Thermodynamic Aspects of Magnetic Molecules

G. Lefkidis

Department of Physics and Research Center OPTIMAS, Kaiserslautern University of Technology,Box 3049, 67653 Kaiserslautern, Germany

San Francisco, Dec 1, 2014

Outlook

• Introduction Hamiltonian Basics of quantum thermodynamics

• Understanding QTD Z system Doublet harmonic oscillator

• Results Otto cycle and magnetism Diesel cycle and magnetism

• Summary

Outlook

• Introduction Hamiltonian Basics of quantum thermodynamics

• Understanding QTD Z system Doublet harmonic oscillator

• Results Otto cycle and magnetism Diesel cycle and magnetism

• Summary

Degrees of Freedom in the Hamiltonian

SOC

electroniccorrelations

static B-field phonons

laser

at atel elel atel

1 11 11 11

2)0( 1

2

1ˆN

a

N

b ba

baN

i

N

j ji

N

i

N

a ia

aN

i

ZZZH

RRrrrR

elelel

111132

eff)1( ˆˆˆˆ

2ˆ

N

ii

N

iS

N

iL

N

i i

ael

Rc

ZH

q

q qBSBLSL

)(ˆ)(ˆ)(ˆ )10()10()2( tttH laserlaser BMEd )()(ˆ)( )2( ttHtdt

di

iE

i

eiiTZ

T )(

1)(

temperature

Electronic Correlations in the Hamiltonian

occupied

virtual

Configuration interaction or coupled cluster expansion

Slater-rules → & Mulliken analysis (localization)

at atel elel atel

1 11 11 11

2)0( 1

2

1ˆN

a

N

b ba

baN

i

N

j ji

N

i

N

a ia

aN

i

ZZZH

RRrrrR

Otto Cycle

Eugenio Barsanti and Felice Matteuccipatented around 1854-1857

Alphonse Beau de Rochas patented 1861

Nicolaus Otto built 1863

1→2 isentropic compression2→3 isochoric heating3→4 isentropic expansion4→1 isochoric cooling (exhaust +intake)

Basics of Quantum Thermodynamics

• quantum levels:• occupations:

• quantum first law of thermodynamics

• internal energy

• work

• heat

H. T. Quan, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. B 76, 031105 (2007)

• volume

• pressure

changing only :quantum adiabatic process

𝑑𝑈=∑𝑛

(𝐸𝑛𝑑𝑝𝑛+𝑝𝑛𝑑𝐸𝑛 )𝑑𝑊=∑

𝑛

𝑝𝑛𝑑𝐸𝑛

𝑑𝑄=∑𝑛

𝐸𝑛𝑑𝑝𝑛

𝐹=−∑𝑛

𝑝𝑛

𝑑𝐸𝑛

𝑑𝐿

(1D system)

→ {𝐸𝑛′ }

{𝑝𝑛 }

𝑈=∑𝑛

𝑝𝑛𝐸𝑛

{𝐸𝑛}

Basics of Quantum Thermodynamics

H. T. Quan, Y.-X. Liu, C. P. Sun, and F. Nori, Phys. Rev. B 76, 031105 (2007)

• quantum first law of thermodynamics

• internal energy

• work

• heat

• volume

• pressure

changing only :Heating/cooling

𝑑𝑈=∑𝑛

(𝐸𝑛𝑑𝑝𝑛+𝑝𝑛𝑑𝐸𝑛 )𝑑𝑊=∑

𝑛

𝑝𝑛𝑑𝐸𝑛

𝑑𝑄=∑𝑛

𝐸𝑛𝑑𝑝𝑛

𝐹=−∑𝑛

𝑝𝑛

𝑑𝐸𝑛

𝑑𝐿

(1D system)

{𝐸𝑛}→ {𝑝𝑛

′ }

𝑈=∑𝑛

𝑝𝑛𝐸𝑛

• quantum levels:• occupations: {𝑝𝑛 }

Outlook

• Introduction Hamiltonian Basics of quantum thermodynamics

• Understanding QTD Z system Doublet harmonic oscillator

• Results Otto cycle and magnetism Diesel cycle and magnetism

• Summary

¿𝑔 ⟩¿𝑒 ⟩

𝐸𝑔=0𝐸𝑒=𝑎𝑥+Δ

𝑥

𝐸

+thermalization if

Z System

𝐸𝑢¿𝑢⟩

¿𝑔 ⟩¿𝑒 ⟩

𝐸𝑔=0𝐸𝑒=𝑎𝑥+Δ

𝑥

𝐸

Z System

𝐸𝑢¿𝑢⟩

𝑄¿=[𝑝𝑒 (𝑥2 ,𝑇2 )−𝑝𝑒 (𝑥1 ,𝑇 1 ) ]𝐸𝑒 (𝑥2 )+[𝑝𝑢 (𝑥2,𝑇 2 )−𝑝𝑢 (𝑥1 ,𝑇1 ) ]𝐸𝑢

𝑄out=[𝑝𝑒 (𝑥2 ,𝑇 2 )−𝑝𝑒 (𝑥1 ,𝑇1 ) ]𝐸𝑒 (𝑥1 )+[𝑝𝑢 (𝑥2 ,𝑇2 )−𝑝𝑢 (𝑥1 ,𝑇 1 ) ]𝐸𝑢

useful work produced by maximized near (not at)crossing/degeneracy

useless energy-move-around due to minimized if→gap+finite temperature

𝜂=𝑄¿−𝑄out

𝑄¿<1

A B

CD

¿𝑔 ⟩¿𝑒 ⟩

𝐸𝑔=0𝐸𝑒=𝑎𝑥+Δ

𝑥

𝐸

Z System

𝐸𝑢¿𝑢⟩

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0T 2

0 .2

0 .4

0 .6

0 .8

1 .0

For and very small we get but very littleenergy conversion per cycle. At room temperature we get

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0T 2

0 .1

0 .2

0 .3

0 .4W o u t𝑇 1→0

𝑇 1→0

A B

CD

𝜂=[𝑝𝑒 (𝑥2 ,𝑇 2 )−𝑝𝑒 (𝑥1 ,𝑇 1 ) ][𝐸𝑒 (𝑥2 )−𝐸𝑒 (𝑥1 )]

[𝑝𝑒 (𝑥2 ,𝑇2 )−𝑝𝑒 (𝑥1 ,𝑇1 ) ]𝐸𝑒 (𝑥2 )+ [𝑝𝑢 (𝑥2 ,𝑇 2)−𝑝𝑢(𝑥1 ,𝑇1 ) ]𝐸𝑢

𝜂=𝑄¿−𝑄out

𝑄¿<1

Doublet Harmonic Oscillator

C. D. Dong, G. Lefkidis and W. Hübner, Phys. Rev. B 88, 214421 (2013)

𝑍=∑𝑛=1

∞

exp¿¿

E=1𝑍

¿

𝐸𝑛

+¿=( 12+𝑛)𝜔 −𝜇𝐵¿

𝐸𝑛−=( 12+𝑛)𝜔+𝜇𝐵Spin up

Spin down

Doublet Harmonic Oscillator

C. D. Dong, G. Lefkidis and W. Hübner, Phys. Rev. B 88, 214421 (2013)

𝐶𝑣=𝜕𝐸 (𝑇 ,𝐵 ,𝜔 )

𝜕𝑇

𝜒𝑀=𝜕𝜕𝐵

¿

Heat capacity

Magnetic susceptibility

Outlook

• Introduction Hamiltonian Basics of quantum thermodynamics

• Understanding QTD Z system Doublet harmonic oscillator

• Results Otto cycle and magnetism Diesel cycle and magnetism

• Summary

Otto Cycle on Ni2: Efficiency and Work Output

W. Hübner, G. Lefkidis, C. D. Dong, D. Chaudhuri, L. Chotorlishvili, and J. BerakdarPhys. Rev. B 90, 024401 (2014)

• Magnetism increases output power at maximum efficiency• Magnetism increases work output

Otto Cycle on Ni2: Equilibrium Distance

W. Hübner, G. Lefkidis, C. D. Dong, D. Chaudhuri, L. Chotorlishvili, and J. BerakdarPhys. Rev. B 90, 024401 (2014)

Only molecules can both expand and compress spontaneously!

Diesel Cycle on Ni2: Isobaric Process in Ni2

Isobaric ranges

C. D. Dong, G. Lefkidis and W. Hübner, J. Supercond. Nov. Magn. 26, 1589 (2013)

𝐹 (𝑅 ,𝑇 )=− 1𝑍 (𝑅 ,𝑇 )∑𝑖

𝑑𝐸 𝑖(𝑅)𝑑𝑅

𝑒−𝐸𝑖 (𝑅)𝐾𝑇

Non-equilibrium, non-thermalized states!

Entropy and Quantum Thermodynamics

Entropy:

• Crossings of lines in PV diagrams due to additional degrees of freedom• Even larger configuration space for magnetic molecules

C. D. Dong, G. Lefkidis and W. Hübner, Phys. Rev. B 88, 214421 (2013)

Heat Capacity and Magnetic Susceptibility of Ni2

C. D. Dong, G. Lefkidis and W. Hübner, Phys. Rev. B 88, 214421 (2013)

𝜒𝑀=𝜕 ⟨𝑀 (𝑇 ,𝑅 ) ⟩

𝜕𝐵𝜒 𝑖𝑗

(𝑥𝑦 )∝∑𝑘

(𝐷𝑖𝑘(𝑥))∗𝐷𝑘𝑗

( 𝑦) 𝑝𝑖−𝑝 𝑗

𝐸𝑖−𝐸 𝑗−𝜔+𝑖 Γ

Singularities indicate level crossings.

Heat Capacity and Magnetic Susceptibility of Ni2

C. D. Dong, G. Lefkidis and W. Hübner, Phys. Rev. B 88, 214421 (2013)

Magnetic susceptibility at 100 K

Magnetic susceptibility at 400 K

Heat capacity

Von-Neumann entropy𝑆=−∑

𝑛𝑖

𝑝𝑛Tr (𝜌𝑛(𝑟𝑒𝑑) ln𝜌𝑛

(𝑟𝑒𝑑))

Summary

• QTD: Effects of energy discretization (in particular level crossings)

• Electronic temperature – non thermalized states

• Quantum Otto Cycle – effects of magnetism

• Isobaric process – Quantum Diesel Cycle

Acknowledgements

Wei Jin (Xi’an) Hongping Xiang (Northridge) Chuanding Dong

Chun Li (Xi‘an)Debapriya Chaudhuri