Pressure-enabled phonon engineering in metals

Nicholas A. Lanzillo; Jay B. Thomas; Bruce Watson; Morris Washington; Saroj K. Nayak

doi:10.1073/pnas.1406721111

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

Significance

Understanding the pressure response of the electrical properties of metals provides a fundamental way of manipulating material properties for potential device applications. In particular, the electrical resistivity of a metal, which is an intrinsic property determined primarily by the interaction strength between electrons and collective lattice vibrations (phonons), can be reduced when the metal is pressurized. In this article, we show that first-principles calculations of the resistivity, as well as experimental measurements using a solid media piston–cylinder apparatus, predict a significant reduction in the electrical resistivity of aluminum and copper when subject to high pressure due primarily to the reduction in the electron–phonon interaction strength. This study suggests innovative ways of controlling transport phenomena in metals.

Abstract

We present a combined first-principles and experimental study of the electrical resistivity in aluminum and copper samples under pressures up to 2 GPa. The calculations are based on first-principles density functional perturbation theory, whereas the experimental setup uses a solid media piston–cylinder apparatus at room temperature. We find that upon pressurizing each metal, the phonon spectra are blue-shifted and the net electron–phonon interaction is suppressed relative to the unstrained crystal. This reduction in electron–phonon scattering results in a decrease in the electrical resistivity under pressure, which is more pronounced for aluminum than for copper. We show that density functional perturbation theory can be used to accurately predict the pressure response of the electrical resistivity in these metals. This work demonstrates how the phonon spectra in metals can be engineered through pressure to achieve more attractive electrical properties.

Strain has proven to be an effective means of modifying the electronic structure in semiconducting materials, particularly band gap modulation in metal-oxide-semiconductor field-effect transistors (1 ⇓⇓⇓⇓–6). Strain also affects the phonon structure and transport properties of metals, which have no band gap to modulate, and may be used to engineer more attractive electrical properties at both the macroscale and the nanoscale.

The nonzero electrical resistivity of a metal has two main contributions: the presence of defects and the vibrations of the lattice atoms about their equilibrium sites (7). Scattering events between electrons and vibrational quanta (phonons) give rise to the finite electrical resistivity in pure samples. First-principles calculations have proven to be remarkably successful in giving accurate descriptions of the phonon-induced electrical resistivity in metals (8 ⇓–10). It also has been shown that the phonon-mediated properties, including the electrical resistivity and the superconducting transition temperature, can be altered under pressure (11 ⇓–13). It has been suggested that the electrical transport properties due to the electron–phonon interaction in aluminum show a particularly strong response to interatomic spacing, particularly when the system is subject to extreme quantum confinement (14, 15).

Studies of the effect of pressure on the superconducting properties of aluminum suggest that superconductivity is suppressed through a reduction in the critical temperature, Tc, as the pressure is increased (11, 12, 16 ⇓–18). It also has been reported that the electron–phonon coupling constant, λ, decreases in aluminum under pressure (11, 16), but a quantitative extension to the electrical resistivity under pressure is lacking. Cheung and Ashcroft (13) suggested a decrease in the electrical resistivity of aluminum under volume compression by using a primitive pseudopotential model with experimentally determined interatomic force constants, but the pressures corresponding to these volume changes are unrealistically large. An in-depth analysis of the changes in the phonon spectra, electron–phonon coupling, and phonon-mediated electrical resistivity at realistic pressures from first-principles theory is lacking.

In this work, we provide a combined first-principles and experimental study of the effect of pressure on the electrical resistivity of 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 surface that makes it easy to treat computationally, whereas Cu typically is used for interconnects in industrial applications and has a lower intrinsic resistivity than aluminum. We show that upon pressurizing each metal, the electrical resistivity decreases. Furthermore, we show that density functional perturbation theory may be used to accurately predict the pressure response of the electrical resistivity in simple metals. First-principles calculations quantitatively match the numerical values of the resistivity changes found by experiment and explain the reduction in terms of shifted phonon frequencies and suppressed electron–phonon scattering as pressure increases.

Theory

Methods.

The density functional calculations were carried out using the ABINIT software package (19 ⇓–21). We consider single-atom unit cells of both fcc aluminum and copper, with equilibrium lattice 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 × 8 q-point grid, and a plane wave cutoff of 10.0 hartree for Al and 40.0 hartree for Cu. We use norm-conserving Martin–Troulliers pseudopotentials (22), the accuracy of which was tested in previous works (8, 14). The electrical resistivity is calculated according to the lowest-order variational solution to the Boltzmann equation, 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 agreement for the equilibrium phonon spectra with previous works (8 ⇓–10). Based on the phonon spectra, the Eliashberg spectral function, α2F(ω), is calculated as a phonon density of states weighted according to interactions between electrons and phonons at the Fermi level. From the spectral function, one can integrate to find the electron–phonon coupling constant, λ:λ=2∫α2F(ω)ωdω,[1]and the electrical resistivity:ρ(T)=∑k,k′3πΩe2g(ϵF)〈vF2〉∫dωxsinh2⁡xα2F(ω)ηk,k′,[2]where Ω is the volume of the unit cell, g(ϵ) is the density of states at the Fermi level, vF is the Fermi velocity, and x is a dimensionless parameter that incorporates temperature; x=ω2kBT. The efficiency factor, ηk,k′, is defined as 1−vk⋅vk′vk2 and accounts for electron scattering in different directions. We note that the transport spectral function, αtr2F(ω) is defined as the regular spectral function (α2F(ω)) 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 to experimentally applied pressures in the range 0–2 GPa are taken from the volume ratios in Vaidya and Kennedy (23) for aluminum and from Wang et al. (24) for copper. In the latter case, the values of volume compression were interpolated linearly for pressures below 2.512 GPa.

Results.

For the cases of unstrained aluminum and copper at zero pressure, our equilibrium values of phonon frequencies, electron–phonon coupling (λ = 0.45 for Al; λ = 0.15 for Cu) and electrical resistivity as functions of temperature agree well with the 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 the electrical resistivity as a function of pressure for both Al and Cu in Fig. 1. We follow the reasoning in Cheung and Ashcroft (13) and plot the scaled resistivity as opposed to the raw values at a given pressure. The scaled resistivity is just the resistivity at a given pressure divided by the zero-pressure value, such that all values fall between 0 and 1. In this way, the effects of any systematic errors in the density functional calculations are cancelled out and the effect of volume compression is observed more clearly.

Fig. 1.

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

Although both metals show a significant decrease in electrical resistivity as the pressure is increased from 0 to 2 GPa, the effect is much more pronounced for Al than in Cu. The resistivity of Al decreases by 8% of the equilibrium value when compressed to 2 GPa, whereas Cu resistivity decreases by only 2% at the same pressure. Both metals show a roughly linear decrease in resistivity as the pressure is increased, and the slopes have been calculated and included in the figure. As evidenced in the plot, Al shows a more aggressive slope with a value of −0.14μΩ-cm/GPa, whereas Cu has a much shallower slope of −0.04μΩ-cm/GPa, indicating that Al shows a more pronounced response to strain.

Experiment

Methods.

Aluminum and copper wires with diameters ranging from 450 to 490 μm (99.999% purity from Alfa Aesar) were used in the experiments. The 19-mm–diameter pressure cell assemblies were designed to have low yield strengths to reduce friction in the piston–cylinder apparatus (Fig. 2). Reported pressures are believed be accurate within <200 bars based on previous calibrations in our laboratory of similar assemblies (Table S1). It was not possible to pressurize an entire length of a wire in an experiment, because it was necessary to conduct resistance measurement using an 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 pressurized extended through the top (15 cm) and bottom (17 cm) components of the high-pressure device. The lengths of wire were varied by wrapping it around a dowel to form a spring-like coil. Care was taken to ensure that individual coils did not touch one another. The coiled lengths of wire were placed into machined tubes of pyrophyllite (9.6 mm i.d., 11.6 mm o.d.) that served as an electrical insulator. A nickel disk was machined with a 2-mm hole to accommodate an Al₂O₃ ceramic insulating tube. The disk was attached to the bottom of the pyrophyllite tube with cyanoacrylate adhesive. An Al₂O₃-based casting material (Aremco Ceramacast 575) was mixed with water and poured into the tube to encase the wire with the insulating ceramic. The assembly cured under a heat lamp for several hours before storage in a 120 °C oven for more than 24 h. Several experiments were heated. In those experiments, pyrophyllite tubes were machined to fit into an electrically conductive graphite heater tube used in standard piston–cylinder assemblies (26). Rings of NaCl were pressed and drilled to fit around the assemblies.

Fig. 2.

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

The straight lengths of wire at the bottoms of the assemblies were covered with Teflon insulators and threaded through holes machined in the 19-mm piston and pusher pieces so that the assembly was sitting on the piston (Fig. 2). The pressure vessel was lowered onto the assembly and piston. Lead-wrapped salt was inserted into the vessel around the assembly. The wire at the top of the assembly was threaded through the steel top plug and top plate. An Al₂O₃ insulator was placed on the wire to extend from the top of the assembly, through the steel top plug and top plate (Fig. 2). The wire was threaded through the top plate, which was lowered onto the Al₂O₃ insulator. A Teflon insulator 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 the experiments using a four-terminal configuration with a 10-kHz test signal. Most experiments were pressurized at room temperature in ∼0.1-GPa increments up to ∼2 GPa, followed by incremental depressurization of ∼0.1 GPa to ∼0.5 GPa. Resistance was measured at each pressure increment. Pressure was cycled up and down several times for each experiment. Several experiments were heated to temperatures 50 °C lower than the wire melting points with a 2-h dwell followed by cooling to room temperature. After cooling, pressure was cycled up and down several additional times. Resistance measurements could not be performed during the heating cycle, because the alternating current passed through the graphite heater tube to heat the assembly (Fig. 2) caused induction in the Cu and Al wires, which produced inaccurate resistance measurements. The Al₂O₃-based ceramic was removed from the wires of several experiments. Cross-sections of the wires were prepared by grinding flat with SiC papers followed by polishing in 1-μm Al₂O₃ and 0.06-μm colloidal SiO₂ suspensions. Backscattered electron imaging was conducted with a Cameca SX100 electron microprobe operating at 15 kV and 20 nA.

Results.

The experiments on Al and Cu wires were initiated by increasing pressure up to 2 GPa, and the results are shown in Fig. 3. Initially, the resistance (R) decreased approximately linearly with increasing pressure (P; slope of R vs. P is dR/dP) until reaching an inflection point, after which the resistance decreased at a lower rate. For Al wires, the inflection point occurred at 1.5 GPa, and for Cu wire, the inflection point occurred at 1.0 GPa. As discussed in Methods, pressure was cycled up and down several times for each experiment. During depressurization and subsequent pressure cycling, dR/dP was lower than during the initial pressurization (Fig. 3). The large decrease in R during initial pressurization is attributed to deformation and microstructural changes of the metals. The wires did not remain perfectly round because during experiments, the Al₂O₃ ceramic pressure medium impinged upon and roughened the surfaces of the wires. Scanning electron microscopy (backscattered electron imaging) of the wire starting materials showed they were composed of crystals 1 mm × 0.2 mm elongated parallel to the length of the wires (Fig. S1). Postexperimental backscattered electron images showed that pressurization recrystallized the wires, producing microstructures that are more equidimensional than the starting materials. After pressurization, the average crystal size was ∼0.3 × 0.3 mm. Measurements during the initial pressurization were not used to evaluate the pressure effect on resistance in the wires. Given the observable changes in wire microstructures and the irreproducible change in resistance during initial pressurization, only subsequent measurements were used to evaluate the pressure effect on resistance in the wires. To evaluate the effect of thermal annealing, several experiments were heated to 600 °C for 1.75 h after pressure cycling several times (Fig. 3A). The slope dR/dP after annealing was similar to the reproducible dR/dP that developed after initial pressurization.

Fig. 3.

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

The change in resistance with increasing pressure for Al and Cu wires is larger for longer lengths of wire in the pressure cell assembly (Table 1 and Fig. 2). The maximum amount of wire that could be included in a ∼45-mm–long pressure cell without adjacent coils touching one another was 60%. The minimum amount of wire included in the pressure cell assemblies was limited by our ability to accurately measure dR/dP with the LCR bridge. For example, increasing the pressure up to 2 GPa on an 8.9-cm length of wire (20%) changed the resistance 0.6 mΩ (precision is 0.1 mΩ). The slopes dR/dP reported in Table 1 were determined from linear fits to pooled resistance measurements excluding data from initial pressurization [e.g., fits were to data from depressurization 1, pressurization 2 depressurization 2, etc. (Fig. 3)]. Table 1 shows that there is a larger pressure effect on resistance for Al than there is for Cu wires. A plot of percentage of wire in the pressure-cell assemblies vs. slope dR/dP (Table 1) may be used to calculate the change in resistance that would result from pressurization of 100% of the Al and Cu wires.

View this table:

Table 1.

Experimental results summarized as change in resistance with increasing pressure (dR/dP) measured for various lengths of aluminum and copper wires in the pressure-cell assembly

The decreased resistances measured in Al and Cu with increasing P up to 2 GPa are similar to changes in resistance measured by Bridgman (27, 28). Bridgman used fluid pressure media (a glucose–glycerin–water mixture, or molasses) to apply pressures 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, respectively, than Bridgman’s results. Agreement between the studies is remarkable because they were conducted nearly a century apart using dissimilar experimental apparatuses.

Discussion

To compare the experimental results with theory, we have plotted the slopes of the R vs. P curves as a function of the percentage of the wire under pressure. We convert the measured values of resistance (R) to resistivity (ρ) given the known lengths and cross-sectional areas of the wires under pressure. We then extrapolate these values of Δρ/ΔP to correspond to a wire with 100% coverage under pressure, which of course was not possible to measure experimentally. We also include the zero point because by definition, the equilibrium resistivity does not change when no pressure is applied. The results are shown in Fig. 4. The extrapolated slopes at 100% under pressure can be compared with the results from theory from Fig. 1. We see close agreement in both cases; in units of microohm centimeters per gigapascal, the experimental value for Al is −0.092 whereas the predicted value from theory is −0.14. For the Cu wire, the experimental value is −0.048 whereas theory predicts a value of −0.04.

Fig. 4.

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

Both theory and experiment show a marked decrease in the electrical resistivity as the pressure is increased up to 2 GPa for Al and Cu metals. This decrease in resistivity is attributed to an overall weakening of the electron–phonon interaction. A measure of the strength of the electron–phonon interaction is given by the Eliashberg spectral function, α2F(ω). We plotted these functions for both Al and Cu at pressures of 0 and 2 GPa in Fig. 5.

Fig. 5.

The calculated Eliashberg spectral functions for Al and Cu at volume compression corresponding to 0 and 2 GPa.

It is clear that upon pressurizing each metal, the spectral functions are blue-shifted (i.e., shifted toward higher frequencies), which corresponds to a typical stiffening of the phonon modes as the volume is decreased. Because the net electron–phonon scattering goes as 1/ω, any increase in phonon frequencies will result in a decrease in electron–phonon coupling. In addition, the heights of the spectral function peaks decrease as they are shifted to higher frequencies, indicating that the higher-frequency phonons are less effective at scattering electrons. These effects combine to give a weaker electron–phonon coupling for compressed metals relative to the equilibrium (zero-pressure) configurations. Any decrease in electron–phonon coupling will be reflected as a decrease in the electrical resistivity.

The hardening of the phonon modes under strain can be understood in terms of the interatomic force constants (IFCs). As the lattice is compressed, the effective “springs” between atoms become more rigid and result in higher-frequency phonons. Although the calculated IFCs for both Al and Cu increase under pressure, the increases are greater for the former than for the latter. Specifically, the head element of the IFC matrix for Al increases by 8.7% in going from 0 GPa to 2 GPa, whereas the same element increases by only 6.3% in Cu. The nearest-neighbor IFC in Al increases by 8.25% in going from 0 to 2 GPa, whereas the same element for Cu increases by only 7.2%. This IFC modulation is the root cause of the phonon blue-shifting that occurs under pressure, which in return, suppresses the net electron–phonon interaction.

It also is worth noting that the Fermi velocity (vF) will be renormalized in the presence of electron–phonon interactions, which are nonadiabatic in nature. These effects will be in addition to the adiabatic change in the Fermi velocity due to lattice compression. We find that the adiabatic compression of the lattice results in an increase in the square of the Fermi velocity (vF2) from 0.58 a.u. at 0 GPa to 0.60 a.u. at 2.0 GPa. However, the net Fermi velocity, which includes both adiabatic and nonadiabatic contributions, is found to decrease as the lattice is compressed from 0.53 a.u. at 0 GPa to 0.52 a.u. at 2.0 GPa. We note that the inclusion of both adiabatic and nonadiabatic effects results in a smaller Fermi velocity because of the presence of the (1+λ) term in the denominator of the equationvF=vF01+λ,[3]which will decrease the renormalized Fermi velocity relative to the bare value (vF0). This indicates that the nonadiabatic effects due to electron–phonon coupling under pressure outweigh the adiabatic changes in the band structure due to lattice compression.

Conclusion

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

Acknowledgments

This work was partially supported by the National Science Foundation Integrative Graduate Education in Research and Traineeship Fellowship, Grant 0333314, as well as the Interconnect Focus Center (Microelectronics Advanced Research Corporation program) of New York. Computing resources were provided by the Computational Center for Nanotechnology Innovations at Rensselaer, partly funded by the State of the New York.

Footnotes

↵¹To whom correspondence should be addressed. E-mail: watsoe{at}rpi.edu.

Author contributions: N.A.L., J.B.T., B.W., M.W., and S.K.N. designed research; N.A.L. and J.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.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1406721111/-/DCSupplemental.

Freely available online through the PNAS open access option.

References

↵
1. Guinea F,
2. Katsnelson MI,
3. Geim AK
(2010) Energy gaps and a zero-field quantum hall effect in graphene by strain engineering. Nat Phys 6(1):30–33.
OpenUrl CrossRef
↵
1. Gui G,
2. Li J,
3. Jianxin Z
(2008) Band structure engineering of graphene by strain: First-principles calculations. Phys Rev B 78(7):075435.
OpenUrl CrossRef
↵
1. Cocco G,
2. Cadelano E,
3. Colombo L
(2010) Gap opening in graphene by shear strain. Phys Rev B 81(24):241412.
OpenUrl CrossRef
↵
1. Gulseren O,
2. Yildirim T,
3. Ciraci S,
4. Kilic C
(2002) Reversible band-gap engineering in carbon nanotubes by radial deformation. Phys Rev B 65(15):155410.
OpenUrl CrossRef
↵
1. Hong KH,
2. Kim J,
3. Lee SH,
4. Shin JK
(2008) Strain-driven electronic band structure modulation of si nanowires. Nano Lett 8(5):1335–1340.
OpenUrl CrossRef PubMed
↵
1. Lee ML,
2. Fitzgerald EA,
3. Bulsara MT,
4. Currie MT,
5. Lochtefeld A
(2005) Strained si, sige and ge channels for high-mobility metal-oxide-semiconductor field-effect transistor. J Appl Phys 97(1):011101.
OpenUrl CrossRef
↵
1. Ashcroft N,
2. Mermin N
(1976) Solid State Physics (Thomson Learning, Tampa, FL).
↵
1. Bauer R,
2. Schmid A,
3. Pavone P,
4. Strauch D
(1998) Electron-phonon coupling in the metallic elements al, au, na, and nb: A first-principles study. Phys Rev B 57(18):11276–11282.
OpenUrl CrossRef
↵
1. Savrasov SY,
2. Savrasov DY,
3. Andersen OK
(1994) Linear-response calculations of electron-phonon interactions. Phys Rev Lett 72(3):372–375.
OpenUrl CrossRef PubMed
↵
1. Savrasov SY,
2. Savrasov DY
(1996) Electron-phonon interactions and related physical properties of metals from linear-response theory. Phys Rev B Condens Matter 54(23):16487–16501.
OpenUrl CrossRef PubMed
↵
1. Profeta G,
2. et al.
(2006) Superconductivity in lithium, potassium, and aluminum under extreme pressure: A first-principles study. Phys Rev Lett 96(4):047003.
OpenUrl CrossRef PubMed
↵
1. Gubser DU,
2. Webb AW
(1975) High-pressure effects on the superconducting transition temperature of aluminum. Phys Rev Lett 35(2):104–107.
OpenUrl CrossRef
↵
1. Cheung J,
2. Ashcroft NW
(1979) Aluminum under high pressure. II. Resistivity. Phys Rev B 20(8):2991–2998.
OpenUrl CrossRef
↵
1. Verstraete MJ,
2. Gonze X
(2006) Phonon band structure and electron-phonon interactions in metallic nanowires. Phys Rev B 74(15):153408.
OpenUrl CrossRef
↵
1. Simbeck AJ,
2. Lanzillo N,
3. Kharche N,
4. Verstraete MJ,
5. Nayak SK
(2012) Aluminum conducts better than copper at the atomic scale: A first-principles study of metallic atomic wires. ACS Nano 6(12):10449–10455.
OpenUrl PubMed
↵
1. Fan W,
2. Li YL,
3. Wang JL,
4. Zou LJ,
5. Zeng Z
(2010) T-c map and superconductivity of simple metals at high pressure. Physica C 470(17-18):696–702.
OpenUrl CrossRef
↵
1. Akahama Y,
2. Nishimura M,
3. Kinoshita K,
4. Kawamura H,
5. Ohishi Y
(2006) Evidence of a fcc-hcp transition in aluminum at multimegabar pressure. Phys Rev Lett 96(4):045505.
OpenUrl CrossRef PubMed
↵
1. Sundqvist B,
2. Rapp O
(1979) Measurement of the pressure-dependence of the electron-phonon interaction in aluminum. J Phys F Met Phys 9(9):L161–L166.
OpenUrl CrossRef
↵
1. Gonze X,
2. et al.
(2009) ABINIT: First-principles approach to material and nanosystem properties. Comput Phys Commun 180(12):2582–2615.
OpenUrl CrossRef
↵
1. Gonze X,
2. et al.
(2005) A brief introduction to the ABINIT software package. Z Kristallogr 220(5-6):558–562.
OpenUrl CrossRef
↵
1. Gonze X
(1997) First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm. Phys Rev B 55(16):10337–10354.
OpenUrl CrossRef
↵
1. Troullier N,
2. Martins JL
(1991) Efficient pseudopotentials for plane-wave calculations. Phys Rev B Condens Matter 43(3):1993–2006.
OpenUrl CrossRef PubMed
↵
1. Vaidya SN,
2. Kennedy GC
(1970) Compressibility of 18 metals to 45 kbar. J Phys Chem Solids 31(10):2329–2345.
OpenUrl CrossRef
↵
1. Wang Y,
2. Chen D,
3. Zhang X
(2000) Calculated equation of state of Al, Cu, Ta, Mo, and W to 1000 GPa. Phys Rev Lett 84(15):3220–3223.
OpenUrl CrossRef PubMed
↵
1. Boyd FR,
2. England JL
(1960) Apparatus for phase equilibrium measurements at pressures up to 50 kb and temperatures up to 1750 c. J Geophys Res 65(2):741–748.
OpenUrl CrossRef
↵
1. Watson EB,
2. Wark DA,
3. Price JD,
4. Van Orman JA
(2002) Mapping the thermal structure of solid-media pressure assemblies. Contrib Mineral Petrol 142(6):640–652.
OpenUrl CrossRef
↵
1. Bridgman PW
(1923) The effect of pressure on the electrical resistance of cobalt, aluminum, nickel, uranium and caesium. Proc Natl Acad Arts Sci 58(4):151–161.
OpenUrl CrossRef
↵
1. Bridgman PW
(1931) The Physics of High Pressure (Dover, New York).

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Proceedings of the National Academy of Sciences: 111 (24)

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Submit

[1] ↵
Guinea F,
Katsnelson MI,
Geim AK
(2010) Energy gaps and a zero-field quantum hall effect in graphene by strain engineering. Nat Phys 6(1):30–33.
OpenUrl CrossRef

[2] Guinea F,

[3] Katsnelson MI,

[4] Geim AK

[5] ↵
Gui G,
Li J,
Jianxin Z
(2008) Band structure engineering of graphene by strain: First-principles calculations. Phys Rev B 78(7):075435.
OpenUrl CrossRef

[6] Gui G,

[7] Li J,

[8] Jianxin Z

[9] ↵
Cocco G,
Cadelano E,
Colombo L
(2010) Gap opening in graphene by shear strain. Phys Rev B 81(24):241412.
OpenUrl CrossRef

[10] Cocco G,

[11] Cadelano E,

[12] Colombo L

[13] ↵
Gulseren O,
Yildirim T,
Ciraci S,
Kilic C
(2002) Reversible band-gap engineering in carbon nanotubes by radial deformation. Phys Rev B 65(15):155410.
OpenUrl CrossRef

[14] Gulseren O,

[15] Yildirim T,

[16] Ciraci S,

[17] Kilic C

[18] ↵
Hong KH,
Kim J,
Lee SH,
Shin JK
(2008) Strain-driven electronic band structure modulation of si nanowires. Nano Lett 8(5):1335–1340.
OpenUrl CrossRef PubMed

[19] Hong KH,

[20] Kim J,

[21] Lee SH,

[22] Shin JK

[23] ↵
Lee ML,
Fitzgerald EA,
Bulsara MT,
Currie MT,
Lochtefeld A
(2005) Strained si, sige and ge channels for high-mobility metal-oxide-semiconductor field-effect transistor. J Appl Phys 97(1):011101.
OpenUrl CrossRef

[24] Lee ML,

[25] Fitzgerald EA,

[26] Bulsara MT,

[27] Currie MT,

[28] Lochtefeld A

[29] ↵
Ashcroft N,
Mermin N
(1976) Solid State Physics (Thomson Learning, Tampa, FL).

[30] Ashcroft N,

[31] Mermin N

[32] ↵
Bauer R,
Schmid A,
Pavone P,
Strauch D
(1998) Electron-phonon coupling in the metallic elements al, au, na, and nb: A first-principles study. Phys Rev B 57(18):11276–11282.
OpenUrl CrossRef

[33] Bauer R,

[34] Schmid A,

[35] Pavone P,

[36] Strauch D

[37] ↵
Savrasov SY,
Savrasov DY,
Andersen OK
(1994) Linear-response calculations of electron-phonon interactions. Phys Rev Lett 72(3):372–375.
OpenUrl CrossRef PubMed

[38] Savrasov SY,

[39] Savrasov DY,

[40] Andersen OK

[41] ↵
Savrasov SY,
Savrasov DY
(1996) Electron-phonon interactions and related physical properties of metals from linear-response theory. Phys Rev B Condens Matter 54(23):16487–16501.
OpenUrl CrossRef PubMed

[42] Savrasov SY,

[43] Savrasov DY

[44] ↵
Profeta G,
et al.
(2006) Superconductivity in lithium, potassium, and aluminum under extreme pressure: A first-principles study. Phys Rev Lett 96(4):047003.
OpenUrl CrossRef PubMed

[45] Profeta G,

[46] et al.

[47] ↵
Gubser DU,
Webb AW
(1975) High-pressure effects on the superconducting transition temperature of aluminum. Phys Rev Lett 35(2):104–107.
OpenUrl CrossRef

[48] Gubser DU,

[49] Webb AW

[50] ↵
Cheung J,
Ashcroft NW
(1979) Aluminum under high pressure. II. Resistivity. Phys Rev B 20(8):2991–2998.
OpenUrl CrossRef

[51] Cheung J,

[52] Ashcroft NW

[53] ↵
Verstraete MJ,
Gonze X
(2006) Phonon band structure and electron-phonon interactions in metallic nanowires. Phys Rev B 74(15):153408.
OpenUrl CrossRef

[54] Verstraete MJ,

[55] Gonze X

[56] ↵
Simbeck AJ,
Lanzillo N,
Kharche N,
Verstraete MJ,
Nayak SK
(2012) Aluminum conducts better than copper at the atomic scale: A first-principles study of metallic atomic wires. ACS Nano 6(12):10449–10455.
OpenUrl PubMed

[57] Simbeck AJ,

[58] Lanzillo N,

[59] Kharche N,

[60] Verstraete MJ,

[61] Nayak SK

[62] ↵
Fan W,
Li YL,
Wang JL,
Zou LJ,
Zeng Z
(2010) T-c map and superconductivity of simple metals at high pressure. Physica C 470(17-18):696–702.
OpenUrl CrossRef

[63] Fan W,

[64] Li YL,

[65] Wang JL,

[66] Zou LJ,

[67] Zeng Z

[68] ↵
Akahama Y,
Nishimura M,
Kinoshita K,
Kawamura H,
Ohishi Y
(2006) Evidence of a fcc-hcp transition in aluminum at multimegabar pressure. Phys Rev Lett 96(4):045505.
OpenUrl CrossRef PubMed

[69] Akahama Y,

[70] Nishimura M,

[71] Kinoshita K,

[72] Kawamura H,

[73] Ohishi Y

[74] ↵
Sundqvist B,
Rapp O
(1979) Measurement of the pressure-dependence of the electron-phonon interaction in aluminum. J Phys F Met Phys 9(9):L161–L166.
OpenUrl CrossRef

[75] Sundqvist B,

[76] Rapp O

[77] ↵
Gonze X,
et al.
(2009) ABINIT: First-principles approach to material and nanosystem properties. Comput Phys Commun 180(12):2582–2615.
OpenUrl CrossRef

[78] Gonze X,

[79] et al.

[80] ↵
Gonze X,
et al.
(2005) A brief introduction to the ABINIT software package. Z Kristallogr 220(5-6):558–562.
OpenUrl CrossRef

[81] Gonze X,

[82] et al.

[83] ↵
Gonze X
(1997) First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm. Phys Rev B 55(16):10337–10354.
OpenUrl CrossRef

[84] Gonze X

[85] ↵
Troullier N,
Martins JL
(1991) Efficient pseudopotentials for plane-wave calculations. Phys Rev B Condens Matter 43(3):1993–2006.
OpenUrl CrossRef PubMed

[86] Troullier N,

[87] Martins JL

[88] ↵
Vaidya SN,
Kennedy GC
(1970) Compressibility of 18 metals to 45 kbar. J Phys Chem Solids 31(10):2329–2345.
OpenUrl CrossRef

[89] Vaidya SN,

[90] Kennedy GC

[91] ↵
Wang Y,
Chen D,
Zhang X
(2000) Calculated equation of state of Al, Cu, Ta, Mo, and W to 1000 GPa. Phys Rev Lett 84(15):3220–3223.
OpenUrl CrossRef PubMed

[92] Wang Y,

[93] Chen D,

[94] Zhang X

[95] ↵
Boyd FR,
England JL
(1960) Apparatus for phase equilibrium measurements at pressures up to 50 kb and temperatures up to 1750 c. J Geophys Res 65(2):741–748.
OpenUrl CrossRef

[96] Boyd FR,

[97] England JL

[98] ↵
Watson EB,
Wark DA,
Price JD,
Van Orman JA
(2002) Mapping the thermal structure of solid-media pressure assemblies. Contrib Mineral Petrol 142(6):640–652.
OpenUrl CrossRef

[99] Watson EB,

[100] Wark DA,

[101] Price JD,

[102] Van Orman JA

[103] ↵
Bridgman PW
(1923) The effect of pressure on the electrical resistance of cobalt, aluminum, nickel, uranium and caesium. Proc Natl Acad Arts Sci 58(4):151–161.
OpenUrl CrossRef

[104] Bridgman PW

[105] ↵
Bridgman PW
(1931) The Physics of High Pressure (Dover, New York).

[106] Bridgman PW

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