Pressure-enabled phonon engineering in metalsNicholas A. Lanzilloa, Jay B. Thomasb, Bruce Watsonb,1, Morris Washingtonc, and Saroj K. Nayaka,d

Departments of aPhysics, Applied Physics and Astronomy and bEarth and Environmental Sciences, and cCenter for Integrated Electronics, RensselaerPolytechnic Institute, Troy, NY 12180; and dSchool of Basic Sciences, Indian Institute of Technology, Bhubaneswar 751007, India

Contributed by Bruce Watson, May 2, 2014 (sent for review August 19, 2013)

We present a combined first-principles and experimental study ofthe electrical resistivity in aluminum and copper samples underpressures up to 2 GPa. The calculations are based on first-principlesdensity functional perturbation theory, whereas the experimentalsetup uses a solid media piston–cylinder apparatus at room tem-perature. We find that upon pressurizing each metal, the phononspectra are blue-shifted and the net electron–phonon interactionis suppressed relative to the unstrained crystal. This reduction inelectron–phonon scattering results in a decrease in the electricalresistivity under pressure, which is more pronounced for alumi-num than for copper. We show that density functional perturba-tion theory can be used to accurately predict the pressure responseof the electrical resistivity in these metals. This work demonstrateshow the phonon spectra in metals can be engineered throughpressure to achieve more attractive electrical properties.

density functional theory | electron-phonon coupling |high-pressure conductivity

Strain has proven to be an effective means of modifying theelectronic structure in semiconducting materials, particularly

band gap modulation in metal-oxide-semiconductor field-effecttransistors (1–6). Strain also affects the phonon structure andtransport properties of metals, which have no band gap tomodulate, and may be used to engineer more attractive electricalproperties at both the macroscale and the nanoscale.The nonzero electrical resistivity of a metal has two main con-

tributions: the presence of defects and the vibrations of the latticeatoms about their equilibrium sites (7). Scattering events betweenelectrons and vibrational quanta (phonons) give rise to the finiteelectrical resistivity in pure samples. First-principles calculationshave proven to be remarkably successful in giving accuratedescriptions of the phonon-induced electrical resistivity in metals(8–10). It also has been shown that the phonon-mediated prop-erties, including the electrical resistivity and the superconductingtransition temperature, can be altered under pressure (11–13). Ithas been suggested that the electrical transport properties due tothe electron–phonon interaction in aluminum show a particularlystrong response to interatomic spacing, particularly when thesystem is subject to extreme quantum confinement (14, 15).Studies of the effect of pressure on the superconducting prop-

erties of aluminum suggest that superconductivity is suppressedthrough a reduction in the critical temperature, Tc, as the pres-sure is increased (11, 12, 16–18). It also has been reported that theelectron–phonon coupling constant, λ, decreases in aluminumunder pressure (11, 16), but a quantitative extension to theelectrical resistivity under pressure is lacking. Cheung andAshcroft (13) suggested a decrease in the electrical resistivityof aluminum under volume compression by using a primitivepseudopotential model with experimentally determined interatomicforce constants, but the pressures corresponding to these vol-ume changes are unrealistically large. An in-depth analysis ofthe changes in the phonon spectra, electron–phonon coupling,and phonon-mediated electrical resistivity at realistic pressuresfrom first-principles theory is lacking.In this work, we provide a combined first-principles and ex-

perimental study of the effect of pressure on the electrical resistivityof aluminum and copper. These metals are chosen because Al

shows a pronounced reaction to changes in interatomic spacing(11, 14, 16, 18) and has a simple, nearly spherical Fermi surfacethat makes it easy to treat computationally, whereas Cu typicallyis used for interconnects in industrial applications and has alower intrinsic resistivity than aluminum. We show that uponpressurizing each metal, the electrical resistivity decreases. Fur-thermore, we show that density functional perturbation theorymay be used to accurately predict the pressure response of theelectrical resistivity in simple metals. First-principles calculationsquantitatively match the numerical values of the resistivitychanges found by experiment and explain the reduction in termsof shifted phonon frequencies and suppressed electron–phononscattering as pressure increases.

TheoryMethods. The density functional calculations were carried out usingthe ABINIT software package (19–21). We consider single-atomunit cells of both fcc aluminum and copper, with equilibriumlattice constants of 7.5 bohr (3.97 Å) and 6.8 bohr (3.60 Å),respectively. We use a 16 × 16 × 16 k-point grid, an 8 × 8 × 8q-point grid, and a plane wave cutoff of 10.0 hartree for Al and40.0 hartree for Cu. We use norm-conserving Martin–Troullierspseudopotentials (22), the accuracy of which was tested in previousworks (8, 14). The electrical resistivity is calculated according tothe lowest-order variational solution to the Boltzmann equa-tion, as outlined in several previous works (8–10).The starting point for our calculations is the computation

of the equilibrium phonon spectra for the metal. We find agree-ment for the equilibrium phonon spectra with previous works(8–10). Based on the phonon spectra, the Eliashberg spectralfunction, α2FðωÞ, is calculated as a phonon density of statesweighted according to interactions between electrons and

Significance

Understanding the pressure response of the electrical proper-ties of metals provides a fundamental way of manipulatingmaterial properties for potential device applications. In partic-ular, the electrical resistivity of a metal, which is an intrinsicproperty determined primarily by the interaction strength be-tween electrons and collective lattice vibrations (phonons), canbe reduced when the metal is pressurized. In this article, weshow that first-principles calculations of the resistivity, as wellas experimental measurements using a solid media piston–cylinder apparatus, predict a significant reduction in the elec-trical resistivity of aluminum and copper when subject to highpressure due primarily to the reduction in the electron–phononinteraction strength. This study suggests innovative ways ofcontrolling transport phenomena in metals.

Author contributions: N.A.L., J.B.T., B.W., M.W., and S.K.N. designed research; N.A.L. andJ.B.T. performed research; N.A.L., J.B.T., and B.W. analyzed data; and N.A.L. and J.B.T.wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: watsoe@rpi.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1406721111/-/DCSupplemental.

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phonons at the Fermi level. From the spectral function, one canintegrate to find the electron–phonon coupling constant, λ:

λ= 2Z

α2FðωÞω

dω; [1]

and the electrical resistivity:

ρðTÞ=Xk;k′

3πΩe2gðeFÞ

�v2F�Z

dωx

sinh2 xα2FðωÞηk;k′; [2]

where Ω is the volume of the unit cell, gðeÞ is the density ofstates at the Fermi level, vF is the Fermi velocity, and x is a di-mensionless parameter that incorporates temperature; x= ω

2kBT.

The efficiency factor, ηk;k′, is defined as 1− vk · vk′v2k

and accountsfor electron scattering in different directions. We note thatthe transport spectral function, α2trFðωÞ is defined as the regularspectral function ðα2FðωÞÞ multiplied by the efficiency factor ηk;k′.The effect of pressure is simulated by decreasing the lattice

constant. The values of the lattice constants corresponding toexperimentally applied pressures in the range 0–2 GPa aretaken from the volume ratios in Vaidya and Kennedy (23) foraluminum and from Wang et al. (24) for copper. In the lattercase, the values of volume compression were interpolated linearlyfor pressures below 2.512 GPa.

Results. For the cases of unstrained aluminum and copper at zeropressure, our equilibrium values of phonon frequencies, elec-tron–phonon coupling (λ = 0.45 for Al; λ = 0.15 for Cu) andelectrical resistivity as functions of temperature agree well withthe literature where comparisons are available (7–10).Whereas the results of electrical resistivity as a function of

temperature have been reported elsewhere (7, 8, 10), we plot theelectrical resistivity as a function of pressure for both Al and Cuin Fig. 1. We follow the reasoning in Cheung and Ashcroft (13)and plot the scaled resistivity as opposed to the raw values at agiven pressure. The scaled resistivity is just the resistivity at a givenpressure divided by the zero-pressure value, such that all valuesfall between 0 and 1. In this way, the effects of any systematicerrors in the density functional calculations are cancelled out andthe effect of volume compression is observed more clearly.Although both metals show a significant decrease in electrical re-

sistivity as thepressure is increased from0 to2GPa, the effect ismuchmore pronounced for Al than in Cu. The resistivity of Al decreasesby 8% of the equilibrium value when compressed to 2 GPa, whereasCu resistivity decreases by only 2% at the same pressure. Bothmetals show a roughly linear decrease in resistivity as the pressure is

increased, and the slopes have been calculated and included in thefigure. As evidenced in the plot, Al shows a more aggressive slopewith a value of −0.14μΩ-cm=GPa, whereas Cu has a much shal-lower slope of −0.04μΩ-cm=GPa, indicating that Al shows a morepronounced response to strain.

ExperimentMethods. Aluminum and copper wires with diameters rangingfrom 450 to 490 μm (99.999% purity from Alfa Aesar) were usedin the experiments. The 19-mm–diameter pressure cell assemblieswere designed to have low yield strengths to reduce friction in thepiston–cylinder apparatus (Fig. 2). Reported pressures are be-lieved be accurate within <200 bars based on previous calibrationsin our laboratory of similar assemblies (Table S1). It was notpossible to pressurize an entire length of a wire in an experiment,because it was necessary to conduct resistance measurement usingan external electrical testing device. A segment of each wire (20–66%) was pressurized in a standard 19-mm–diameter piston–cylinder pressure vessel (25); lengths of wire that were not pres-surized extended through the top (15 cm) and bottom (17cm) components of the high-pressure device. The lengths ofwire were varied by wrapping it around a dowel to form a spring-like coil. Care was taken to ensure that individual coils did nottouch one another. The coiled lengths of wire were placed intomachined tubes of pyrophyllite (9.6 mm i.d., 11.6 mm o.d.) thatserved as an electrical insulator. A nickel disk was machined witha 2-mm hole to accommodate an Al2O3 ceramic insulating tube.The disk was attached to the bottom of the pyrophyllite tubewith cyanoacrylate adhesive. An Al2O3-based casting material(Aremco Ceramacast 575) was mixed with water and poured intothe tube to encase the wire with the insulating ceramic. The as-sembly cured under a heat lamp for several hours before storagein a 120 °C oven for more than 24 h. Several experiments wereheated. In those experiments, pyrophyllite tubes were machinedto fit into an electrically conductive graphite heater tube usedin standard piston–cylinder assemblies (26). Rings of NaCl werepressed and drilled to fit around the assemblies.The straight lengths of wire at the bottoms of the assemblies were

coveredwith Teflon insulators and threaded through holesmachinedin the19-mmpiston andpusherpieces so that theassemblywas sittingon the piston (Fig. 2). The pressure vessel was lowered onto the as-sembly and piston. Lead-wrapped salt was inserted into the vesselaround theassembly.Thewireat the topof theassemblywas threadedthrough the steel top plug and top plate. An Al2O3 insulator wasplacedon thewire toextend fromthe topof theassembly, through thesteel top plug and top plate (Fig. 2). The wire was threaded throughthe top plate, which was lowered onto the Al2O3 insulator. A Tefloninsulator covered the wire to the edge of the pressure vessel.An inductor, capacitor, resistor (LCR) bridge (Hewlett-Packard

4275A) was used to measure the resistance of the wires used in theexperiments using a four-terminal configuration with a 10-kHz testsignal. Most experiments were pressurized at room temperaturein ∼0.1-GPa increments up to ∼2 GPa, followed by incrementaldepressurization of ∼0.1 GPa to ∼0.5 GPa. Resistance was mea-sured at each pressure increment. Pressure was cycled up and downseveral times for each experiment. Several experiments wereheated to temperatures 50 °C lower than the wire melting pointswith a 2-h dwell followed by cooling to room temperature. Aftercooling, pressure was cycled up and down several additionaltimes. Resistance measurements could not be performed duringthe heating cycle, because the alternating current passed throughthe graphite heater tube to heat the assembly (Fig. 2) caused in-duction in the Cu and Al wires, which produced inaccurate re-sistance measurements. The Al2O3-based ceramic was removedfrom the wires of several experiments. Cross-sections of the wireswere prepared by grinding flat with SiC papers followed bypolishing in 1-μm Al2O3 and 0.06-μm colloidal SiO2 suspensions.

Fig. 1. The scaled electrical resistivity (resistivity divided by zero-pressureresistivity at room temperature) as a function of pressure for Al and Cucalculated from first principles.

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Backscattered electron imaging was conducted with a CamecaSX100 electron microprobe operating at 15 kV and 20 nA.

Results. The experiments on Al and Cu wires were initiated byincreasing pressure up to 2 GPa, and the results are shown inFig. 3. Initially, the resistance (R) decreased approximately lin-early with increasing pressure (P; slope of R vs. P is dR/dP) untilreaching an inflection point, after which the resistance decreasedat a lower rate. For Al wires, the inflection point occurred at 1.5GPa, and for Cu wire, the inflection point occurred at 1.0 GPa.As discussed in Methods, pressure was cycled up and down sev-eral times for each experiment. During depressurization andsubsequent pressure cycling, dR/dP was lower than during theinitial pressurization (Fig. 3). The large decrease in R duringinitial pressurization is attributed to deformation and micro-structural changes of the metals. The wires did not remain per-fectly round because during experiments, the Al2O3 ceramicpressure medium impinged upon and roughened the surfaces ofthe wires. Scanning electron microscopy (backscattered electronimaging) of the wire starting materials showed they were composedof crystals 1 mm × 0.2 mm elongated parallel to the length of thewires (Fig. S1). Postexperimental backscattered electron imagesshowed that pressurization recrystallized the wires, producingmicrostructures that are more equidimensional than the starting

materials. After pressurization, the average crystal size was∼0.3 × 0.3 mm. Measurements during the initial pressurizationwere not used to evaluate the pressure effect on resistance inthe wires. Given the observable changes in wire microstructuresand the irreproducible change in resistance during initial pres-surization, only subsequent measurements were used to evaluatethe pressure effect on resistance in the wires. To evaluate theeffect of thermal annealing, several experiments were heated to600 °C for 1.75 h after pressure cycling several times (Fig. 3A). Theslope dR/dP after annealing was similar to the reproducible dR/dPthat developed after initial pressurization.The change in resistance with increasing pressure for Al and

Cu wires is larger for longer lengths of wire in the pressure cellassembly (Table 1 and Fig. 2). The maximum amount of wirethat could be included in a ∼45-mm–long pressure cell withoutadjacent coils touching one another was 60%. The minimumamount of wire included in the pressure cell assemblies waslimited by our ability to accurately measure dR/dP with the LCRbridge. For example, increasing the pressure up to 2 GPa on an8.9-cm length of wire (20%) changed the resistance 0.6 mΩ(precision is 0.1 mΩ). The slopes dR/dP reported in Table 1 weredetermined from linear fits to pooled resistance measurementsexcluding data from initial pressurization [e.g., fits were to data

Fig. 2. Schematic drawing of a solid media pressure-cell assembly and part of the piston–cylinder apparatus. See text for details.

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from depressurization 1, pressurization 2 depressurization 2, etc.(Fig. 3)]. Table 1 shows that there is a larger pressure effect onresistance for Al than there is for Cu wires. A plot of percentageof wire in the pressure-cell assemblies vs. slope dR/dP (Table 1)may be used to calculate the change in resistance that wouldresult from pressurization of 100% of the Al and Cu wires.The decreased resistances measured in Al and Cu with in-

creasing P up to 2 GPa are similar to changes in resistancemeasured by Bridgman (27, 28). Bridgman used fluid pressure

media (a glucose–glycerin–water mixture, or molasses) to applypressures up to 1.2 GPa on Al and Cu. For Al and Cu at 1.2 GPa,we obtain resistance values that are 0.5% and 1% higher, re-spectively, than Bridgman’s results. Agreement between thestudies is remarkable because they were conducted nearly acentury apart using dissimilar experimental apparatuses.

DiscussionTo compare the experimental results with theory, we haveplotted the slopes of the R vs. P curves as a function of thepercentage of the wire under pressure. We convert the measuredvalues of resistance (R) to resistivity (ρ) given the known lengthsand cross-sectional areas of the wires under pressure. We thenextrapolate these values of Δρ=ΔP to correspond to a wire with100% coverage under pressure, which of course was not possibleto measure experimentally. We also include the zero point be-cause by definition, the equilibrium resistivity does not changewhen no pressure is applied. The results are shown in Fig. 4. Theextrapolated slopes at 100% under pressure can be comparedwith the results from theory from Fig. 1. We see close agreementin both cases; in units of microohm centimeters per gigapascal,the experimental value for Al is −0.092 whereas the predictedvalue from theory is −0.14. For the Cu wire, the experimentalvalue is −0.048 whereas theory predicts a value of −0.04.Both theory and experiment show a marked decrease in the

electrical resistivity as the pressure is increased up to 2 GPa forAl and Cu metals. This decrease in resistivity is attributed to anoverall weakening of the electron–phonon interaction. A mea-sure of the strength of the electron–phonon interaction is givenby the Eliashberg spectral function, α2FðωÞ. We plotted thesefunctions for both Al and Cu at pressures of 0 and 2 GPa inFig. 5.It is clear that upon pressurizing each metal, the spectral

functions are blue-shifted (i.e., shifted toward higher frequen-cies), which corresponds to a typical stiffening of the phononmodes as the volume is decreased. Because the net electron–phonon scattering goes as 1=ω, any increase in phonon fre-quencies will result in a decrease in electron–phonon coupling.In addition, the heights of the spectral function peaks decreaseas they are shifted to higher frequencies, indicating that thehigher-frequency phonons are less effective at scattering elec-trons. These effects combine to give a weaker electron–phononcoupling for compressed metals relative to the equilibrium (zero-pressure) configurations. Any decrease in electron–phonon cou-pling will be reflected as a decrease in the electrical resistivity.The hardening of the phonon modes under strain can be un-

derstood in terms of the interatomic force constants (IFCs). Asthe lattice is compressed, the effective “springs” between atomsbecome more rigid and result in higher-frequency phonons. Al-though the calculated IFCs for both Al and Cu increase under

Fig. 3. Experimental results for electrical resistance as a function of pressurein the range 0–2 GPa for Al (A) and Cu (B).

Table 1. Experimental results summarized as change inresistance with increasing pressure (dR/dP) measured for variouslengths of aluminum and copper wires in the pressure-cellassembly

Experiment MetalPercentage of wire inpressure-cell assembly

Slope(dR/dP) SE

AlW-15 Al 66.3 −3.01 0.11AlW-17 Al 50.0 −1.75 0.03AlW-19 Al 32.1 −0.89 0.03AlW-22 Al 56.3 −2.02 0.07AlW-23 Al 20.0 −0.40 0.03AlW-18 Cu 50.0 −0.33 0.03AlW-21 Cu 60.8 −0.74 0.03AlW-24 Cu 20.0 −0.20 0.02

Fig. 4. A linear fit of the values for the pressure coefficients of resistivitycorresponding to different segments of the Al and Cu wires under pressure.Linear extrapolation to 100% gives values to compare with theory.

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pressure, the increases are greater for the former than for thelatter. Specifically, the head element of the IFC matrix for Alincreases by 8.7% in going from 0 GPa to 2 GPa, whereas thesame element increases by only 6.3% in Cu. The nearest-neighborIFC in Al increases by 8.25% in going from 0 to 2 GPa, whereasthe same element for Cu increases by only 7.2%. This IFCmodulation is the root cause of the phonon blue-shifting thatoccurs under pressure, which in return, suppresses the netelectron–phonon interaction.It also is worth noting that the Fermi velocity ðvFÞ will be

renormalized in the presence of electron–phonon interactions,which are nonadiabatic in nature. These effects will be in addi-tion to the adiabatic change in the Fermi velocity due to latticecompression. We find that the adiabatic compression of thelattice results in an increase in the square of the Fermi velocity

ðv2FÞ from 0.58 a.u. at 0 GPa to 0.60 a.u. at 2.0 GPa. However, thenet Fermi velocity, which includes both adiabatic and nonadiabaticcontributions, is found to decrease as the lattice is compressedfrom 0.53 a.u. at 0 GPa to 0.52 a.u. at 2.0 GPa. We note that theinclusion of both adiabatic and nonadiabatic effects results ina smaller Fermi velocity because of the presence of the ð1+ λÞterm in the denominator of the equation

vF =v0F

1+ λ; [3]

which will decrease the renormalized Fermi velocity relative to thebare value ðv0FÞ. This indicates that the nonadiabatic effects due toelectron–phonon coupling under pressure outweigh the adiabaticchanges in the band structure due to lattice compression.

ConclusionTo summarize, we have shown that the electrical resistivity of bothaluminum and copper decreases under pressure. The change inresistivity is more pronounced in aluminum under pressure than incopper, because the phonons are blue-shifted to higher frequencies,which suppresses the net electron–phonon interaction. Experimentsconfirm the trends predicted by theory, and quantitative agreementis found in comparing the slopes of resistivity vs. pressure curves.This demonstrates how phonons can be engineered through strainto achieve more attractive electrical properties with applica-tions ranging from interconnects to integrated circuits.

ACKNOWLEDGMENTS. This work was partially supported by the NationalScience Foundation Integrative Graduate Education in Research and Trainee-ship Fellowship, Grant 0333314, as well as the Interconnect Focus Center(Microelectronics Advanced Research Corporation program) of New York. Com-puting resources were provided by the Computational Center for Nanotechnol-ogy Innovations at Rensselaer, partly funded by the State of the New York.

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