Thermodynamic studies of strongly correlated 2D electron system - PowerPoint PPT Presentation

Thermodynamic studies of strongly correlated 2D electron system Vladimir Pudalov, Ginzburg Center, LPI Alexander Kuntsevich, LPI Igor Burmistrov, Landau ITP Landau ITP Michael Reznikov, Technion, Haifa 1

Thermodynamic studies of strongly correlated 2D electron system V.M. Pudalov, A.Yu. Kuntsevich, M.E. Gershenson, I.S. Burmistrov, M. Reznikov, Phys. Rev. B 98 , 155109 (2018). L.A.Morgun, A.Yu. Kuntsevich, and V.M.P, Phys. Rev. B 93, 235145 (2016). N.Teneh, A.Yu. Kuntsevich, V.M.P, M.Reznikov, Phys. Rev. Lett . 109 , 226403 (2012). A.Yu.Kuntsevich, Y.V.Tupikov, V.M.P., I.S.Burmistrov, Nature Comm . 6, 7298 (2015). Y.Tupikov, A.Yu.Kuntsevich, V.M.Pudalov, I.S.Burmistrov, JETP Lett. 101 , 125 (2015) 2

Motivation • Exper data shows strong growth of χ s with r s σ → -1). Stoner instability in the 2D FL-state ? (i.e. F 0 • 2D systems are probed mainly by transport. Can the thermodynamics be measured when the number of particles ~ 10 8 ? • Transport studies reveal inconsistency with homogeneous FL concepts. Can the thermodynamics shed a light ? E 1 = ∝ ee r s E λ F n F 3

Strong growth of χ * ∝ m*g* with density lowering ( r s growing) n = 3 × 10 12 см -2 n = 8 × 10 10 см -2 2 4 6 8 r s V.M.Pudalov, et al., PRL 88 , 196404 (2002); PRB 2008 4

a | with lowering n Strong growth of | F 0 (increase of r s ) g = b g * σ + 1 F 0 Towards Stoner (or Bloch) instability N. Klimov, D. Knyazev, O. Omelyanovskii, V. Pudalov, H.Kojima, M. Gershenson, PRB 78, 195308 (2008) 5

Ground state energy of the 2D system  Variational and fixed- node MC calculations have insufficient accuracy  No way to measure E g crystal  Constructive approach: to measure ∂ E/ ∂ x r s = U / E F ∝ n -1/2 Tanatar, Ceperley, PRB 1989 6

 First Derivatives ∂ E/ ∂ x : ∂ E / ∂ n = µ chemical potential ∂µ / ∂ n compressibility ∂µ / ∂ B magnetization ∂µ / ∂ T entropy 7

1. С ompressibility ∂µ / ∂ n For the ideal Fermi gas 2D F-gas 2D FL 8

1. С ompressibility ∂µ / ∂ n For the ideal Fermi gas 2D F-gas 2D FL VP et al, JETPLett. (1985) 9

2. Spin magnetization ∂µ / ∂ B. Current Amplifier Principle of measurements + Gate V G _ Out SiO 2 Modulated magnetic field B+ δΒ Si Ohmic contact 2D electron gas δ B ~ = 0.03T, 6Hz Advantages M.Reznikov, A.Yu.Kuntsevich, •High sensitivity (10 8 spins) N.Teneh, V.M.P, JETP Lett. (2010). •Measures thermodynamic N.Teneh, A.Yu. Kuntsevich, V.M.P, magnetization M.Reznikov, Phys. Rev. Lett. 109, •Accessibility of the Insulating 226403 (2012). phase •Low-field measurements 10

Electric circuitry Maxwell relation V Modulated magnetic field Current Amplifier B+ δΒ V=Q/C 0 + ∆µ /e V Out SiO 2 Si Ohmic contact ∂ µ 2 e dn = − 2D electron gas ~ ∂ C dB B N.Teneh, AK, VP, M.R., PR L 109 , 226403 (2012) 11

Principle of measurements e φ W 2D = ∆ ϕ + ∆ µ U / e µ ∂ µ ∆ ω W Al eV G B cos( t ) ∆ ϕ = −∆ µ = − ∂   / e ε 0 B e Al ∂ µ ∆ ω ω SiO 2 B C sin( t )  = ∂ Si I B e z ∂ µ ∂ M = − Maxwell relation: ∂ ∂ B n n B F(n,B) – free energy 12

dM / dn , expectations for the degenerate Fermi-gas (no interactions) dM dn µ µ B 2 E F / g µ B g µ Β Β B Polarization field 13

Earlier measurements (high fields) . ∂ µ n = 1.5 · 10 11 cm -2 1 µ B ∂ B g µ B B ~2 E F B (Tesla) O.Prus, Y.Yaish, M.Reznikov, U.Sivan, V.Pudalov, PRB , 67, 205407 (2003) 14

Low field measurements : B < T n = 1.5 × 10 11 cm -2 ∂ µ 1 µ B ∂ B kT g µ B B~2E F B (Tesla) N.Teneh, A.Yu. Kuntsevich, V. M. Pudalov, and M. Reznikov, 15 Phys.Rev.Lett. 109, 226403 (2012).

dM / dn > μ B FM - b * interaction ! Mean field simulation t = T / T c , T c ∝ n k J ~ 1/2 b* ~ 2 dM/dn changes sign with T ! 16

dM / dn > μ B n=0.5x10 11 FM - interaction ! Mean field simulation t = T / T c , T c ∝ n k dM/dn changes sign with n ! 17

Two phase state FL 18

Sign reversal of dM / dn 2 × 10 11 cm -2 T=1.7K 0.5 × 10 11 cm -2 - dM / dn ( µ B ) 6.8K ∂ M / ∂ n <0 at n > n c ⇒  each electron added to the 2D system causes decrease in the number of SDs ∂ M / ∂ n > 0 for n → 0  ∂ M / ∂ n → 0 at n = n c ⇒ A critical behavior of ∂ M / ∂ n ∂ M / ∂ n < 0 for n > n c 19

Thermodynamic spin susceptibility This is the response of the overall electrons T 1.7 8K Susceptibility of the localized spins greatly exceeds and masks that of the itinerant electrons 20

Density dependence of d χ / dn 1.7K 1.8K 3.5 2K 2.2K 3 2.4K 2.7K 2.5 2.9K ∂χ / ∂ n [ µ B /T] 3.1K 2 3.3K 3.5K 1.5 3.8K n c 4K 1 4.2K 4.6K 0.5 5.1K 5.7K 0 6.9K 8K -0.5 9.2K 13.1K 0 2 4 6 8 10 n [10 11 cm -2 ] 21

Sign change of d χ / dn (and dM / dn ): a critical behavior 22

Sign change of dM / dn : critical behavior n c 23

Thermodynamic spin susceptibility: T-dependence This is the response of the overall electrons Susceptibility of the localized spins diverges as ~(1/T) 2 24

Part 2: Entropy 25

Entropy per electron Problem: n =0 is inaccessible 26

Differential entropy per electron Problem: n =0 is inaccessible 27

Experimental set-up & principle of measurements ∂ µ ∆ = ω i ( t ) TC sin( t ) ∂ 0 T sample ∆ d ( T ) = − c dt ∆ J = j 0 cos( ω t/2) heat sink α d ( T ) 1 − α ∆ = π 2 C T i r cos( 2 ft ) 0 dt 2 Liquid He bath For f > α / C ~0.1 Hz, ∆ T 0 ~1/ Cf and i ≠ i ( f ) 28

Samples and their parameters T = 2.5-25K B =0-9Tesla ∆ T ~0.05-0.25K f ~ 0.15 - 5 Hz 29

Expectations: Entropy for the 2D case S=- Σ { f . ln(f)+(1-f) . ln(1-f)} For a degenerate 2D Fermi-gas with D = Const dS / dn = - d µ / dT = 0. When dS/dn ≠ 0 ? D depends on the Non-degenerate system carrier density Interacting system 30

31 Entropy magneto-oscillations Ideal degenerate 2D gas VALLEY VALLEY CYCLOTRON SPIN GAP GAP GAP GAP

Ideal 2D gas in GaAs/AlGaAs dS / dn vs B Y. Tupikov et al, JETPL 2015 32

What is expected in zero field ? n=10 11 cm -2 U~e 2 /<r>~n 1/2 70K E F ~n 7K r s ~U/E F ~n -1/2 10 T ~ T F . Non-degenerate Fermi-gas > 0 T << T F . Degenerate Fermi liquid < 0 Small corrections ? n, E F / T 33

Positive & negative ( ∂ S/ ∂ n) E F ≈ 10K In accord with FL: (i) The higher the temperature, the larger is the entropy (ii) As n increases, (d S/dn) decreases to 0 (iii) For the lowest T ’s and high densities, (dS/dn) gets negative (iv) The effective mass agrees with that extracted from SdH 34

Negative ( ∂ S/ ∂ n) In accord with FL : (i) The higher the temperature, the larger is the entropy (ii) As n increases, S decreases to 0 (iii) For the lowest T ’s and high densities, S gets negative (iv) The effective mass agrees with that extracted from SdH 35

Positive ( ∂ S/ ∂ n) However, (dS/dn) exceeds the value calculated for the ideal Fermi-gas 36

Role of the disorder “Dirty sample” Clean sample Disorder does not affect the entropy behavior!! 37

Checking the 3rd low: Entropy integration The 3rd low of thermodynamics in the Fermi-liquid 38

Thermodynamic effective mass The effective mass m*(n) shows a reentrant behavior. It tends to m b as n → 0. m*(n) from SdH in the FL regime 39

Thermodynamic effective mass The effective mass m*(n) shows a reentrant behavior. It tends to m b as n → 0. Strongly correlated plasma regime: m *( n,T ) can be scaled using effective parameter E F < T << U 40

Thermodynamic effective mass & Plasma regime parametrization 41

Summary:  One can measure ∂ S/ ∂ n for a system with n>10 8 electrons.  High densities, low temperature — Fermi-liquid  Low densities — strongly correlated plasma : Novel state of the electronic matter, where interaction parameter is T- and n- dependent . Thank you for attention ! 42