极性三元混晶中的Wannier激子结合能

!光物理!
B i ndi n g Ener g ies of W annier Excit ons i n Pol ar Ternar y
M i xed C r y st als
Z~AO Guo-un L I ANG X i-xi a BAN Shi-li an g
D e p art m ent of Ph y sics Inner M on g oli a Uni versit y~ohhot010021Chi na
Abstract B i ndi n9ener9i es Of W anni er exc i t Ons i n t er nar y m i xed cr y st al s TMGs Of P O l ar cr y st al s Wer e vari-
ati Onal l y cal cul at ed b y t aki n9t h e i nt er acti On be t Ween t h e exc i t On and t WO br anch es Of O P ti cal P h OnOn mOdes
i nt O accOunt i n t h e mOdi fi ed r andO m-e l ement-i sOdi s P l acement MREl mOde l.Th e nu meri cal r esul t s Of t h e ex-
c i t Oni c bi ndi n9ener9i es as f uncti Ons Of t h e cO m P Os i ti On1Wer e9i ven f Or sever al TMc s.Th e cOntri buti Ons Of
t h e t WO br anch es Of P h OnOns t O t h e exc i t Oni c bi ndi n9ener9i es Wer e di scussed.l t i s sh O Wn t h at t h e e l ectr On-
P h OnOn e-P cOu P l i n9P l a y s an i m P Ort ant r O l e f Or t h e bi ndi n9ener9i es Of t h e exc i t Ons W i t h l ar9er e l ectr On-
h O l e mass di ff er ences.A nOn-l i near de P endence Of t h e P h OnOn cOntri buti Ons t O t h e bi ndi n9ener9i es On t h e
cO m P Os i ti On1i s f Ound.Th e val i di t y Of t h e e ff ecti ve P h OnOn mOde a PP r Oxi mati On EP MA t O f Or mul at e t h e ex-
c i t On-P h OnOn i nt er acti On Was di scussed.
K e y wrods t er nar y m i xed cr y st al TMc W anni er exc i t On Haken*s P Ot enti al exc i t On bi ndi n9ener9y
CLC nu mber O471.3Docu m ent code A Article code1005-0086200404-0487-05
极性三元混晶中的W annier激子结合能
赵国军梁希侠班士良
内蒙古大学物理系内蒙古呼和浩特010021
摘要在无序元胞孤立位移MRE I模型下考虑激子与混晶中的两支光学声子的相互作用利用变分法
计算了极性三元混晶TMC s中的W anni er激子的结合能数值计算给出几种混晶材料的结合能随组
份1的变化关系讨论了混晶中的两支光学声子对激子结合能的贡献结果表明当电子和空穴有效
质量相差较大时电声子相互作用对激子结合能有着重要的贡献声子对结合能的影响随混晶组份非线
性变化的同时讨论了有效声子近似EP MA在计算激子结合能时的适用范围
关键词三元混晶W anni er激子~aken势激子结合能
1Introducti on
l n r ecent y ear s much att enti On h as been P ai d t O t h e ex P eri ment al and t h eOr e ti cal st u di es On t er nar y m i xed cr y st al s T Mc s.Th e exci t On st at es P l a y a cr u-ci al r Ol e f Or t h e O P ti cal P r O P erti es Of t h e T Mc s.Th e l atti ce vi br ati Ons and t h e i r cOu P l i n9W i t h t h e e l ectr Ons i n t h e T Mc s h ave been st u di ed t h eOr e ti cal l y and ex-P eri ment al l y1~7.HO Wever t h e e ff ect Of t h e e l ectr On-P h OnOn e-P i nt er acti On On t h e exci t On st at es Whi ch h as been W i de l y st u di ed f Or t h e bi nar y cr y st al s8~11 Was l ess i nvesti9at ed f Or t h e T Mc s t O Our knO W l ed9e. As i s We l l knO Wn di ff er ent f r O m9en er al bi nar y cr y s-t al s t h er e ar e t WO br anch es Of O P ti cal P h OnOn mOdes cOu P l i n9W i t h t h e exci t Ons i n a P Ol ar T Mc.
光电子·激光
第15卷第4期2004年4月Jour nal o f O p t oelect ronics Laser Vol.15No.4A p r.2004
Received dat a2003-07-18Revised date2003-11-03
Foundation ite m W or k f unded b y t he Nati onal Nat ural S ci ence Foundati on of Chi na1016400310264003
E-m ail stz hao g @m ail.i
l n t hi s P a P er We st u d y t h e e ff ect s Of t h e e-P cOu-P l i n9On W anni er exci t Oni c st at es i n a P Ol ar T Mc A1B1-1c.Th e cOntri buti Ons Of t h e t WO-br anch es Of l On9i t u di nal O P ti cal<L O>P h OnOn mOdes ar e t aken i nt O accOunt.Th e mOdi fi ed r andO m-e l ement-i sOdi s P l ace-ment<MREl>mOde l i s used t O cOns i der t h e e-P Fr Ohl i ch i nt er acti On.An e ff ecti ve Ha m i l t Oni an f Or t h e exci t On i s deri ved f Ol l O W i n9a LLP-l i ke me t h Od t h at Was used f Or exci t Ons i n bi nar y cr y st al s b y Haken[12]. Th e9r Ound st at e en er9y Of an exci t On i s cal cul at ed b y a vari ati Onal me t h Od.Th e nu meri cal r esul t s Of t h e ex-ci t Oni c bi ndi n9en er9i es f Or sever al P Ol ar T Mc s ar e Obt ai n ed as f uncti Ons Of t h e cO m P Os i ti On1and t h e r a-ti O Of masses Of t h e e l ectr On and h Ol e r es P ecti ve l y.A nOn-l i n ear de P endence Of t h e P h OnOn cOntri buti Ons t O t h e bi ndi n9en er9i es On t h e cO m P Os i ti On1i s f Ound.An e ff ecti ve P h OnOn mOde a PP r Oxi mati On<EP MA>[13]i s al sO used t O f Or mul at e t h e e-P i nt er acti Ons i n t h e T Mc s and cO m P ar ed W i t h t h e MREl r esul t.Th e val i di-t y Of t h e EP MA f Or exci t On P r Obl ems i s al sO di s-cussed.
2H a m ilt oni an
W e cOns i der an exci t On exci t ed i n a T Mc A1B1-1 c and i nt er acti n9W i t h L O P h OnOns.Th e exci t On-P h O-nOn s y st em Ha m i l t Oni an can be W ri tt en as
~=P 2 e
2m e +
P2h
2m h
-
e2
4T E0E O r
+Z
j g
H U j L a+jg a jg+
Z j g [C
jg
a+jg+h.c.]<1>
Her e t h e r e f er ence en er9y l eve l s f Or e l ectr Ons and h Ol es h ave been ch Osen r es P ecti ve l y at t h e bOtt O m Of t h e cOnduct Or-band and t h e t O P Of t h e val ence-band. m e and m h ar e t h e e ff ecti ve masses Of t h e e l ectr On and t h e h Ol e r es P ecti ve l y.r r e~r h r e<r h>and p e <p
h
>ar e r es P ecti ve l y t h e P Os i ti On-vect Or and mO men-t a Of t h e e l ectr On<h Ol e>.Th e band masses Of t h e e-l ectr On and t h e h Ol e i n t h e T Mc ar e9i ven b y
m e<h>=1m e<h>A+<1-1>m e<h>B<2> Wh er e m e<h>A and m e<h>B r e f er t O t h e band masses Of t h e e l ectr On<h Ol e>f Or t h e end-member bi nar y cr y st al s Ac and Bc r es P ecti ve l y.Th e hi9h-f r e G u enc y di e l ec-tri c cOnst ant E O Of t h e T Mc i s de t er m i n ed b y t h e f Ol-l O W i n9f Or mul a[12]
E O=
1+2[1Y A+<1-1>Y B]
1-[1Y A+<1-1>Y B]
<3> W i t h
Y l=
<E
O l-1
>
<E
O l+2
>
<4> ~ere l=A and B r e f er t O t h e end member bi nar y cr y st al s Ac and Bc r es P ecti ve l y.E O l i s t h e hi9h-f r e-G u enc y di e l ectri c cOnst ant s Of cr y st al0i s t h e vacuu m di e l ectri c cOnst ant.
Th e l ast t er m i n E G.<1>descri bes t h e i nt er acti On Ha m i l t Oni an be t Ween t h e exci t On and t WO br anch es Of L O-P h OnOns i n t h e f r a meWOr k Of t h e MREl mOde l[6] and
C jg=
g j
g V1/2
<e-i r e-e-i r h><5> W i t h
g+=
i e
E0+b33
H
2U+
<>L1/21
T11+2B1T12+B21T
<>221/2
<b
31+B1b32
>
g-=
i e
E0+b33
H
2U-
<>L1/21
T11+2B1T12+B21T
<>221/2
<b
31+B1b32
><6> j=+and~st and f Or t h e hi9h er-and l O Wer-f r e-G u enc y br anch es Of L O-P h OnOn mOdes Wh Ose f r e-G u enci es ar e9i ven b y t h e f Ol l O W i n9e G u ati On
U2L=
-<b*11+b*22>[<b*11-b*22>2+4b*12b*21]1/
<}2/2<7> I n aboze e g ueations t he P ara m eters b B and T Wer e 9i ven b y t h e P r evi Ous WOr k[6]and We h ave nOt l i st ed t h em h er e f Or sh Ort.
3Vari ati onal calcul ati on
B y us i n9a me t h Od s i m i l ar t O t h at used b y Haken f Or bi nar y cr y st al s We h ave Obt ai n ed an e ff ecti ve Ha m i l t Oni an f Or t h e exci t On-P h OnOn s y st em i n t h e mass-cent er s y st em as f Ol l O Ws
~eff=P
2
2J
-
e2
4T E0E O r
-E
j
<O
e j H U j L+O h j H U j L
>+E
j
g j2
2T H U j L r
1-e
-J e j r+e-m h j r
<>
2
<8> Wh er e m e and m h ar e r es P ecti ve l y t h e e l ectr On-and h Ol e-P Ol ar On masses9i ven b y
m e h=m e h1+
O e h++O e h-
<>
6
<8a> W i t h
-
884
-光电子·激光2004年第15卷
O e h j=g j2J e h j 4T H2U2j L
u e h j=
2m e h U j L
<>
H
1/2
J=
m e m h
m e+m h
l n abOve Ha m i l t Oni an We h ave P ut t h e mO ment u m Of t h e mass-cent er t O be Zer O s i nce We ar e Onl y cOn-cer n ed abOut t h e r e l ati ve mOti On Of t h e e l ectr On and h Ol e.
W e nO W st art W i t h a vari ati Onal tr eat ment f Or t h e e ff ecti ve Ha m i l t Oni an H e ff.Th e tri al Wave f uncti On f Or t h e9r Ound st at e Of t h e exci t On i s ch Osen as
r>=<>3/8T>1/2e->r/2<9> Th e vari ati Onal P ar a me t er>and9r Ound st at e en er9y E G ar e bOt h de t er m i n ed b y m i ni m i Zi n9t h e vari ati Onal en er9y
E z=<r~eff r>=H 2>2
8J -
e2>
8T E0E O
+
E j
g j2>
4T H U j L
1-
1
2
1+
u e j
<>>-2+1+u h j
<>>-2-
E
j
<O
e j H U j L+O h j H U j L
><10> Th e bi ndi n9en er9y Of t h e exci t On WOul d be cal-
cul at ed b y su btr acti n9t h e9r Ound st at e en er9y f r O m t h e t Ot al en er9y f O t h e f r ee e l ectr On-h Ol e P Ol ar On P ai r
E f ree=E
j <O
e j H U L+O h j H U j L
><11>
Fi nal l y We Obt ai n ed t h e bi ndi n9en er9y Of t h e exci t On as
E B=-H 2>2
8J +
e2>
8T E0E O
-
E j
g j2>
4T H U j L
1-1
2
1+
u e j
<>>-2+1+u h j
<>>-2
<12> FOr cO m P ari sOn We al sO adO P t t h e EP MA13t O
cal cul at e t h e bi ndi n9en er9y Of t h e exci t On i n Whi ch t h e t WO br anch es Of L O-P h OnOn mOdes h ave been s i m P l i fi ed as a br anch Of e ff ecti ve P h OnOn mOdes Wh Ose f r e G u enc y i s9i ven b y
U EL=1U+L+<1-1>U-L<13> Th e vari ati Onal bi ndi n9en er9y i s t h en de t er m i n ed as
E B=-H 2>2
8J +
e2>
8T E0E E
s
+
e2>
16T E0
1
E O
-
1
E E
<>s
1+u E
e
<>X-2+1+u E h
<>X-2<14>
Wh er e E E
s
i s t h e e ff ecti ve st ati c di e l ectri c cOnst ant13
and u E
s i s de t er m i n ed s i m i l ar t O E G.<8a>.
4Nu m erical results and discuss i on
W e h ave P erf Or med t h e vari ati Onal cal cul ati Ons
f Or sever al T Mc s y st ems.Th e P ar a me t er s used i n Our
cal cul ati Ons ar e l i st ed i n Tab.1and t h e r esul t s ar e
sh O Wn i n Fi9s.1-3.
Fi9.1P l Ot s t h e cur ves Of t h e bi ndi n9en er9i es Of
t h e h eav y-h Ol e exci t Ons i n T Mc A l1Ga1-x As
Zn Se1Te1-1A9c l1Br1-1and A l1Ga1-x N.l t i s Obvi Ous
t h at t h e bi ndi n9en er9i es W i t h P h OnOn i nfl u ences ar e
l ess t h an t h at W i t h Out P h OnOn i nfl u ences.l t i s ex P ect ed
t h at e l ectr On L O-P h OnOn i nt er acti On i n P Ol ar mat eri al s
scr een sur e l y t h e cOul O mb P Ot enti al be t Ween t h e e-
l ectr On and t h e h Ol e sO t h at t h e bi ndi n9en er9i es Of
t h e exci t Ons ar e r educed.Nat ur al l y t h e bi ndi n9en er-
9i es de P end evi dentl y On t h e cO m P Os i ti On1t hr Ou9h
t h e r educed masses Of t h e exci t Ons and t h e exci t On-
P h OnOn i nt er acti On.
Th e cOntri buti Ons Of t h e exci t On-P h OnOn i nt er ac-
ti On t O t h e bi ndi n9en er9i es f Or t h e sa me s y st ems ar e
i l l ustr at ed i n Fi9.2.Th e t Ot al cOntri buti On and t h Ose
f r O m t WO br ach es Of t h e O P ti cal P h OnOn mOdes ar e al l
P l Ott ed.S i m i l arl y t O t h e P Ol ar On case t h e t WO-mOde
beh avi Or i s sh O Wn f Or sO me s y st ems<A l1Ga1-x As and
A9c l1Br1-x>but On e-mOde beh avi Or f Or Ot h er s
<A l
1Ga1-x As and Zn Se1Te1-x
>213.l t i s al sO f Ound
t h at t h e cur ves Of t h e t Ot al e-P cOntri buti Ons t O t h e
bi ndi n9en er9i es ar e Obvi Ous l y nOn-l i n ear On t h e cO m-
P Os i ti On1f Or e i t h er t h e t WO-mOde Or On e-mOde be-
h avi Or s y st ems.
Tab.l0P tical P honon ener g ies dielectric const ants
band-m asses of electrons and holes of several
bi nar y cr y st als.H ere ener gy is m easured i n
m e V and m i n t he electron rest m ass
H U TO H U LO E O E s m e m hh m l h
A l A s844.8850.598.1610.060.15000.760.15010
Ga A s833.2936.2510.8913.180.06570.620.07410
A l N982.84113.024.688.500.33001.43
G a N969.5592.035.359.500.19000.86
ZnS e825.6630.495.908.330.17100.600.14510
Zn T e821.9525.427.289.860.16000.600.15410
A g C l814.8823.023.979.500.30202.09814
A g
B r811.3517.074.6810.60.21501.25014
984
第4期Z~AO Guo-un et al.B i ndi n g Ener g i es of W anni er Excit onsi n
F i g .l B i ndi n g ener g ies !E B "of t he heav y -hole excit ons w it h !soli d li ne "and w it hout !
lon g dashed li ne "i ncl udi n g t he excit on-P honon i nteraction #calcul ated b y E M PA !short dashed li ne "#and w it hout i nvolvi n g UCVV !g ri d li nes "
as f unctions of t he co m P os ition 1f Or sever al TMc s y st ems $!a "A l 1Ga l -1As #
!b "Zn Se 1Te l -x #!c "A 9c l 1Br l -x #!d "A l 1Ga l -1
N
F i g .2Tot al contri butions !soli d "of t he excit on-P honon i nteraction and t he contri butions of t wo branches of P honon m odes !dashed and g ri d li nes "se P aratel y
t o t he bi ndn g ener g ies of t he excit ons i n several T M C s y ste m s $!a "A l X G a l -X A s #
!b "ZnS e X Te l -X #!c "A g C l X B r l -x #!d "A l X G a l -X
N
F i g .3Contri butions of t he excit on P honon i nteraction t o t he bi ndi n g ener g ies of t he excit ons i n t he li g ht !dashed "
and heav y !soli d "hole cases f or !a "A l X G a l -X A s and !b "ZnS e X Te l -X s y ste m s
Th e e ff ect Of t h e uni t-ce l l vOl u me vari ati On
U cVV i s al sO cOns i der ed i n Our cal cul ati On .D i ff er -ent f r O m t h e P Ol ar On P r Obl ems 13 9t h e U cVV i s nOt r e-mar kabl e On t h e exci t On bi ndi n 9en er 9y f Or al l t h e cO m P ut ed mat eri al s see Fi 9.1 .l t i s t h e case t h at t h e P Ol ari Zati On Ori ent ati On Of t h e e l ectr Oni c and h Ol e *
s ar e O PP Os i t e 9sO t h at t h e t Ot al e ff ect i s al mOst can-ce l l ed .FOr cO m P ari sOn 9
We h ave al sO P l Ott ed t h e cur ves Of t h e bi ndi n 9en er 9i es i nc l u di n 9t h e e -P cOntri buti Ons cal cul at ed b y t h e EP MA i n Fi 9.1.On e can see f r O m t h e fi 9ur es t h at t h e di ff er ence Of t h e t WO cur ves Ob-t ai n ed b y EP MA and MREl mOde l s i s Obvi Ous 9i n P arti -
.
094.光电子·激光2004年第15卷
cul ar f Or t h e t WO-mOde beh avi Or T Mc s,such as A l1Ga1-x As,and A9c l1Br1-x s y st ems.l t f Ol l O Ws t h at t h e EP MA i s nOt a9OOd a PP r Oxi mati On i n cOns i deri n9 t h e exci t On-L O P h OnOn i nt er acti On f Or t h e T Mc s even f Or t h e Weaker e-P cOu P l i n9s y st ems I-V cO m P O-n ent s.Th er e f Or e t h e ch ar act eri sti cs Of t WO-mOde O P ti-cal vi br ati On i n T Mc s sh Oul d be t aken i nt O accOunt i n exci t On P Ol ar On P r Obl ems.
TO under st and t h e i nfl u ence Of t h e r ati O Of e l ec-tr On-h Ol e masses t O t h e cOntri buti On Of t h e e l ectr On <h Ol e>-P h OnOn i nt er acti On,as exa m P l es,We h ave i l-l ustr at ed t h e exci t On bi ndi n9en er9i es as f uncti Ons Of t h e cO m P Os i ti On1f Or A l1Ga1-x As,and Zn Se1Te1-x s y st ems i n t h e h eav y and l i9ht h Ol e cases,r es P ecti ve-l y.l n t h e cal cul ati On We ch OOse t h e l i9ht-and h eav y-h Ol e masses<i n uni t Of t h e bar e e l ectr On>r es P ecti ve-l y as0.15and0.76f Or t h e bi nar y cO m P Ound A l As,0. 074and0.620f Or GaAs,0.145and0.750f Or Zn Se, 0.154and0.60f Or Zn Te.Th e cO m P ut ed r esul t s ar e sh O Wn i n Fi9.3.l t i s ex P ect ed t h at t h e bi ndi n9en er-9i es Of t h e h eav y-h Ol e exci t Ons ar e ar Ound t WO-ti mes 9r eat er t h an t h at Of t h e l i9ht-h Ol e exci t Ons f Or bOt h t h e t WO s y st ems.Th e L O-P h OnOn cOntri buti Ons t O t h e bi ndi n9en er9i es var y i n t h e r an9e f r O m ar Ound34% Of t h e t Ot al bi ndi n9en er9i es<f Or h eav y-h Ol e A l1Ga1-x As>t O ar Ound46%<f Or l i9ht-h Ol e Zn Se x Te1-1>.l t i s t hus c l ear t h at t h e e-P cOu P l i n9P l a y s an i m P Ort ant r Ol e f Or t h e exci t On bi ndi n9en er9i es i n e i t h er t h e h eav y-h Ol e case Or t h e l i9ht-h Ol e case.
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B io g ra P h y=
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-
194
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第4期Z~AO Guo-un et al.=B i ndi n g Ener g i es of W anni er Excit onsi n-
极性三元混晶中的Wannier激子结合能
作者：赵国军， 梁希侠， 班士良
作者单位：内蒙古大学物理系,内蒙古,呼和浩特,010021
刊名：
光电子·激光
英文刊名：JOURNAL OF OPTOELECTRONICS·LASER
年，卷(期)：2004,15(4)
被引用次数：5次
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引用本文格式：赵国军.梁希侠.班士良极性三元混晶中的Wannier激子结合能[期刊论文]-光电子·激光 2004(4)。