ WLSD�

• 'RQRUVNL WLS SROXSURYRGQLND � GRGDMX VH SHWRYDOHQWQH SULPHVH �3� $V� 6E��6O��

-RQL]DFLMRP SULPHVD HOHNWURQL VDGRQRUVNRJ QLYRD SUHOD]H XSURYRGQX ]RQX L WLPH SRYH`DYDMXNRQFHQWUDFLMX SURYRGQLK HOHNWURQD�9H`LQVNL QRVLRFL QDHOHNWULVDQMD VXHOHNWURQL�

6O� �� 3RORCDM GRQRUVNRJ QLYRD X Q�WLSX SROXSURYRGQLND


Eg

Ec

Ev

Ea

Akceptorski nivo

Ec

Ev

Ed

Ea

n

p

• $NFHSWRUVNL WLS SROXSURYRGQLND � GRGDMX VH WURYDOHQWQH SULPHVH �%� *D��6O��

-RQL]DFLMRP SULPHVD HOHNWURQL L]YDOHQWQH ]RQH SUHOD]H QDDNFHSWRUVNL QLYR L WLPH VHSRYH`DYD NRQFHQWUDFLMD ^XSOMLQD XYDOHQWQRM ]RQL� 9H`LQVNL QRVLRFLQDHOHNWULVDQMD VX ^XSOMLQH�

6O� �� 3RORCDM DNFHSWRUVNRJ QLYRD X S�WLSX SROXSURYRGQLND

=D SROXSURYRGQLNH YDCL MHGQDaMHGQDaLQD HOHNWURQHXWUDOQRVWL�LQD HOHNWURQHXWUDOQRVWL�

dDaA nNppNn −+=−+ �

JGH MH 1'� NRQFHQWUDFLMD GRQRUVNLK SULPHVD�

1$� NRQFHQWUDFLMD DNFHSWRUVNLK SULPHVD

QG� NRQFHQWUDFLMD HOHNWURQD QD GRQRUVNRP QLYRX �QHMRQL]RYDQL GRQRUL�

SD� NRQFHQWUDFLMD ^XSOMLQD QD DNFHSWRUVNRP QLYRX �QHMRQL]RYDQL DNFHSWRUL�

dDD nNN −=+

−+

=

kT

EE

Nn

fd

Dd

exp2

11

aAA pNN −=−

−+

=

kT

EE

Np

af

Aa

exp21

7LSRYL SROXSURYRGQLND7LSRYL SROXSURYRGQLND

�� =D VRSVWYHQL �EHVSULPHVQL� SROXSURYRGQLN 1' 1

$ Q

G S

D �� L L] MHGQDaLQH

HOHNWURQHXWUDOQRVWL VOHGL�

pn =


n

pvcf

m

mkTEEE ln

4

3

2+

+=

$NR MH PQ P

S)HUPLMHY QLYR MH QD VUHGLQL

]DEUDQMHQH ]RQH �6O� ��

6O� �� 3RORCDM )HUPLMHYRJ QLYRD X VRSVWYHQRP SROXSURYRGQLNX

�� 'RQRUVNL SROXSURYRGQLN 1$ S

D � �Q � WLS SROXSURYRGQLND�

-HGQDaLQD HOHNWURQHXWUDOQRVWL LPD REOLN�

dD nNpn −+=

D� 1D QLVNLP WHPSHWDWXUDPD XJODYQRP HOHNWURQL VD (GSUHOD]H X SURYRGQX ]RQX

MHU HOHNWURQL L] YDOHQWQH ]RQH QHPDMX GRYROMQX HQHUJLMX GD SUHGMX X SURYRGQX]RQX ⇒ S ≈ �

dD nNn −=

E� 1D YLVRNLP WHPSHUDWXUDPD L HOHNWURQL L] YDOHQWQH ]RQH PRJX GD SUHGMX XSURYRGQX ]RQX

++=−+= DdD NpnNpn

$NR VX VYH SULPHVH MRQL]RYDQH QG �� D YDCL Q!!S� RQGD MH�

DNn ≈

AHVWR VH YH` VREQH WHPSHUDWXUH VPDWUDMX YLVRNLP WHPSHUDWXUDPD�

�� $NFHSWRUVNL SROXSURYRGQLN 1'� Q

G � �S � WLS SROXSURYRGQLND�

ppNn aA =−+

Ec

Ef

Ev


D� 1D QLVNLP WHPSHUDWXUDPD Q ≈ �

ppN aA =−

E� 1D YLVRNLP WHPSHUDWXUDPD

pNn A =+−

$NR VX VYH SULPHVH MRQL]RYDQH SD �� YDCL GD MH S!!Q� RQGD MH S ≈ 1

$

�� 3RWSXQR NRPSHQ]RYDQL SROXSURYRGQLN

1' 1

$

�� 'HOLPLaQR NRPSHQ]RYDQL SROXSURYRGQLN

1$! 1

'→ S � WLS SROXSURYRGQLND

1'! 1

$→ Q � WLS SROXSURYRGQLND

�� ,]UDaXQDWL NRQFHQWUDFLMX�� ,]UDaXQDWL NRQFHQWUDFLMX SULPHVD X 6L SR PSULPHVD X 6L SR P�� DNR MH NRQFHQWUDFLMD SULPHVDDNR MH NRQFHQWUDFLMD SULPHVD

�SSP� G�SSP� G6L6L ��J�FP ��J�FP�� 0� 06L6L ��J�PRO� ��J�PRO�

5H^HQMH�5H^HQMH�

�SSP � � DWRP SULPHVD QD PLOLRQ DWRPD VLOLFLMXPD

�FP� → ��J 6L → 16L

�PRO → ��J 6L → 1$

�� 16L �� 1

$

molg

molcmg

molg

NcmgN ASi

/1.28

/11002.6/3.2

/1.28

/3.2 2333 ⋅⋅=

⋅= �

328

322

1093.4

1093.4−

−

⋅=

⋅=mN

cmN

Si

Si

3226

1093.410

−⋅== mN

N Siprim

�� 8 X]RUNX *H QDOD]L�� 8 X]RUNX *H QDOD]L VH ��VH �� DWRPD 6E SR PDWRPD 6E SR P�� 8]LPDMX`L GD VX SUL VREQRM� 8]LPDMX`L GD VX SUL VREQRMWHPSHUDWXUL VYL DWRPL 6E MRQL]RYDQL RGUHGLWL NWHPSHUDWXUL VYL DWRPL 6E MRQL]RYDQL RGUHGLWL NRQFHQWUDFLMX HOHNWURQDRQFHQWUDFLMX HOHNWURQD L ^XSOMLQD�L ^XSOMLQD�>LULQD ]DEUDQMHQH ]RQH X JHUPDQLMXPX MH (J ��H9�>LULQD ]DEUDQMHQH ]RQH X JHUPDQLMXPX MH (J ��H9�



6E � GRQRUVND SULPHVD → 1' ��P��

Q ≈ 1' ��P��

n

npnnp ii

22 =⇒=⋅

−

=

−=

kT

E

h

kTm

h

kTm

kT

ENNn gpngvci exp

22

22exp

2/3

2

2/3

22 ππ

VPDWUDPR GD MH PQ PS P�

−

=

kT

E

h

kTmn gi exp

24

3

202 π

314

3

20

2

102.6

exp2

4−⋅=

−

== mN

kT

E

h

kTm

N

np

D

g

D

i

π

Q !! S

�� =D�� =D *H NRML VDGUCL �*H NRML VDGUCL �⋅��PP�� DWRPD $V L ��DWRPD $V L ��PP�� DWRPD *D L]UDaXQDWL SRORCDMDWRPD *D L]UDaXQDWL SRORCDM)HUPLMHYRJ QLYRD X RGQRVX QD GQR)HUPLMHYRJ QLYRD X RGQRVX QD GQR SURYRGQH ]RQH QD 7 ��.� (IHNWLYQDSURYRGQH ]RQH QD 7 ��.� (IHNWLYQDJXVWLQD VWDQMD X SURYRGQRMJXVWLQD VWDQMD X SURYRGQRM ]RQL MH 1]RQL MH 1&& ��

��PP�� D NRQFHQWUDFLMD VRSVWYHQLK� D NRQFHQWUDFLMD VRSVWYHQLK

QRVLODFD QQRVLODFD QLL � �⋅��PP�� 6PDWUDWL GD VX QD GDWRM WHPSHUDWXUL� 6PDWUDWL GD VX QD GDWRM WHPSHUDWXUL VYH SULPHVHVYH SULPHVHMRQL]RYDQH�MRQL]RYDQH�


dDaA nNppNn −+=−+ ← MHGQDaLQD HOHNWURQHXWUDOQRVWL

$NR VX VYH SULPHVH MRQL]RYDQH ⇒ SD Q

G �� MHGQDaLQD HOHNWURQHXWUDOQRVWL

LPD REOLN�

DA NpNn +=+

n

npnnp ii

22 =⇒=⋅


nNn

nNn D

iA ⋅+=+

2

( ) 022 =−−− iAD nnNNn

( )2

4 22

2,1i

nNNNNn

ADAD +−±−=

322104 −⋅=− mNN AD ( ) 22 4 iAD nNN >>−322104 −⋅=−= mNNn AD

−−=

kT

EENn fcc exp

JeVn

NkTEE CfC

201024.214.0ln −⋅===−

(OHNWULaQD SURYRGQRVW SROXSURYRGQLND(OHNWULaQD SURYRGQRVW SROXSURYRGQLND

6 RE]LURP GD VX QRVLRFL QDHOHNWULVDQMD L HOHNWURQL L ^XSOMLQH� JXVWLQD VWUXMH

MH GDWD L]UD]RP�

Kjjj pn ⋅=+= σ �

D VSHFLILaQD SURYRGQRVW SROXSURYRGQLND MH

pn epen µµσ +=

=D VRSVWYHQL SROXSURYRGQLN�

inpn == �

SD MH VSHFLILaQD HOHNWULaQD SURYRGQRVW�

( ) ( ) ( ) kTE

pnVCpni

g

eNNeen 22/1 −+=+= µµµµσ

=DYLVQRVW VSHFLILaQH HOHNWULaQH SURYRGQRVWL RG WHPSHUDWXUH VH PRCH GDWL

L]UD]RP�

kT

Eg

e 20−

⋅= σσ JGH MH ( ) ( )pnVCNNe µµσ += 2/10


�� 6RSVWYHQD VSHFLILaQD SURYRGQRVW SROXSURYRGQLND QD�� 6RSVWYHQD VSHFLILaQD SURYRGQRVW SROXSURYRGQLND QD WW�� °& MH& MH σ�� ΩP�P��D QD WD QD W�� °& MH& MH σ�� ΩP�P��

D�D� 2GUHGLWL ^LULQX ]DEUDQMHQH2GUHGLWL ^LULQX ]DEUDQMHQH ]RQH L ]DYLVQRVW]RQH L ]DYLVQRVW σ I��7�� I��7��

E�E� RGUHGLWLRGUHGLWL σ��

F�F� QD`L VSHFLILaQX HOHNWULaQX RWSRUQRVW QD W ��QD`L VSHFLILaQX HOHNWULaQX RWSRUQRVW QD W ��°&�&�


D� kTEg

e 20−

⋅= σσ

Tk

Eg 1

2lnln 0 −= σσ

56.213lnln 1 ==σ KT 3001 =1

1

003.01 −= KT

34.496lnln 2 ==σ KT 3502 =1

2

0028.01 −= KT

21

12

11lnln

2TT

k

Eg

−

−= σσ

eV

TT

kEg 73.011lnln

2

21

12 =−

−= σσ

6O� �� 2GUHGMLYDQMH ^LULQH HQHUJHWVNRJ SURFHSD QD RVQRYX ]DYLVQRVWL σ I�7�

E�

−=

101 2exp

kT

Egσσ ⇒ ( ) 171

10 109.12exp −Ω⋅=

= m

kT

Egσσ

F�

==

kT

Eg2

exp11

0σσρ

mΩ= 028.0ρ

~ k

Eg

2

σOQ

2σOQ

1σOQ

T

1

2

1

T 1

1

T


�� 6RSVWYHQD VSHFLILaQD�� 6RSVWYHQD VSHFLILaQD HOHNWULaQD RWSRUQRVW *H QD 7 ��. MHHOHNWULaQD RWSRUQRVW *H QD 7 ��. MH ρ �� ΩP�P�3RNUHWOMLYRVW HOHNWURQD L ^XSOMLQD3RNUHWOMLYRVW HOHNWURQD L ^XSOMLQD NRG *H ]DYLVL RG WHPSHUDWXUH QD VOHGH`LNRG *H ]DYLVL RG WHPSHUDWXUH QD VOHGH`L

QDaLQ�QDaLQ� µQQ �� ⋅��77��PP��9V L�9V L µSS �� ⋅��77��PP��9V�9V

,]UDaXQDWL VRSVWYHQX NRQFHQWUDFLMX QRVLODFD QDHOHNWULVD,]UDaXQDWL VRSVWYHQX NRQFHQWUDFLMX QRVLODFD QDHOHNWULVDQMD�QMD�


( )ρ

µµσ 1=+= pnien

( )pni en µµρ += 1

( )Vs

mn

267.13 255.0300105.3 =⋅= −µ

( )Vs

mp

23.24 183.0300101.9 =⋅= −µ

3181036.30 −⋅= mni

�� 8 SROXSURYRGQRP PDWHULMDOX VX SRNUHWOMLYRVWL HOHNWURQD L ^XSOMLQD�� 8 SROXSURYRGQRP PDWHULMDOX VX SRNUHWOMLYRVWL HOHNWURQD L ^XSOMLQD µQQ LL µSS ]D]DNRMH YDCLNRMH YDCL µQQ !! µSS� .DGD VH SURYRGQRVW SRVPDWUD NDR IXQN� .DGD VH SURYRGQRVW SRVPDWUD NDR IXQNFLMD NRQFHQWUDFLMHFLMD NRQFHQWUDFLMH^XSOMLQD SRND]DWL GD VH PLQLPDOQD YUHGQRVW SURYRGQRVWL PRCH^XSOMLQD SRND]DWL GD VH PLQLPDOQD YUHGQRVW SURYRGQRVWL PRCH L]UD]LWLL]UD]LWL

VD�VD�( )

pn

pni µµ

µµσσ

+⋅

=2/1

min

2NDGD MHNDGD MH

2/1

=

p

ninp µ

µJGH MHJGH MH iσ SURYRGQRVWSURYRGQRVW

VRSVWYHQRJ SROXSURYRGQLND�VRSVWYHQRJ SROXSURYRGQLND�


pn epen µµσ +=

.RG VRSVWYHQRJ SROXSURYRGQLND Q S QL

⇒ ( )pnii en µµσ +=

p

nnnpn ii

22 =⇒=⋅

pni epp

ne µµσ +=

2


pni

pni

p

ne

p

ne

dp

d µµµµσ =⇒=+−=2

2

2

2

0

2/1

=

p

ninp µ

µ

pp

nin

p

ni

i en

n

ne µ

µµµ

µµ

σ ⋅

+

=

2/1

2/1

2

min

( ) ( )pni

ipnie

nenµµ

σµµσ+

=⋅= 2/1min 2

( )pn

pni µµ

µµσσ

+⋅

=2/1

min

2

�� ,]UDaXQDWL RGQRV�� ,]UDaXQDWL RGQRV WHPSHUDWXUQLK NRHILFLMHQDWD VSHFLILaQH HOHNWWHPSHUDWXUQLK NRHILFLMHQDWD VSHFLILaQH HOHNWULaQH RWSRUQRVWLULaQH RWSRUQRVWLSROXSURYRGQLND $ L % DNR MHSROXSURYRGQLND $ L % DNR MH ^LULQD ]DEUDQMHQH ]RQH SROXSURYRGQLND $^LULQD ]DEUDQMHQH ]RQH SROXSURYRGQLND $((J$J$ ��H9� D SROXSURYRGQLND % ( ��H9� D SROXSURYRGQLND % (J%J% ��H9� ��H9�


3R GHILQLFLML WHPSHUDWXUQL NRHILFLMHQW VSHFLILaQH HOHNWULaQH RWSRUQRVWL MH�

dT

dρρ

α 1=

6SHFLILaQD HOHNWULaQD SURYRGQRVW MH�

kT

Eg

e 20−

= σσ

kT

Eg

e20

11

σσρ == kT

Eg

e20ρρ =

−=

−= 22

20 22 kT

Eg

kT

Ege

dT

d kTEg

ρρρ


22 22

11

kT

Eg

kT

Eg

dT

d −=

−== ρ

ρρ

ρα

18095.0

7.1

2

2

2

2

===−

−=

eV

eV

Eg

Eg

kT

EgkT

Eg

B

A

B

A

B

A

αα

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S

lR ρ=

ρρρ ∆=∆⇒∆=∆

R

R

S

lR

−=⇒=

22

0 2kT

Eg

dT

de kT

Eg

ρρρρ

TkT

Eg∆

−=∆

22ρρ T

kT

Eg

R

R∆

−=

∆22

šš

kT

Uš

š n

NkTUeNn ln22 =⇒⋅=

−

JMn

dNkTUAAM

M

dNN

š

AšSePb

A 20106.4ln2 −⋅==⇒+==

4.5=∆R

R


x

z

y

+RORY HIHNDW X SROXSURYRGQLFLPD+RORY HIHNDW X SROXSURYRGQLFLPD

3RVPDWUDMPR SROXSURYRGQLN VD 1 UD]OLaLWLK YUVWD QRVLODFD QDHOHNWULVDQMD�

1DHOHNWULVDQMH L�WH YUVWH QDHOHNWULVDQMD MH TL I

L⋅H JGH MH I

L≠ � � ]QDN QDHOHNWULVDQMD�

� D QL² NRQFHQWUDFLMD L µ

L� SRNUHWOMLYRVW L�WH YUVWH QDHOHNWULVDQMD �6O� ��

kBB

iKK

z

x&&

&&

=

=

6O� �� >HPDWVNL SULND] +RORYRJ HIHNWD X SROXSURYRGQLFLPD

iKfKfv xiiiiix&&

& µµ ==

∑=

=N

ixix Kj

1

σ

iii enµσ =

( )BvqFixi

&

&

&

×=

jKK HyH&&

=

jKqKqF HyiHie&&&

==

( )BvqKqFFFixiHiei

&

&

&&&&

×+=+=

iii KqF&&

⋅=

( ) jKjBKfKK iyzxiiHyi&&&

=−= µ

∑ ∑∑= ==

=−==N

i

N

iiiizxi

N

iHyiyiy fBKKKj

1 11

0σµσσ

zxN

iii

N

iiii

zxN

ii

N

iiii

Hy BKne

nfe

BK

f

K

∑

∑

∑

∑

=

=

=

= ==

1

1

2

1

1

µ

µ

σ

σµ

B&

K HK


i

N

ii

xN

ii

xx

ne

jjK

µσ ∑∑==

==

11

zxN

iii

N

iiii

Hy Bj

ne

nfeK

2

1

1

2

=

∑

∑

=

=

µ

µ

2

1

1

2

==

∑

∑

=

=

N

iii

N

iiii

zx

HyH

ne

nfe

Bj

KR

µ

µ

=D HOHNWURQH =D ^XSOMLQH

ni

i

i

nn

f

µµ ==

−= 1

pi

i

i

pn

f

µµ === 1

( )222

npe

npR

np

npH µµ

µµ+−

=

+RORYD NRQVWDQWD ]D GRQRUVNL WLS SROXSURYRGQLND ⇒ Q�WLS SROXSURYRGQLNDNDGD MH Q!!S�

enRH

1−=

=D DNFHSWRUVNL WLS SROXSURYRGQLND ⇒ S�WLS SROXSURYRGQLND NDGD MH S!!Q�

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Documents

3ROXSURYRGQL PDWHULMDOLmaterijali.elfak.ni.ac.rs/poluprovodnici.pdf · 3ROXSURYRGQL PDWHULMDOL 49 ln(niC*T-3/2) 1/T ~Eg/2k Eg Ec Ev Ed Donorski nivo DNR MH 3 / 2 2 2 2 = h mk C π