Measurement of the Electronic Gru¨ neisen Constant Using Femtosecond Electron Diffraction
Shouhua Nie, Xuan Wang, Hyuk Park, Richard Clinite, and Jianming Cao
Physics Department and National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA
(Received 7 September 2005; published 19 January 2006)
We report the first accurate measurement of the electronic Gru¨neisen constant
e
using a novel method
employing the new technique of femtosecond electron diffraction. The contributions of the conduction
electrons and the lattice to thermal expansion are differentiated in the time domain through transiently
heating the electronic temperature well above that of the lattice with femtosecond optical pulses. By
directly probing the associated thermal expansion dynamics in real time using femtosecond electron
diffraction, we are able to separate the contributions of hot electrons from that of lattice heating, and make
an accurate measurement of
e
of aluminum at room temperature. This new approach opens the
possibility of distinguishing electronic from magnetic contributions to thermal expansion in magnetic
materials at low temperature.
DOI: 10.1103/PhysRevLett.96.025901 PACS numbers: 65.40.b, 63.20.Kr, 78.47.+p, 82.53.k
The electronic Gru¨neisen constant (
e
) defines the di-
mensional changes of a solid in response to the heating of
its conduction electrons [1,2]. Like the electronic specific
heat capacity,
e
is an important physical quantity directly
related to the density of electronic states at the Fermi level
(n
E
F
) [3]. Conventional means of measuring
e
utilize
either high precision dilatometry [4,5] or thermoelastic
stress pulses [6,7] at a sample temperature of a few tens
of Kelvin or less. At such low temperatures, however,
dimensional changes associated with magnetic ordering
set in, which make the measurement of
e
in many mag-
netic materials virtually impossible [1,4,8].
Here, we report a new approach to circumvent these
limitations. Instead of cooling down a sample under ther-
mal equilibrium conditions, we transiently heat its conduc-
tion electrons well above the lattice temperature using
femtosecond optical pulses. By directly probing the asso-
ciated thermal expansion dynamics in real time using
femtosecond electron diffraction [9–11], we are able to
differentiate the contributions of hot electron from that of
lattice heating [12], and make an accurate measurement of
e
of aluminum at room temperature. This method opens
the possibility of distinguishing electronic from magnetic
contributions to thermal expansion in magnetic materials at
low temperature.
When subject to any temperature variation, a solid re-
sponds by changing its geometrical parameters through
expansion and/or contraction. This dimensional change is
driven by the minimization of system free energy and
occurs at the microscopic level through rearrangement of
crystallographic cell dimensions and mean positions of
atoms within the unit cell [2,4]. Contributions to the free
energy come from the lattice, itinerant electrons, electric
dipoles, magnetic ions, nuclear spins, and their mutual
interactions. Accordingly, the behavior of thermal expan-
sion is inherently related to the physics governing these
subsystems and their interactions. Among them, the elec-
tronic thermal expansion is of particular importance, as
e
is associated with the derivative of n
E
F
with respect to the
sample volume V ,
e
@ lnn
E
=@ lnV
T;EE
F
[3].
As is the case for specific heat, measurement of
e
is
confronted with the difficult problem of isolating it from
the contributions of other subsystems. For a metal without
magnetic ordering, including a magnetic metal at tempera-
tures above its Curie point, the stress responsible for ther-
mal expansion consists of two independent contributions:
the stress related to the lattice anharmonicity (
l
) and the
pressure of hot electrons (
e
) [4]. Assuming the electrons
and lattice maintain separate states of equilibrium charac-
terized by temperature deviations of T
e
and T
l
after a
thermal perturbation, the combined stress can be written as
[13,14]
e
l
e
C
e
T
e
l
C
l
T
l
; (1)
where C
e
and C
l
are heat capacities for electrons and
phonons,
e
and
l
are the corresponding Gru¨neisen con-
stants. For most metals,
e
and
l
have nearly the same
magnitude; for example, for Al at room temperature,
l
2:16,
e
1:6 [5]. In traditional static thermal measure-
ments, electrons and lattice are equally perturbed and al-
ways in thermal equilibrium with T
e
T
l
. The con-
tributions to thermal expansion from each component are,
as a result, weighed by the magnitudes of their heat ca-
pacities (subsystem thermal energy). At room temperature,
since C
l
is nearly 2 orders of magnitude larger than C
e
[15], the thermal expansion is completely dominated by the
lattice contribution, thus obstructing measurement of
e
.A
traditional approach to getting around this obstacle relies
on lowering the sample temperature below a few tens of
Kelvin, at which C
e
(/ T) becomes comparable to or bigger
than C
l
(/ T
3
). At such low temperatures, however, ferro-
magnetic materials are magnetically ordered [15]. The
associated dimensional changes, such as magnetostriction
[16], display the same or similar temperature dependence
as electronic thermal expansion. This makes the analysis of
low temperature data difficult and prevents a reliable mea-
PRL 96, 025901 (2006)
PHYSICAL REVIEW LETTERS
week ending
20 JANUARY 2006
0031-9007= 06=96(2)=025901(4)$23.00 025901-1 © 2006 The American Physical Society