BEC-BCS cross-over in the exciton gas

Transcription

1 BEC-BCS ross-over in the exiton gas Monique Combesot Institut des NanoSienes de Paris, CNRS Université Pierre et Marie Curie Evora ot

2 1)Dilute limit: BEC ondensate of linearly polarized dark exitons 2)Under a density inrease: - mixture of dark and bright ondensates - phase separation between BEC and BCS ondensates 2

3 Part 1 dilute limit 3

4 In the dilute limit, eletrons and holes form exitons. At low T, they undergo a Bose-Einstein ondensation in a linearly polarized dark state 4

5 Semiondutor ondution band k photon valene band k v photon Q p ondution eletron k = k v + Q p k v valene eletron 5

6 eletron k e photon hole k h eletron k e = k v + Q p photon Q p hole k h = "k v 6

7 hole: full valene band minus one eletron formidable redution of the many-body problem! 7

8 Naive view of an exiton "e eletron + e hole exiton similar to Hydrogen atom light effetive mass dieletri onstant m e * " 0.1m e " s #10 R X = me4 $ #13.6eV 0.1 ' 2h 2 2 & ) " s % 10 2 ( R X "10meV a X = h2 " s $ 10 ' # 0.53A 2 & ) me % 0.1( a X " 50A 8

9 The true story is far more omplex 9

10 Coulomb interation in a periodi lattie Bloh states r nk = eik.r L 3 2 u nk (r) u nk(r) = u nk(r + a) + nk = a nks v 10

11 V Coul = V e"e "V e"e " n 2 k 2 # q " n 1 k 1 + q n 2 k 2 n 1 k 1 = 1 2 $ +...a " + n 1 k 1 +qa n " 2 k 2 #q a n2 k 2 a n1 k 1 v q ( n 1 ",n 1 )v #q ( n " 2,n 2 ) v q (n,n) = 4"e2 v q ( n " # n) = O(q 0 ) L 3 q 2 repulsive sattering between ondution and valene eletrons 11

12 Intraband Coulomb proesses 4"e 2 L 3 q 2 v v v v v v dressed by a dieletri onstant whih results from many-body effets indued by «boiling valene band» 12

13 2 1 v 2 v 1 v 1 v v 2 13

14 4"e 2 L 3 q 2 1 q v 1 q... q 2 v q (n,n) = 4"e2 L 3 q 2 v q ( n " # n) = O(q 0 ) 1 q q 0 v... q 2 1 q v 14

15 n 2 n 1 n 2 2 4"e n # s L 3 q 2 1 Diret satterings between valene and/or ondution eletrons: repulsive as between free eletrons redued by the semiondutor dieletri onstant 15

16 repulsion between ondution and valene eletrons attration between eletrons and holes + a k1 +q + a vk2 "qa vk2 a k1 + "a vk2 a vk2 "q We turn from ondution/valene eletrons to eletrons/holes a k = a k a vk = b "k V v = "4#e 2 & $ s L 3 q 2 q%0 & + a k1 +q k 1 k 2 b + "k2 b "k2 +qa k1 "k 2 + q = # k 2 16

17 h e k 1 + q k 2 " q k 1 k 2 e h " 4#e2 $ s L 3 q 2 Attration between one eletron and one hole redued by dieletri onstant 17

18 One Wannier exiton e e e h h h e e h h e h k e + k h = Q onserved through Coulomb satterings Wannier exiton ",Q = # k e,k h... energy " # + Q 2 2(m e + m h ) k h,k e ",Q wave funtion r e,r h ",Q = eiq.r L 3 2 # " (r) R = m er e + m h r h m e + m h r = r e " r h 18

19 Exiton reation operator ",Q = # k e,k h k h,k e ",Q B + "Q 0 a + ke b + kh 0 B + "Q = # a + ke b + kh k h,k e ",Q 19

20 Exitons are boson-like partiles i = (" i,q i ) [ B + + j,b ] i " = B + j B + i " B + i B + j = 0 as bosons [ + B m,b ] i " = # mi " D mi but not exatly [ + D mi,b ] j = # + " {... }B n n n j n j m i + m i 20

21 Being boson-like partiles, exitons, as old atoms, must undergo Bose-Einstein ondensation Yet, the preise nature of omposite boson ondensate is not known! lose to (B 0 + ) N but surely not exatly (B + ) N 0 21

22 Exiton BEC searhed for deades. but never evidened! Reason: the ondensate must be dark and searh has been done through photoluminesene experiments 22

23 ondution band l = 0 s =1/2 j e z = ± 1 2 valene band l =1 l h z = ±1,0 s =1/2 s h z = ± 1 2 j h z = ± 3 2,± 1 2 pair z J eh = j e z + j h z = ±2,±1,0 23

24 Although small, interband Coulomb interation also exists v v v 1 v 1 v 2 v 2 e 1 e 2 h 2 h 1 emission and reabsorption of a virtual photon exists for bright pairs J= (+1, 0, -1) only 24

25 z J eh = ±1,0 bright oupled to " ±,# photons z J eh = ±2 dark unoupled to photons no valene-ondution Coulomb proesses 25

26 Coulomb intraband exists for all e Coulomb interband exists for bright only h (repulsive) interband Coulomb proesses push bright exiton above dark exiton 26

27 Dark exitons have the lowest energy just beause they are dark! Exiton BEC must be dark No hope to see it (diretly) with photoluminesene experiments Better to know where ondensation takes plae! 27

28 Pairs are reated in bright states by photon absorption However, dark / bright exitons are linked by arrier exhange +2 1/2-3/2-1 dark -2 3/2-1/2 +1 bright 28

29 Dark ondensate must have a linear polarization H N = 0 BN HB +N 0 0 B N B +N 0 a " 2 (+2) + a " #2 (#2) B + = a 2 B a "2 B "2 a 2 (+2) + a "2 ("2) a " 2 (+2) + a " #2 (#2) a 2 (+2) + a "2 ("2) a " 2 (+2) a 2 (+2) a # "2 ("2) a "2 ("2) + a " a 2 (+2) 2 (+2) a # "2 ("2) a "2 ("2) H N minimum for a 2 2 = a "2 2 =1 2 linear polarization 29

30 So, in the dilute limit, eletrons and holes form exitons. At low T, they undergo a Bose-Einstein ondensation in a linearly polarized dark state 30

31 How an we see a dark ondensate? paraboli trap lowest state dark lowest trap level we predit the appearane of a dark spot at the enter of the trap when BEC takes plae 31

32 32

33 Optial trap: standing wave (+Q, -Q) K exiton K +Q photon +Q K + 2Q exiton K - Q photon +Q K exiton beomes superposition of (K, K +2Q, K- 2Q) so, it is trapped. Potential depth: a fration of mev 33

34 Part 2 Under a density inrease 34

35 (A) a bright omponent appears in the ondensate 35

36 dark exitons bright exitons D k + B k + H kin = k 2 " D + k D k + (# 0 + k 2 " )B + k B k 2M 2M lowest state (D 0 + ) N 0 dark ondensate 36

37 When the density inreases, we must inlude interations k 4 1 "2 k 2 k 3 "1 2 the Hamiltonian then reads k 2 H = " D + k D k + "(# 0 + k 2 )B + k B k + 2M 2M k 1 g " B + k4 B + k3 D k2 D k1 + h.. mean field treatment D + D " N d B + B " N b D " B " N d e i# d N b e i# b H " # 0 N b + 2g N b N d os2($ b %$ d ) minimum " g 37

38 H " # 0 N b $ 2 g N b N d minimum "H "N b = 0 N b (0) = 2 g N "# 0 4 g " 0 #10µeV m " 0.1 threshold N th L 2 "10 9 m #2 N " N b N th = " 0 2 g " # 0 $ exh ( ) " # 0 R X ( L a X ) D for N > N th the lowest state (D + 0 ) N"N ( 0) b (B + 0 ) N ( 0) b 0 ondensate has a bright omponent 38

39 Josephson osillations " = " b #" d and "N = (N b # N d ) /2 are onjugate variables Hamilton equations d("n) = #H dt #$ d" dt = # $H $(%N) lose to equilibrium "N # "N (0) +...os$ J t h 2 " 2 J = 32g 2 bd N (0) b N (0) d = 2# 2 0 ( N 2 N $1) 2 th " 0 #10µeV " J #10 10 N /N th 39

40 (B) a phase separation takes plae between exiton gas and eletron-hole plasma 40

41 Mott dissoiation a X Exiton gas Eletron-hole plasma " = N( a X L )D Exitons dissoiate into an eletron-hole plasma for " #1 41

42 dilute exiton gas E N = NR X "1+ (...)# +... [ ] = N$ X (#) " = N( a X L )D positive to avoid ollapse dense eletron-hole plasma # E N = N (...) k 2 F 2m " & % (...)e2 k F +...( = N) eh (*) $ ' N " (k F L) D " eh (#) $ %n 2 3 & 'n 1 3 " #n $ %n 1 2 in 3D in 2D the average pair energy has a minimum in the dense limit 42

43 "(#) «eletron-hole droplets» in Ge and Si " 0 " = N( a X L )D "R X T 0 x eh phase separation eh " 0 ( L a X ) D N 43

44 "(#) " 1 " 2 " = N( a X L )D "R X N = N 1 + N 2 dn 1 = "dn 2 E N = N 1 "(# 1 ) + N 2 "(# 2 ) 0 = de N dn 1 = "(# 1 ) + # 1 $ " (# 1 ) %"(# 2 ) %# 2 $ " (# 2 ) for #"($ 1 ) = #"($ 2 ) "(# 2 ) $"(# 1 ) # 2 $# 1 = % " (# 1 ) = % " (# 2 ) Phase separation along double tangente 44

45 T 0 x " 1 ( L a X ) D eh eh phase separation " 2 ( L N ) D a X 45

46 (C) a BCS ondensation ours in the dense eletron-hole plasma 46

47 Even if strongly sreened eletrons and holes still attrat eah other #...a k b "k = B Cooper Formation of «eletron-hole Cooper pairs» neutral, so not superonduting but still superfluid 47

48 Again, dark pairs have the lowest energy Exhange oupling must however bring a bright omponent to the ondensate sine it is dense Moreover, degeneray between the two dark and the two bright pairs must lead to a linear polarisation as optimum state (B B + "2 ) N"N ( 0) 1 (B B + "1 ) N ( 0)

49 T exiton BEC eh dark ondensate x phase separation «gray» ondensate " 2 ( L N " ) D 1 ( L ) D a X a X exiton BCS 49

50 So, under a density inrease, -a bright omponent appears in the ondensate. -a phase separation ours between exiton gas and e-h plasma -a BCS ondensation of exitoni Cooper pairs takes plae in the e-h plasma 50

51 Conlusion Eletron-hole systems are quite rih! 1) At low density, exitons are formed. They suffer BEC ondensation into a dark state with linear polarisation 2) Under a density inrease, - a bright omponent appears in the ondensate - a phase separation ours between exiton gas and eletron-hole plasma - a BCS ondensation of exitoni Cooper pairs takes plae in the dense plasma 51

52 Referenes Bose-Einstein ondensation of exitons: the key role of dark exitons Monique Combesot, Odile Betbeder-Matibet, Roland Combesot Phys. Rev. Lett. 99, (2007) Optial traps for dark exitons Monique Combesot, Mihael Moore, Carlo Piermarohi Phys. Rev. Lett. 106, (2011) «Gray» BCS ondensate of exitons and internal Josephson osillations Roland Combesot, Monique Combesot Phys. Rev. Lett. 109, (2012) 52