Kondo vs Fano resonances in Quantum Dot

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Lydia Hamilton
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Giuliao ( iv. Calabria, Italy) P.Luigao, B.ouault, A.T., B.L.Altshulr, PRB7,3(R)(5) P.

1 ivrsita Frio II i Napoli Italy Koo vs Fao rsoas i Quatum Dot Capri Capri 4/5 4/5 P.tfasi, B.Bula (Poza) A.T., P.Luigao, A.Nao B.ouault (CNR Motpllir) D.Giuliao ( iv. Calabria, Italy) P.Luigao, B.ouault, A.T., B.L.Altshulr, PRB7,3(R)(5) P.tfasi, A.T., B.Bula, PRL 93,8685(4) P.tfasi, A.T., B.Bula, o-mat/5385

2 Go rs (MIT) Diffrt ool ow V g Liar outa Fu hr (aovr)

3 Th thr rgims of th prvious sli: CB, Koo a Fao orrspo to irasig of hybriizatio of th ot with th otats. owvr thr is o otiuity... tatmt: Koo rsoat trasmissio is strogly pt o hargig ffts Fao rsoas our i a vi i whih Coulomb Bloa has b wash out. imultaous prs of th two is just a spulatio. (whih is partly isuss hr also)

4 CB First ltur: basis o Koo outa Koo rsoa o ltur: -- basis o Fao -- what is s i ots

5 ourtsy of M.Kastr B

6 lt us start with! attrig amplitus: f < f > L ot R Tulig aross th ot

7 igl lvl ot with o itratig ltros ( ) Γ α α πν V ( ).. ˆ, ˆ, h V CV N N R L T g D L T D L α α α α α α α α o orr prturbatio thory i th tulig :

8 ( ) Γ i i i i V πδ ( ) V πν Γ ( ) t t i t Γ ψ Quasi bou stat i th ot i th abs of itratios Prturbativ orrtio to th ot ltro stat: ( ) ( ) δ ν

9 Now : Isolat sigl lvl i prs of hargig ˆ D µ ˆ ( µ )ˆ, ( ˆ N ) µ, N CV g hmial pottial : µ N ( N ) ( N ) o o N µ µ aitio rgy : ( ) N N Arso mol: N ; ; ( ) ( N ) ( N ) o o ( N ) o

10 Va r Wil (Dlft) () v-o fft:

11 xtrm limit: sigl oupay of th impurity. at low tmpraturs: if < µ :, o allow µ a forbi o A sigl spi ½ suffis!

12 Quhig of th harg gr of from o th ot lvl : milassial limit: ( ) µ µ β Tr Z ; xp ( ) ( ) [ ] 4 ( ) ( ) [ ] ( ) 4 4 β β β β β δ

13 δ : symmtri Arso mol: µ δ ( ) β Costrait of sigl sit oupay! ( ) ( ) β β β 4 δ( ) z is th oly yamial variabl!

14 µ ( ) [ ( ) ] wh T, itratio btw spis bom sstial 3 4; 4; tot tot χ ( T ) µ β ; ; β β 3 µ 4 4 : G is a siglt if >! Why th itratio shoul b AF? ot ltro spi / outio ltros spis

15 Is th itratio AF? Costrai ilbrt spa is Ξ, full Fo spa is,,, Projtio oto Ξ rsults i a AF ouplig virtual oubl oupais ar grat with tulig to a fro th otats.

16 {[ ] ( ) ( ) } δ ( ) V V : : hoos x ( ) ( ) ( ) ( ) ( ) ( ) * * V V V V

17 ( ) ( ) { } Ξ Ξ * V V P P * * VV VV Q ( ) ( ) Ξ Ξ z L ff Q P P trm with z : spi ½

18 ( ) ( ) ( ) [ ] Q z z L ff V V Q ta I th symmtri Arso mol V Q 4,

19 Log sigularity i th T-matrix xpasio Cosir oly spi-flip prosss: ff [ ( ) ( ) ] Las ( ) ( r i Las ) G ( i Las V) G r [ ] ( ) ( ), V ( ) G T r G r G VG r ( ) r i V VG V... r G r G TG r r

20 T a attrig of a ba ltro ( ) r i VG V... N ( f ( )) i _ b N i ( ) f ( i ) a b... T _

21 Log sigularity is a quatum fft: ab lassially z ( ) D D z i N b a quatum: z ( ) f os ot isappar

22 ( ) D D f f b a l l F B T 4 l... summig th sris F B ff T ν 4 ()l (p T fiit...)

23 prour shoul b mor arful: Poor ma s salig Dal with high rgy D stats prturbativly: ( D D δd), µ -D ( D D δd),

24 alig: D D δd implis: δ j ν () δ ν () δd D ouplig is ruig with D: j o j( Do) Itgrat from D o to D<D o j( D) j D l o D o or:

25 j j( D) o, j > D o j l o D o j sals to wh D. D a sal ow to B T This quatity is sal ivariat: j o j D o D T K alig (prturbativ) approah is vali up to T >> TK

26 Koo grou stat: Atifrromagti ouplig : T Rsrvoir siglt ot Rsrvoir siglt

27 Nozirs Frmi liqui hypotsis: trog ouplig stat has o mmory of spi but a rsoa at µ - sattrig approah: ta th sattrig tr at th origi ψ (x) [ os r sig( x) f ir [ ] o ir i si r f v o r x sattrig amplitu

28 i-out sattrig stats: out l i ψ ( r) ( ) ψ ( r) l l out ( ) l ( r) ψ l ψ l (r) out r x i ( ) l ψ ( r) l poitli sattrr ψ l (r) i O hal vaishs at th ot loatio: it is fully trasmitt ayhow. o ( )

29 matrix : l ( ) l f iδ l uitary matrix: R T l l ( ) TTrasmissio t l T o iδ π si δ l l l l πi ( ) l t π v v ( ) t si δ o o Couta: t-matrix ~ G ~ g o h 4Γ ( Γ Γ ) L g ~ L o T Γ R R

30 Koo rsoa T << T K at tmpraturs at µ with of th rsoa : πt K o µ o Im [ Σ ] K (orvati[87]) alulat via IP

31 O rsoa at µ : v o δ δ π T si v δ ( uitarity limit) Couta: ~ G ~ h g o Fril sum rul: imp δ v ( µ π ) ν ( ) ImTr π imp µ ν ( ω ) ω ( r r G G ) π µ o δ ω v Im π ω ω l t tˆ π δ ω v

32 Tmpratur p of th outa T a) : los to th uitarity limit T K << T b) : i th prturbativ rgim a) ~ G h T ω iπtk iδ ω iπt si K δ v tg δ tg δ δ ( ω ) artg ω π T K πt ω K ( ω T ) B fiit T as ilasti prosss : as T agai.

33 b) : i th prturbativ rgim ~ ~ π ~ π ( ) ~ π Go t Go ν () Go l o FL FL G T K << T I fat : optial thorm ( ) π o t πi R t o T T K t wg w δ ω i I { } t πν( ) m from th fiitio of T K : ν () l D T K with ( D T ) B a ν () L hv F

34 G/G o T/T K of th first part

Lecture contents. Density of states Distribution function Statistic of carriers. Intrinsic Extrinsic with no compensation Compensation

Lecture contents. Density of states Distribution function Statistic of carriers. Intrinsic Extrinsic with no compensation Compensation Ltur otts Dsity of stats Distributio futio Statisti of arrirs Itrisi trisi with o ompsatio ompsatio S 68 Ltur #7 Dsity of stats Problm: alulat umbr of stats pr uit rgy pr uit volum V() Larg 3D bo (L is