Theory of angle-resolved photoemission experiments on a two-band model

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1 Theoy o gle-esolved photoesso expeets o two-bd odel T De Co * Deptet o Physs, jg Uvesty o Ioto See & Tehology, jg 0044, Ch Abstt Cosdeg the eleto sttes sde d outsde the sold, we deve oul o photoesso testy. A geel theoetl wy to detee eleto stutues o solds o ARPES expeets s outled. It s show tht the spetl uto sde the solds ot be esued detly by ARPES, the eets o ee eleto sttes o the eleto stutue obseved by ARPES esueets ust be osdeed, d the esults o ARPES expeets ot be udestood utl these esults hve bee de osstet wth theoetl lulto. Keywods: photoesso; eleto stutue PACS ube: I; 74.5.Jb; h. Itoduto Agle-esolved photoesso spetosopy (ARPES s epted s ledg tool the vestgto o hgh-t supeodutos [], d lot o wos bout ARPES usg oe-bd odel hve bee epoted [-5]. We hve obseved tht ARPES expeets ot vey oplex pheoe, soe o whh be ttbuted to the oes ow othe expeets. o exple, photoeleto ssoted wth the deepe oe-levels be dsted o the oe ssoted wth the loh bds, d eleto stte sde the tel ust be dstgushed o the oe o the sue. Although the eleto sttes sde the solds ould be obseved wth photoeletos, thee s o ethod to ly how to elte the eets o the eleto sttes outsde the solds. I t, the usul suggesto ltetues s tht the eleto spetl uto wth the sold s detly obseved by ARPES, but we d tht ths suggesto s ot ppopte o the spetl utos o eleto systes stogly *Coespodg utho. *E-l ddess: tdo@ust.edu. (T. D. Co. *Tel:

2 oelted tels. Ths wo deves the oul o photoesso testy o two-bd odel. It s show tht the eleto stutues the stogly oelted tels ust be deteed o the bss o obg the ARPES expeet wth the soluto o the theoet odel le the peod Adeso odel. Ths es tht ARPES expeets ot be udestood utl they hve bee de osstet wth theoet lulto.. oul I ode to d the eleto stutue below the e eegy wth the photoeletos outsde the sold, the wve utos both wth d outsde the sold ust be desbed wth the se Hlto. Povded the eets o the sue o ystl ould be elted o the esults obseved expeets, we osde the eleto sttes both sde d outsde the sold, d te the Hlto o H = H 0 H t ( The teto wth photo s teted s the petubto gve by jeh Ht. A ( whee A =0, A s the extel lssl veto potetl. A oo bss set o the sttg pot o the y-body lulto s just ee-ptle wve uto (x j x = e o good etls. Howeve, the wve utos both wth d outsde the stogly oelted tel should be desbed wth othe bs utos o the bss o the so-lled supeposto pple o sttes. These wve utos ould be ( x ( x, s = d φ q q ( x χ e j x χ q,, (3 whee d q destoy eleto q stte o sp sde the sold, d destoy eleto outsde the sold. Thus we te the o H0 =, E ( M d h..,, ε d d d d H (4 whee H d d epeset the tetos betwee eletos wth the tel, whle the ult-bd odel, suh s the oe o upte supeodutos, eques

3 ( x x s = d x p x e j x χ (, φ q q ( χ ϕ q q ( χ q, q,, (5 d so o. Ths wll gve oe oplex odel th Eq.(4. Howeve, ult-bd odel ould be edued to oe-bd odel o the eleto popetes o these tels, thus ou dsusso s lted to the odel (4. Moeove, Eq.(6-(5 below be exteded to othe odels. We pply the Golde Rules qutu ehs to the y-ptle theoy, d wte the tsto te betwee the -eleto tl stte > w π < Ht. > = h δ ( E d the l stte > E hν (6 It should be oted tht both the l stte Hlto H 0, thus d the tl stte s oe o ll possble tl sttes { e the sttes ssoted wth the se } Eq.(6. euse the eets o sue sttes e elted, the wve veto s see s the oseved oe. Ptully, ths should be esoble whe s ppoxtely pllel to the sue o the sold. We wte < Ht > = < φ Ht φ < (7 s dsussed the ltetues, the l stteφ d the tl stteφ o sgle ptle be sged by the wve veto, thus we ewte Eq.(7 s < Ht > = M t < (8 whee expesses (,. euse w ll = V w 3 ll ( π d 3 d d 3 = e ddω = dedω o the ee h eleto sttes outsde the sold, the outg te o the photoeletos pe seod pe sold gle pe eegy tevl s expessed the o I, E = C p < H > δ ( E E hν (, t. = M p < > ( E E h t, δ ν (9 3

4 whee p = e β ( μ Ω E s the pobblty o oe tl stte, d, do ot ot the suto ove the wve vetos. It s eessy to ote tht < > δ beuse > d > ould ot be the, egeutos o the se Hlto. Se eh tsto o the tl stte to the l oe hs photoeleto whh hs wve veto, thus we wte > ~ >, should be ethe d o beuse > s ot the egeuto o d d o. The s the qus-ptle destuto opeto, d > s the egeuto o. Eq.(8 beoes I, E = C M ( t p < >, δ ( E E φ E hν (0 whee E oespods to the eegy o the ( --eleto stte, othe sybols ollow Re. []. The expesso o o the testy be hged o Shödge to Hesebeg te-depedet opetos wth the d o the tegl δ ( E E hν = d( t t' e πh j( E E hν ( t t '/ h ( d ths ples < > δ ( E E E φ hν j( E φ hν ( t t' / h = d( t t' e < ( t > πh < ( t' > ( Usg Eq. (, we ewte Eq.(0 s I(, E = C πh ( E h ( t ' ( ' e j φ ν d t t t e β ( E μ Ω < ( t ( t' > (3 Itodug the uto < G (,, t t' = e β ( E μ Ω < oe wll d tht ths uto s elted to the spetl uto [6] by G < (,, ω =j (ω (,, ω A thus we get the expesso ( t ( t' > (4 4

5 I, E = C' M ( E φ hν A (,, E φ hν = C' M ( E ( t t A (,, (5 E whee E s the eegy o eleto ( E =0 t E sde the sold. I the expessos bove, we hve used the totl φ eegy osevto E = hν s gve Re. []. It sees tht Eq. (5 s sl to the o the E ltetues, but t s show tht wht ARPES expeets esue s the spetl uto elted to the eletos both sde d outsde the sold, sted o the oe suggested by soe uthos. A o qus-ptles whh e 3. Results d dsussos Howeve, we wll d tht the eleto stutues o the sold be udestood Eq. (5. To d the spetl uto (,, ω, we should lulte the Gee s uto A G (,, τ τ ' = < T ( τ ( τ ' > τ wth whh we get the etded uto G (,, ω = hgeg (,, jω (7, et jω ω δ d the spetl uto A (,, ω = IG, (,, ω (8 et To expess ou des, stly, we te H d d =0 Eq.(4. Ths Hlto be dgolzed, o we get (6 G,, jω = ( jω E M /( jω ε (9 Let ω E /( ω ε =0, we d two solutos M 0, ω = ε ( ± = [ E ε ± ( E 4 M (0 ε thus the dgolzed Hlto s H = 0 ε α α ( 0, ε 0, β β ( ( It sees tht should be te s ethe α o β. I t, should be β due to the estto o the e uto ( E Eq.(5 whe we osde the eleto popetes o tels, d the A (,, = ( ( 0, πδ E ε ths exple. euseε E ( 0, ε, due to the eet o the ovelp tx eleet 5

6 M, the eleto stutue sde the sold ould ot be deteed detly by ARPES expeets. Howeve, oe-bd odel o H0 ould be te o good etls, ths se, A (,, ω s just the spetl uto o the eleto systes these etls. o tul tels, espelly o soe stogly oelted tels, we should osde othe tetos H d d 0. I ths se, geelly speg, oe ds G,, jω = ( jω E M /( jω ε Σ(, jω ( I we deote byω the soluto o the equto ω E /( ω ε ReΣ(, ω =0 (3 M thee should be two el solutos o ω, be wtte s ( ε d ε, d the spetl uto egos whee IΣ (, ω ( =0 A = ( ( A ( A ( A = γ (,, ωπδ ( ω ε ( ( A = γ (,, ωπδ ( ω ε The spetl uto the eegy ego whee IΣ(, ω 0 lso be obseved the expeets, but t s ot (4 ou ous ths wo se t does ot oespod to the qus-ptle spetl uto. The spetl uto ( ( ( A (,, ω should be te s A. Moeove, A s ot the ext spetl uto sde the sold, dε s lso eted by M. Ths shows tht the eleto stutues stogly oelted tels ust be deteed o the bss o obg ARPES expeet wth the soluto o the theoet odel. Soe odels sl to Eq.(4 hve bee solved s epoted the ltetues, suh s these ltetues [7-0]. Could they be ppled to expl the eleto stutues o stogly oelted tels? Ths hs to be dsted by expeets wth Eq.(5. 4. Suy Ths wo deves the oul o photoesso testy o the two-bd odel d dsusses how to detee the 6

7 eleto stutues stogly oelted tels by ARPES expeets. Eqs.(-(5 do ot deped o the ollowg dsusso ths wo, d these equtos show tht the eets o the ovelp tx eleet M ust be elted whe we ted to udestd the esults o ARPES expeets. ACKOWLEDGMETS The utho ths jg Uvesty o Ioto See & Tehology o l suppot. 7

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