arXiv:0802.3354v2 [cond-mat.str-el] 24 Feb 2008 rsuei re olwradfietn h nryo the of energy of the application fine-tune and the lower been to order be has in must approach pressure time Our life their [5]. within decay achieved in of the not generation and excitons optical are between the particles. excitons equilibrium excited state laser the steady but A materials state of lowest How- energetically class the this semiconductors. in in excitons ever, on concentrated has and effects new promise [4] and applications features excitons tempera- the new of initiate high statistics would coupling Bose the The strong understanding superconductors. to ture for weak tem- essential lower from a crossover became at The (BC) condensation Bose perature. whereas, a Weakly and to theory; temperatures subject high BCS are at [3]. form within excitons form as coupled formation strongly could coherence, simultaneous phase a state reveal and ground should new excitons bound a mechanism how possible the of into insights provided exci- insulators the of coined. notion been The has or (EI) than gap. insulator larger energy tonic is exciton the exci- the to of of comparable energy formation binding also the spontaneous might if the (SC) tons that against semiconductor realized unstable a [2] of be Knox structure not band does attraction. electrons the Coulomb free hole of the electron- density Fig. unsta- screen bound small is a (see the such (SM) because one of semimetal pair formation or a the that zero against suggested spin ble [1] since to Mott boson add a 1). is spins quasiparticle pair individual a electron-hole such the bound exciton; an A due called is attraction. electron stays Coulomb charged hole the out charged negatively to excited positively the is a attracting electron band, behind, character an valence if fermionic the However, the from of electrons. the consequence of the is sulators h xeietlraiaino ooi rudstate ground bosonic a of realization experimental The excitonic of concept theoretical the of elaboration The in- and semiconductors metals, of structure band The n oebudt arwt hi pn de oitgrval integer to added spins their TmSe with in pair holes a and to electrons bound that hole a and rudsaei odne matter. condensed d in thermodynamically gap state The signi indirect ground experimentally. a an found implies in hence, as excitons and, lattice, phonons of of number means large by conservation a of formation The nte16sseuain rs fagon tt xssi so in exists state ground a if arose speculations 1960s the In hroyaia intrso nEctncInsulator Excitonic an of Signatures Thermodynamical a.fu et¨repyi,EHZuih 03Zuih Sw Z¨urich, 8093 Z¨urich, f¨ur Festk¨orperphysik, ETH Lab. o lmsNtoa aoaoy o lms M855 USA 87545, NM Alamos, Los Laboratory, National Alamos Los S ohcuefu ehi,84 aprwl Switzerland Rapperswil, 8640 f¨ur Technik, Hochschule HSR uo ak .D Thompson D. J. Park, Tuson 0 . 45 Te Dtd a 9 2018) 29, May (Dated: 0 . 55 ee Wachter Peter en Bucher Benno ofr xiosa h hroyaia rudstate. ground thermodynamical the as excitons form do adcue f5 yrdzto ihtersl faso- Tm a state of conduction result valent the the intermediate with from called hybridization states en- 4f-5d a 4f small causes the the band actually, of but, separation regime ergetic binding ionic the for thdottcpesr.Wt hstcnqe ml si- small a calorimetry technique, ac this With An pressure. transitions hydrostatic phase [9]. the studying at for on adapted pressure. was depends [10] starting technique that down the path cooling and pressure on cell a constant follows pressure not rather the was but pressure as The employed was supercon- gauge. manometer A lead fluid. ducting transmitting pressure silicon-based a phase. thermodynamical to new and a capacity of new heat existence This the a the measure establish of thermodynamics. to formation motivation by the the corroborated was Yet be trans- must 2a). the phase of phase (Fig. data A these properties state. of port semimetal out the deduced in been has than diagram higher below is conductivity that heat K 20 giving order resistivity, first the a like showed jump, [8] conductivity Thermal transport. electrons 4f 1). quasi-localized (Fig. the kbar of 5 a around states the by crossing of demanded level density as energy high their a) 2 with excitons Fig. of to local- formation (inset effect to kbar, electrons due Hall is 8 of anomaly gap. ization and the the that of 5 reveal [7] closing between measurements anticipated increases range the resistivity of pressure the despite the K, in 250 strongly transition below gap But a energy temperatures semimetal. is ferromagnetic the at there a to K and semiconductor 300 [6] a At from pressure of applying tens band. on (some valence closes narrow 4f a wide acquire meV) states 4f localized erally sr ol ugs h aecsTm valences the suggest would istry xiost h nrylvlo lcrn ntevalence the in electrons 1). of Figure level (see energy band the to excitons efcapn uepesr elhsbe sdwith used been has cell pressure CuBe self-clamping A heat from come have phase new a to hints Further TmSe rvdpaedarmssan bosonic a sustains diagram phase erived cn hneo h etcpct fthe of capacity heat the of change ficant 0 . 45 i tt aeil iha electron an with materials state lid eiodco eursmomentum requires semiconductor Te e,ie xios eew show we Here excitons. i.e. ues, 0 . 55 sanro a eiodco.Chem- semiconductor. gap narrow a is itzerland +2 +2 . 2 [Se hr h gen- the where 0 . 45 Te 0 . 55 ] − 2 2

∆ 2’ (5d) (TmSe 0.45 Te 0.55 ) exciton e- 300

21

K 10 L M N

T = 5K )

-3

20

10 (cm + 250 19 n h phonon 10

18

10 ENERGY ENERGY

B

A

200

2 4 6 8 10 12

0 kbar PRESSURE (kbar) exciton level meV

150

13 5 kbar

120 2+ 4f (Tm ) -

= phonon semiconductor semimetal e exciton level

100 E ∆ h+ TEMPERATURE (K)

50

A Γ WAVE VECTOR k X B

0

0 2 4 6 8 10 12 14 16 FIG. 1: Schematic illustration of the band structure of PRESSURE (kbar) TmSe0.45Te0.55 on applying external pressure. Exciton levels below the X point of the Brillouin zone reduce their separa- tion to the quasi-localized 4f13 electrons of Tm2+. Around

5 kbar, it becomes energetically favorable for an electron to occupy the excitonic energy level, leaving behind a positively

charged hole. Conservation of momentum requires the bind- 50 ing of a phonon with a wave vector from the Γ to the X point.

T (K)

50 100 150 200 250 300

nusoidal heat input is supplied to the sample (1x1x0.2 25 3

11 kbar mm ) through a constantan wire bent to a meander-like (J/mole K per f.u.)

12 kbar m

13 kbar c structure attached to one side of the sample. The result- 14 kbar ac ac HEAT CAPACITY (a.u.) 14.5 kbar

15 kbar

ing temperature oscillation was detected by a Chromel- 17 kbar @ RT ac HEAT CAPACITY (a.u.) CAPACITY HEAT ac Au/Fe thermoelement attached to the other side of the 0 single crystal. At steady state, the time-average tempera- 0 50 100 150 200 250 300 TEMPERATURE (K) ture T1 of the sample is larger than the bath temperature (pressure medium). In addition, there is a small temper- ature oscillation about T1 with a peak-to-peak tempera- FIG. 2: Heat capacity of TmSe0.45Te0.55 at high pressures a) ture ∆Tac. Under optimal conditions of the ac frequency, Phase diagram constructed from a number of different mea- • N ∆Tac is directly related to the heat capacity. However surements: (gray) onset of resistivity increase; maximum the calculation of the absolute value of the heat capacity of resistivity;  maximum of ac heat capacity; ⋆ first or- H would require the knowledge of thermal resistances to the der transition of heat conductivity; first order transition of resistivity and  the onset of deviation of ac heat capacity bath and the thermometer. In the absence of this infor- of pressure run L. The inset shows the pressure-dependent mation, the absolute value of heat capacity was estimated Hall number n at 5 K [7], where n decreases in phase A and by scaling the high-temperature ac heat capacity to the electrons start to delocalize again in phase B. b) ac heat ca- Dulong-Petit value of TmSe0.45Te0.55 of 52 J/molK per pacity data for different pressure runs: black (zero pressure), f.u. However, some influence of the pressure transmitting red (run M), blue (run N, metallic), green (run L) and ma- fluid can not be prevented; thus, conclusions are based genta (run K). A ferromagnetic phase transition at 6 K is mainly on differential changes of the heat capacity. With discernible in the metallic phase. The inset to Fig. 2 b) rep- resents the experimental results for seven pressure runs that this technique, phase transitions also manifest themselves cross the semimetal to phase B boundary. Peaks in Cac () through a marked phase shift of the thermo-signal across are denoted in Fig. 2 a). The green arrow indicates the phase the sample. transition from phase A to phase B.

Figure 2 b) shows the ac heat capacity Cac for selected pressure runs. The pressure inside the self-clamped cell reduces on cooling, as determined in independent mea- change in slope of the heat capacity of run K around surements, and, consequently, the heat capacity could be 250 K, signalling a second order phase transition. The measured neither at constant pressure nor at constant reduced heat capacity of run K below 250 K (20 meV) in- volume. Depicted are the ac heat capacities of repre- dicates thermally no longer activated lattice vibrations. sentative pressure runs K, L, M, N and zero pressure The quasi-localization of free carriers starts below the (black). The heat capacities show a conventional cur- same temperature [6]. Indeed, wave-vector conservation vature for a Debye-like phonon density of state at zero upon the formation of excitons in an indirect gap semi- pressure and in the metallic regime. The phase boundary conductor implies the binding of high energy phonons at from semiconducting to phase A is reflected as a large the boundary of the Brillouin zone to form excitons (Fig. 3

1). The number of excitons is then given approximately

by the decrease in free-carrier density, measured by the 20 − 3 Hall number [7]: ∆nmax ≃ 10 e /cm . The heat ca-

pacity of run K is nearly linear between 100 to 250 K, 6.0

1.6 % which points to a constant phonon density of states. The

5.9 reduced heat capacity also implies a stiffer lattice [11], 8.5 LATTICE CONSTANT (Å) [8]. 50 100 150 200 250 300

TEMPERATURE (K)

Pressure run L enters phase B from phase A around

200 K. Only 1 kbar higher in the semimetal phase, the C (J/mole K ) heat capacity changes its shape qualitatively (inset to Fig. 2b). Pressure run M enters phase B from the semimetal side and a giant change takes place below 200

K. In the metallic region (run N) the heat capacity recov- 0 ers a conventional Debye-like curvature but with a lower 50 100 150 200 250 TEMPERATURE (K) Debye temperature ΘD, as expected for intermediate va- lent systems at higher pressures [8]. By subtracting the heat capacity of run N, the specific FIG. 3: Heat capacity for the violet pressure run (11.8 kbar at heat of the runs from the metallic region into phase B 5 K, Fig. 2b) after subtracting the conventional background can be decomposed into a phononic part and a contribu- characterized in run N. A fit of the extra heat capacity ∆C tion ∆C belonging to the phase transition. A fit of the reveals a Gaussian peak (dark gray) together with a broader extra heat capacity ∆C reveals a Gaussian peak together contribution at lower temperatures (light gray). The dark gray area reflects a latent heat. The accompanying expansion with a broader contribution at lower temperatures (Fig. of the lattice increases the pressure inside the self-clamped 3). A Gaussian shape of the heat capacity is known for pressure cell counterbalancing the phase transition. The inset first order transitions [12] as a temperature gradient is al- shows the large lattice expansion at the same pressure [6]. ways present [13]. Furthermore, neutron measurements [6] show an isostructural phase transition with an expan- sion of the volume by 4.8% at the phase boundary from Let us first discuss the first order phase transition into semimetal to phase B (inset to Fig.3). This expansion of phase B. The spin entropies for the thulium ions are +2 7 the lattice increases the pressure inside the self-clamped Sspin/kB = ln(2J +1) = 2.08 for Tm (J = 2 ) and pressure cell, counterbalancing the phase transition. A 2.56 for Tm+3 (J = 6), respectively. A gain of entropy first order transition also has been observed in the resis- by a valence change from Tm+3 to Tm+2 would result in tivity [6], Hall constant [7] and heat conductivity [8]. ∆Sspin = 0.48·kB = 3.98 J/mol K , in good agreement The broadening of the transition allows calculation of with the experimental result shown in the inset of Fig. 4. the entropy change and, therefore, a verification of the The first order expansion of the lattice might then solely Clausius- Clapeyron relation: be a consequence of the larger Tm+2 ions. The f-electron intra-site Coulomb repulsion does not favor this picture dp ∆S ∆Q and, particularly, the Hall measurements show that elec- = = (1) dT ∆V T · ∆V trons in phase B are not localized as they are in phase A [7]. The electrons in phase B seem to compensate the lo- dp The slope dT on the left side of Eq. (1) can be de- cal magnetic moment of the Tm ions but are not localized termined from the phase boundary shown in Fig. 2a and on the Tm site itself. The idea of electrons redistributed gives −5.2·105 J/m3 K at 200 K. For the right side of Eq. over d-orbitals that are bound to its original anion site, (1), the corresponding change of entropy at the bound- i.e. excitons, also has been used to explain the golden ary (dark gray shaded peak in Fig. 3) provides ∆S=0.8 phase of SmS [14],[15]. − J/mol K. With a volume expansion of 1.56·10 6 m3/mol, The expansion of the crystal at the first order phase the resulting ∆S/∆V = −5.1 · 105 J/m3 K is in excellent transition against the pressure medium (≈ 12 kbar) re- agrement with the slope of the phase boundary. The ad- quires an energy ∆Eelastic = p · ∆V ≈ 1800 J/mol = ditional broad heat capacity at lower temperatures com- 20 meV/f.u. This value is to be compared with the la- prises ∆S=1.6 J/mol K and, hence, gives a total change tent heat L = ∆S · T = 165 J/mol = 1.7 meV/f.u. The of entropy of 2.4 J/mol K (inset to Fig. 4). spin entropy which usually drives the Kondo and Kondo Figure 4 shows the additional contribution ∆C/T for - lattice effects in highly correlated electron systems can the pressure runs into phase B. The change of entropy not explain the phase transition. Only an electronic en- due to entering phase B from phase A was estimated by ergy could account for the large energy per formula unit; subtracting the heat capacity of run K from run L (green the binding energy of excitons is in this range. This curve in Figure 4). The total change of entropy is similar formation of excitons has been predicted for electrons for all runs. in TmSe0.45Te0.55 interacting via a statically screened 4

TmSe0.45Te0.55 at high pressure have revealed a pro- nounced change in the phonon spectrum up to 250 K.

4 ) The binding of a large number of high energy phonons is in agreement with the formation of an excitonic insulator S (J/mole K (J/mole S

) phase. The gain of spin entropy points to the formation 2

0

0 50 100 150 200 250 of singlet excitons. 0.04

T (K)

The authors thank F. Bronold, H.Fehske, and A. T T (J/mole K

/ Schilling for valuable discussions. Work at Los Alamos

C was performed under the auspices of the US DOE, Office of Science. Correspondence and requests for further in- formation. should be addressed to Benno Bucher (email:

0.00 [email protected]).

0 50 100 150 200 250 300

TEMPERATURE (K)

FIG. 4: ∆C/T for different pressure runs. The pressures [1] N. Mott, Philos. Mag. 6, 287 (1961). at low temperatures are from right to left: cyan (10.7 kbar), [2] R.S. Knox, Solid State Physics, Eds. F. Seitz, D. Turnbull violet (11.8 kbar), red (11.9 kbar), dark yellow ( 13.5 kbar), and H. Ehrenreich (Academic Press Inc, New York, 1963) brown (13.7 kbar), black (13.9 kbar) and green (7.6 kbar). Suppl. 5, p 100. The change of entropy due to entering phase B from phase A [3] For example see: B. Halperin and T. Rice, Solid State was calculated by subtracting the heat capacity of run K from Physics, Eds. F. Seitz, D. Turnbull and H. Ehrenreich run L (green curve). The inset shows the integrated area of (Academic Press Inc, New York, 1968) Vol. 21. ∆C/T , which corresponds to the involved change of entropy [4] M. Rontani and L. J. Sham, Phys. Rev. Lett. 94, 186404 and is about the same for all pressure runs. (2005). [5] For a review see: P.B. Littlewood et. al., J. Phys.: Con- dens. Matter 16, S3597 (2004). Coulomb potential [16]. The broad contribution to the [6] J. Neuenschwander and P. Wachter, Phys. Rev. B 41, heat capacity in phase B (Fig. 3) could be attributed to 12693 (1990). the continuous formation of bound excitons. But the for- [7] B. Bucher, P. Steiner, and P.Wachter, Phys. Rev. Lett. mation of a large number of excitons with heavy holes is 67, 2717 (1991). 69 subject to crystallization [3] and the first order transition [8] P. Wachter, B. Bucher, and J.Malar, Phys. Rev. B , could indicate a melting process. 094502 (2004). [9] J.D. Thompson, Rev. Sci. Instr. 55, 231 (1984). The localization of the electrons in phase A [7] is [10] P. Sullivan and G. Seidel, Phys. Rev. 173, 679 (1968). accompanied by the disappearance of phonons (below [11] For example see: T.H.K. Barron and G.K. White, Heat 250 K of run K in Fig. 2b) as expected at the for- Capacity and Thermal Expansion at Low Temperatures mation of a large number of excitons in an indirect (Kluwer Academic, New York, 1999) gap semiconductor. Hence, the insulating phase A [12] A. Schilling et.al., Phys. Rev. Lett. 78, 4833 (1997). could be interpreted by the opening of a ”many-body [13] This phenomenon is similar to boiling a pot of water: gap”, i.e. the proposed Bose condensate of excitons the water does not instantly turn into gas, but forms a turbulent mixture of water and water vapor bubbles. [17]. The ground state of an excitonic insulator in an Furthermore if a substance as water and TmSe0.45Te0.55 indirect gap semiconductor competes with an instability expands upon cooling the effect called regelation takes to an electron - hole liquid [3]. But the asymmetry place: melting under pressure and freezing again when of electron to hole mass of mh/me ≈ 80 may prevent the pressure is reduced. 76 the e-h liquid instability in TmSe0.45Te0.55 and favors [14] K. Matsubayashi et.al., J. Jap. Phys. Soc. , 033602 crystallization in a two-component Coulomb system (2007). [15] K.A. Kikoin, Sov. Phys. JETP 58, 582 (1983). [18]. Even more significant might be the fact that the [16] F.X. Bronold and H. Fehske, Phys. Rev. B 74, 165107 movement of holes is inherently related to a change of (2006). the local magnetic moment. A magnetic binding energy [17] Ji-Min Duan, D.P. Arovas, and L. J. Sham, Phys. Rev. may then support the formation of hole - electron bosons. Lett. 79, 2097 (1997). [18] M.Bonitz, V.S.Filinov, V.E.Fortov, P.R.Levashov, and In summary ac heat capacity measurements of H.Fehske, Phys. Rev. Lett. 95, 235006 (2005).