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Thermodynamic studies of strongly correlated 2D electron system

Vladimir Pudalov, Ginzburg Center, LPI

Landau ITP

Michael Reznikov, Technion, Haifa Alexander Kuntsevich, LPI Igor Burmistrov, Landau ITP

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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)

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Motivation

• Exper data shows strong growth of χs with rs

(i.e. F0

σ→ -1). Stoner instability in the 2D FL-state ?

• 2D systems are probed mainly by transport. Can the

thermodynamics be measured when the number of particles ~ 108 ?

• Transport studies reveal inconsistency with

homogeneous FL concepts. Can the thermodynamics shed a light ?

λF

3

n E E r

F ee s

1 ∝ =

Strong growth of χ* ∝ m*g* with density lowering (rs growing)

2 4 6 8 rs

n =8×1010см-2

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n =3×1012см-2

V.M.Pudalov, et al., PRL 88, 196404 (2002); PRB 2008

Strong growth of |F0

a | with lowering n

(increase of rs)

• N. Klimov, D. Knyazev, O. Omelyanovskii, V. Pudalov,

H.Kojima, M. Gershenson, PRB 78, 195308 (2008)

σ

1 * F g g

b

+ =

Towards Stoner (or Bloch) instability

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Ground state energy of the 2D system

Tanatar, Ceperley, PRB 1989

crystal

 Variational and fixed- node MC calculations have insufficient accuracy  No way to measure Eg  Constructive approach: to measure ∂E/∂x

rs= U/EF ∝ n-1/2

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

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For the ideal Fermi gas

• 1. Сompressibility ∂µ / ∂n

2D F-gas 2D FL

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For the ideal Fermi gas VP et al, JETPLett.(1985)

• 1. Сompressibility ∂µ / ∂n

2D F-gas 2D FL

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• 2. Spin magnetization ∂µ / ∂B.

Principle of measurements

_ + VG Out Modulated magnetic field B+δΒ Current Amplifier Ohmic contact Gate SiO2 Si 2D electron gas

Advantages

• High sensitivity (108 spins)
• Measures thermodynamic

magnetization

• Accessibility of the Insulating

phase

• Low-field measurements

δB~ = 0.03T, 6Hz

M.Reznikov, A.Yu.Kuntsevich, N.Teneh, V.M.P, JETP Lett. (2010). N.Teneh, A.Yu. Kuntsevich, V.M.P, M.Reznikov, Phys. Rev. Lett. 109, 226403 (2012).

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Electric circuitry

N.Teneh, AK, VP, M.R., PRL 109, 226403 (2012)

Maxwell relation

V=Q/C0+∆µ/e

V

B dB dn C ~ e ∂ µ ∂ − =

2

V Out

Modulated magnetic field B+δΒ

Current Amplifier Ohmic contact SiO2 Si

2D electron gas

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µ eVG

Al

ε0 eφ WAl W2D

/ cos( ) / sin( )

n B

U e B t e B e B C t I B e M B n ϕ µ µ ω ϕ µ µ ω ω µ = ∆ + ∆ ∂ ∆ ∆ = −∆ = − ∂ ∂ ∆ = ∂ ∂ ∂ = − ∂ ∂   

SiO2 Si z

Principle of measurements Maxwell relation:

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F(n,B) – free energy

dM/dn, expectations for the degenerate Fermi-gas (no interactions) B µ gµΒΒ

Polarization field

µB

dM

dn 2EF/gµB

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B

B ∂

∂µ µ 1

gµBB ~2EF

B (Tesla)

O.Prus, Y.Yaish, M.Reznikov, U.Sivan, V.Pudalov, PRB, 67, 205407 (2003)

Earlier measurements (high fields).

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n = 1.5·1011 cm-2

B

B ∂

∂µ µ 1

gµBB~2EF B (Tesla) kT

N.Teneh, A.Yu. Kuntsevich, V. M. Pudalov, and M. Reznikov, Phys.Rev.Lett. 109, 226403 (2012).

Low field measurements: B < T

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n = 1.5×1011 cm-2

dM/dn > μB FM - interaction ! t = T/Tc, Tc∝ nk J ~ 1/2b* ~ 2 b*

Mean field simulation

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dM/dn changes sign with T !

dM/dn > μB FM - interaction ! t = T/Tc, Tc∝ nk dM/dn changes sign with n !

Mean field simulation

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n=0.5x1011

FL Two phase state

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Sign reversal of dM/dn

T=1.7K 6.8K

2×1011cm-2 0.5×1011cm-2

• dM/dn (µB)
• ∂M/∂n <0 at n>nc ⇒

each electron added to the 2D system causes decrease in the number

• f SDs
• ∂M/∂n > 0 for n→ 0

∂M/∂n → 0 at n = nc ⇒ A critical behavior of ∂M/∂n

∂M/∂n < 0 for n > nc

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Thermodynamic spin susceptibility

This is the response

• f the overall

electrons

Susceptibility of the localized spins greatly exceeds and masks that of the itinerant electrons

T

1.7 8K

2 4 6 8 10

• 0.5

0.5 1 1.5 2 2.5 3 3.5 n [1011 cm-2] ∂χ/∂n [µB/T]

1.7K 1.8K 2K 2.2K 2.4K 2.7K 2.9K 3.1K 3.3K 3.5K 3.8K 4K 4.2K 4.6K 5.1K 5.7K 6.9K 8K 9.2K 13.1K

Density dependence of dχ/dn

nc

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Sign change of dχ/dn (and dM/dn): a critical behavior

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Sign change of dM/dn: critical behavior

nc

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Thermodynamic spin susceptibility: T-dependence

This is the response of the overall electrons Susceptibility of the localized spins diverges as ~(1/T)2

Part 2: Entropy

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Entropy per electron Problem: n =0 is inaccessible

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Differential entropy per electron Problem: n =0 is inaccessible

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Experimental set-up & principle of measurements

Liquid He bath heat sink α

For f > α/C ~0.1Hz, ∆T0 ~1/Cf and i ≠ i(f)

sample

J = j0cos(ωt/2)

− = ∆ dt T d c ) (

) 2 cos( 2 1 ) (

2

ft r i T dt T d C π α = ∆ − ∆

) sin( ) ( t TC T t i ω µ ∆ ∂ ∂ =

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Samples and their parameters

T = 2.5-25K B =0-9Tesla ∆T~0.05-0.25K f ~ 0.15 - 5 Hz

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For a degenerate 2D Fermi-gas with D = Const dS/dn = - dµ/dT= 0.

D depends on the carrier density Non-degenerate system Interacting system

S=-Σ{ f.ln(f)+(1-f).ln(1-f)}

Expectations: Entropy for the 2D case

When dS/dn ≠ 0 ?

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Entropy magneto-oscillations

SPIN GAP VALLEY GAP VALLEY GAP CYCLOTRON GAP

Ideal degenerate 2D gas

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dS/dn vs B

• Y. Tupikov et al, JETPL 2015

Ideal 2D gas in GaAs/AlGaAs

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What is expected in zero field?

T<<TF. Degenerate Fermi liquid T~TF. Non-degenerate Fermi-gas

Small corrections

n, EF/T

?

U~e2/<r>~n1/2 EF~n rs~U/EF~n-1/2 n=1011 cm-2

70K 7K 10

> 0 < 0

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Positive & negative (∂S/ ∂n)

EF≈10K

In accord with FL: (i) The higher the temperature, the larger is the entropy (ii) As n increases, (dS/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

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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

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Positive (∂S/ ∂n)

However, (dS/dn) exceeds the value calculated for the ideal Fermi-gas

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Role of the disorder Disorder does not affect the entropy behavior!!

“Dirty sample” Clean sample

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Checking the 3rd low: Entropy integration

The 3rd low of thermodynamics in the Fermi-liquid

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Thermodynamic effective mass

The effective mass m*(n) shows a reentrant behavior. It tends to mb as n→0.

m*(n) from SdH in the FL regime

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Thermodynamic effective mass

The effective mass m*(n) shows a reentrant behavior. It tends to mb as n→0.

Strongly correlated plasma regime: EF<T<<U

m*(n,T) can be scaled using effective parameter

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Thermodynamic effective mass & Plasma regime parametrization

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Summary:

 One can measure ∂S/∂n for a system with n>108 electrons.  High densities, low temperature — Fermi-liquid  Low densities — strongly correlated plasma: Novel state

• f the electronic matter, where interaction parameter is

T- and n- dependent.

Thank you for attention!

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