NANO EXPRESS Open Access

Surface plasmon resonance of Au-Cu bimetallicnanoparticles predicted by a quasi-chemicalmodelYen-Hsun Su1,2* and Wen-Lin Wang1

Abstract

Au-Cu alloys are functional materials with nonlinear optical applications. However, the optical properties of suchalloys are difficult to predict due to the random mixing of materials. In this paper, we present a quasi-chemicalmodel to simulate the optical properties of Au-Cu alloy systems based on the mixing of Gibbs free energy. Thismodel is also able to predict the position of the surface plasmon resonance peaks for Au-Cu alloy nanoparticles.The model can be applied to predict the optical properties of alloy systems in the fields of plasmonics andnanophotonics.

Keywords: Dielectric properties; Gibbs free energy; Nanostructures; Metals; Optical properties

BackgroundMetal nanoparticles (NPs) have attracted much researchinterest due to their unusual chemical and physicalproperties, such as catalytic activity, novel electronics,optics, and magnetic properties, and they have potentialapplications in solar cells and biosensors [1-7].Alloy nanoparticle systems have been found to exhibit

optical limiting properties due to surface plasmon reson-ance and have been used in biodiagnostic applications[8,9]. Alloy nanoparticles are materials used to tune theposition of surface plasmon resonance, and thus help toproduce materials for use in nonlinear optical applica-tions [10-14]. Au-Cu alloy system is a completely dissol-uble alloy. The position of surface plasmon resonancefor Au NPs is about 520 nm. The position of surfaceplasmon resonance for Cu NPs is 570 ~ 580 nm [15]. Atlow temperatures, Au, Au3Cu, AuCu, AuCu3, and Cuexist and order easily in Au-Cu alloys system. The pre-diction of the optical properties of such alloy systems isdesirable if they are to be used in the design of opticaldevices. However, the optical properties of alloy systems

are difficult to predict because of the random mixing ofmaterials.The quasi-chemical method is a statistical approach

for predicting the short-range-order of Au-Cu alloys sys-tem according to Gibbs free energy. While the opticalproperties of Au-Cu alloys can be computed by thequasi-chemical model based on the energy potential be-tween the electric field and induced dipole, few workshave attempted to do this.In this study, we thus simulate the optical properties

of Au and Cu using a quasi-chemical model, based onthe energy potential between the electric field and in-duced dipole. We then used this quasi-chemical methodto modify the statistics for the short-range-order of Au-Cu alloy system. Then the optical properties are simu-lated by combining the Gibbs free energy and electricpotential energy. The light extinction of nanoparticles iscalculated by using Mie theory. The results show thatthe model is suitable for predicting the position of sur-face plasmon resonance peaks.

MethodsModelRegular solutionAu-Cu alloy system refers to a solid solution. Propertiesof a regular solution are best examined based on theconcept of excess function [16]. The excess value of an

* Correspondence: kistic@gmail.com1Department of Materials Science and Engineering, National Cheng KungUniversity, Tainan, Taiwan2Advanced Optoelectronic Technology Center, National Cheng KungUniversity, Tainan, Taiwan

© 2013 Su and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

Su and Wang Nanoscale Research Letters 2013, 8:408http://www.nanoscalereslett.com/content/8/1/408

extensive thermodynamic solution property is simply thedifference between its actual value and the value that itwould have if the solution were ideal based on the Gibbsfree energy of the solution,

G ¼ Gid þ GXS ð1ÞIn which G is the molar Gibbs free energy of the solu-

tion, Gid is the molar Gibbs free energy that the solutionwould have if it were ideal, and GXS is the excess molarGibbs free energy of the solution. Because the two com-ponents have equal molar volumes and do not exhibit achange in molar volume when mixed, their regular solu-tion behavior can be understood by the application of astatistical mixing model, i.e., a quasi-chemical model.

Quasi-chemical modelThe energy of the solution is the sum of its interatomicbond energies. Consider 1 mol of a mixed crystalcontaining NA atoms of A and NB atoms of B such that

XA ¼ NA

NA þ NB¼ NA

NOand XB ¼ NB

NO; ð2Þ

where NO is Avogadro’s number. The mixed crystal, orsolid solution, contains three types of atomic bond: A-Abonds, B-B bonds, and A-B bonds. A-A bonds the en-ergy of each of which is UAA, B-B bonds the energy ofeach of which is UBB, A-B bonds the energy of each ofwhich is UAB. If in the solution, there are PAA A-Abonds, PBB B-B bonds, and PAB A-B bonds, the energyof the solution U is obtained as the linear combination

U ¼ PAAUAA þ PBBUBB þ PABUAB ð3Þand the problem of calculating U becomes one of calcu-lating the values of PAA, PBB, and PAB. Thus,

ΔUM ¼ PAB½UAB−12

UAA þ UBBð Þ�: ð4Þ

The change in volume is negligible. Since ΔVM = 0,

ΔHM ¼ ΔUM ¼ PAB½UAB−12

UAA þ UBBð Þ�: ð5Þ

Ideal mixing requires the condition UAB = UAA = UBB.If ΔHM = 0, the mixing of the NA atoms with the NB

atoms of B is random.

ΔSM ¼ ΔSM;id ¼ −R XA lnXA þ XB lnXBð Þ: ð6ÞThe quasi-chemical model is a statistical mixing model

in Gibbs free energy. According to Equations 5 and 6,the mixing Gibbs free energy will be presented. In the‘Results and discussion’ section, the dipole energy inGibbs free energy was utilized to consider the opticalproperties with different frequencies of incident light.

Results and discussionThe probability that a neighboring pair of sites containsan A-B pair is 2XAXB, an A-A pair is XA

2 , and B-B pair

is XB2, and X2

A þ 2XAXB þ X2B ¼ XA þ XBð Þ2 ¼ 1: The

quasi-chemical model is a statistical mixing model thatdescribes the mixing cluster. The difference in Gibbs en-ergy is presented as follows:

ΔGM ¼ ΔHM−TΔSM: ð7ÞCombining Equations 5 and 6 with Equation 7 pro-

duces the following:

ΔGM ¼ PAB½UAB−12

UAA þ UBBð Þ�þ RT XA lnXA þ XB lnXBð Þ: ð8Þ

Because PAB = 2XAXB,

ΔGM ¼ 2XAXB½UAB−12

UAA þ UBBð Þ�þ RT XA lnXA þ XB lnXBð Þ: ð9Þ

The Gibbs free energy of the solution is as follows:

G ¼ G0 þ ΔGM ¼ XA2G0

A þ XB2G0

B

þ 2XAXB½UAB−12

UAA þ UBBð Þ�

þRT XA lnXA þ XB lnXBð Þ: ð10Þ

After applying the electric field ⇀E ,

G−⇀pmixing ⋅⇀E ¼ XA

2 G0A−

⇀pA⋅⇀E� �

þXB2 G0

B−⇀pB⋅

⇀E

� �

þ2XAXB

�UAB−⇀pAB⋅

⇀E

� �−12

�UAA−⇀pA⋅

⇀E

� �

þ UBB−⇀pB⋅

⇀E

� ���

þRT XA lnXA þ XB lnXBð Þ: ð11Þ

where ⇀pmixing is the induced dipole moment of

metamaterial, ⇀pA is the induced dipole moment of ma-terial A, ⇀pB is the induced dipole moment of material B,and ⇀pAB is the induced dipole moment due to the inter-action of materials A and B.The Gibbs energy was subtracted when applying an

electric field from that without applying one, as follows:

⇀pmixing⋅⇀E ¼ XA

2 ⇀pA⋅⇀E

� �þXB

2 ⇀pB⋅⇀E

� �

þ2XAXB

�⇀pAB⋅

⇀E

� �−12

⇀pA⋅⇀E

� �þ ⇀pB⋅

⇀E

� �� ��:

ð12Þ

Su and Wang Nanoscale Research Letters 2013, 8:408 Page 2 of 6http://www.nanoscalereslett.com/content/8/1/408

Because ⇀p ¼ ε−ε0ð Þ⇀E , the above equation can be re-written as follows:

�εmixing−ε0

�⇀E ⋅

⇀E ¼ XA

2�εA−ε0

� �⇀E⋅⇀E�

þ XB2�εB−ε0

� �⇀E⋅⇀E�

þ2XAXB

� �εAB−ε0

� �⇀E⋅⇀E�−12

�εA−ε0ð Þ⇀E⋅⇀E

� �

þ�εB−ε0

� �⇀E⋅⇀E���

ð13Þ

εmixing−ε0� � ¼ XA

2 εA−ε0ð Þ þ XB2 εB−ε0ð Þ

þ2XAXB½ εAB−ε0ð Þ− 12

εA−ε0ð Þ þ εB−ε0ð Þð Þ�

εmixing−ε0� � ¼ XA

2 εA−ε0ð Þ þ XB2 εB−ε0ð Þ

þ2XAXB

�εAB−

12

�εA þ εB

� �

εmixing ¼ εAXA2 þ εBXB

2 þ 2XAXB

�εAB−

12

εA þ εBð Þ−ε0� �

:

ð14Þ

The dielectric function of the mixed material includesthe interaction term εAB− 1

2 εA þ εBð Þ−ε0��

and independ-

ent terms εAXA2 and εBXB

2. When εAB− 12 εA þ εBð Þ−ε0�

�is assumed to be an experience constant, Λ, the dielec-tric function of mixing material is reduced to the fol-lowing form:

εmixing ¼ εAXA2 þ εBXB

2 þ 2XAXBΛ ð15Þ

Wavelength (nm)

300 400 500 600 700 800

r

-25

-20

-15

-10

-5

0

AuAu3CuAuCuAuCu3Cu

(a)

Wavelength (nm)

300 400 500 600 700 800

i

0

1

2

3

4

5

6

7

8

AuAu3CuAuCuAuCu3Cu

(b)

Figure 1 Effective dielectric complex of the alloy. (a) Real part,εr. (b) Imaginary part, εi, of the dielectric complex of Au-Cu alloy.

Wavelength (nm)

300 400 500 600 700 800

Qa

bs

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Au

Au3Cu

AuCu

Au Cu3

Cu

(a)

Wavelength (nm)

300 400 500 600 700 800

Qabs

0.0

0.2

0.4

0.6

0.8

1.0

Au

Au3Cu

AuCu

Au Cu3

Cu

(b)

Wavelength (nm)

300 400 500 600 700 800

Qabs

0.0

0.5

1.0

1.5

2.0

2.5

Au

Au3Cu

AuCu

Au Cu3

Cu

(c)

Figure 2 Extinction of Au-Cu alloy nanoparticles. Extinction ofAu-Cu alloy nanoparticles (10 nm) when (a) n = 1, (b) n = 1.4, and(c) n = 1.8 (Qabs is the extinction coefficient).

Su and Wang Nanoscale Research Letters 2013, 8:408 Page 3 of 6http://www.nanoscalereslett.com/content/8/1/408

The Newton formula [17] is used to apply these con-cepts to the clustered material. The dielectric functionsrefer to these clusters and the embedding matrix,

εNewton‐mixing ¼ f εA þ 1−fð ÞεB: ð16ÞThe quasi-chemical model describes the mixing of

clusters in which the interaction term is approximatedby Newton formula mixing. Combining the probabilityof neighboring pairs with the Newton formula, the op-tical model of the regular solution is as follows:

εeff ¼ X2AεA þ 2XAXB εNewton‐mixing

� �þ X2BεB: ð17Þ

The effective dielectric complex of the alloy ispresented in Figure 1.According to Mie theory [18,19], the resonances de-

noted as surface plasmon were relative with the onset ofthe quantum size and shape effects of Au NPs. There is

one SPR band for metal NPs, and this is shown as fol-lows [20,21]:

εi ¼ εh þ fεh εh−εmð Þ

Γ iεm þ 1−Γ ið Þεh ; ð18Þ

where εh is the dielectric constant of the host mediumembedding Au NPs, εm is the dielectric constant of AuNPs, f is the volume fraction of Au NPs, εi is the total di-electric constant, and Γi is a set of three parameters de-fined along the principal axes of the particlecharacterizing its shape. Γ1 + Γ2 + Γ3 = 1 and the otherparameters range from 0 to 1. The frequencies of thesurface plasmon of nonspherical metal NPs have two orthree bands, depending on their shape. The extinctioncoefficients of alloy metal NPs with different sizes andenvironments are presented in Figures 2, 3, 4.

Wavelength (nm)

300 400 500 600 700 800

Qab

s

0

2

4

6

8

101 nm4nm10 nm20 nm30 nm50 nm100 nm

Wavelength (nm)

300 400 500 600 700 800

Qab

s

0

2

4

6

8

101 nm4nm10 nm20 nm30 nm50 nm100 nm

Wavelength (nm)

300 400 500 600 700 800

Qab

s

0

2

4

6

81 nm4nm10 nm20 nm30 nm50 nm100 nm

Wavelength (nm)

300 400 500 600 700 800

Qab

s

0

2

4

6

81 nm4nm10 nm20 nm30 nm50 nm100 nm

Wavelength (nm)

300 400 500 600 700 800

Qab

s

0

2

4

6

81 nm4nm10 nm20 nm30 nm50 nm100 nm

(a) (b)

(c) (d)

(e)

Figure 3 Extinction of different sized NPs. (a) Au, (b) Au3Cu, (c) AuCu, (d) AuCu3, and (e) Cu alloy nanoparticles (n = 1; Qabs is theextinction coefficient).

Su and Wang Nanoscale Research Letters 2013, 8:408 Page 4 of 6http://www.nanoscalereslett.com/content/8/1/408

The quasi-chemical model is used to calculate the op-tical properties of Au-Cu alloys. The real part of the di-electric complex is negative for Au-Cu alloy system. Theimaginary part of dielectric constant for Au-Cu alloysystem shows the peaks that appear in range from 430to 520 nm due to the electronic transition between the dband and sp band. The real and imaginary parts of thedielectric complex for Au-Cu alloys system are as shownin Figure 1a,b, respectively.We use Mie theory to predict the spectrum and pos-

ition of surface plasmon resonance. Figure 2b shows theextinction of a 10-nm diameter Au-Cu nanoparticle indifferent refractive index surroundings. For n = 1.4, thesurface plasmon resonance peaks are 532, 538, 561, 567,and 578 nm for Au, Au3Cu, AuCu, AuCu3, and Cu, re-spectively, and these results which are in agreement withthose of other experimental results [22].The extinction spectra of Au-Cu bimetallic nanoparti-

cle with size effect are presented in Figure 3. As the size

of nanoparticles increase, the peak of surface plasmonresonance red-shifts. When the size is less than 50 nm,the size effect becomes more significant. The higher theratio of Cu to Au of is, the more the surface plasmonresonance red-shifts. When the size is greater than 50nm, the size effect is less significant due to the small in-crease in the cross section. The refractive index effect isshown in Figure 4. As the refractive index increases, thesurface resonance peak will red-shift and become in-creasingly sharp. Based on this, it is possible to predictthe surface plasmon resonance peaks of regular solutionalloys, such as Au-Cu, Cu-Ag, Ag-Cu, and Au-Cu-Agsystems.

ConclusionIn this work we used the quasi-chemical model to com-pute the optical properties of Au-Cu alloy system. Theresults show that it is possible to use this approach topredict the positions of surface plasmon resonance

Wavelength (nm)

300 400 500 600 700 800

Qa

bs

0.0

0.5

1.0

1.5

2.0

2.5

n=1.0

n=1.2

n=1.4

n=1.6

n=1.8

Wavelength (nm)

300 400 500 600 700 800

Qa

bs

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

n=1.0

n=1.2

n=1.4

n=1.6

n=1.8

Wavelength (nm)

300 400 500 600 700 800

Qa

bs

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

n=1.0

n=1.2

n=1.4

n=1.6

n=1.8

Wavelength (nm)

300 400 500 600 700 800

Qa

bs

0.0

0.2

0.4

0.6

0.8

1.0

1.2

n=1.0

n=1.2

n=1.4

n=1.6

n=1.8

Wavelength (nm)

300 400 500 600 700 800

Qa

bs

0.0

0.2

0.4

0.6

0.8

1.0

n=1.0

n=1.2

n=1.4

n=1.6

n=1.8

(a) (b)

(c) (d)

(e)

Figure 4 Extinction of different refractive index. (a) Au, (b) Au3Cu, (c) AuCu, (d) AuCu3, and (e) Cu alloy nanoparticles.

Su and Wang Nanoscale Research Letters 2013, 8:408 Page 5 of 6http://www.nanoscalereslett.com/content/8/1/408

peaks. This model is thus a useful tool in the develop-ment of for future applications of alloy nanoparticles forplasmonics and nanophotonics.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsYHS and WLW contribute in writing and model setting in all these works.Both authors read and approved the final manuscript.

Authors’ informationYHS is an assistant professor and WLW is a student in the Department ofMaterials Science and Engineering in National Cheng Kung University,Taiwan.

AcknowledgementsThis work was financially supported by the National Science Council ofTaiwan (nos. 100-2218-E-259-003-MY3 and 102-2221-E-006-293-MY3) which isgratefully acknowledged. This research was, in part, supported by theMinistry of Education, Taiwan, Republic of China and the Aim for the TopUniversity Project of the National Cheng Kung University (NCKU).

Received: 17 July 2013 Accepted: 28 August 2013Published: 2 October 2013

References1. Banholzer MJ, Osberg KD, Li S, Mangelson BF, Schatz GC, Mirkin CA: Silver-

based nanodisk codes. ACS Nano 2010, 4:5446.2. Wustholz KL, Henry AI, McMahon JM, Freeman RG, Valley N, Piotti ME, Natan

MJ, Schatz GC, Van Duyne RP: Structure-activity relationships in goldnanoparticle dimers and trimers for surface-enhanced Ramanspectroscopy. J Am Chem Soc 2010, 132:10903.

3. Zhang XL, Song JF, Li XB, Feng J, Sun HB Sun: Optical Tamm statesenhanced broad-band absorption of organic solar cells. Appl Phys Lett2012, 101:243901.

4. Sen A, Lin CJ, Kaun CC: Single-molecule conductance through chiral goldnanotubes. J Phys Chem C 2013, 117:13676.

5. Su YH, Ke YF, Cai SL, Yao QY: Surface plasmon resonance of layer-by-layergold nanoparticles induced photoelectric current in environmentally-friendly plasmon-sensitized solar cell. Light Sci Appl 2012, 1:e14.

6. Stratakis E, Kymakis E: Nanoparticle-based plasmonic organic photovoltaicdevices. Mater Today 2013, 16:133.

7. Su YH, Hsu CY, Chang CC, Tu SL, Shen YH: Ultra-thin titanium nanolayersfor plasmon-assisted enhancement of bioluminescence of chloroplast inbiological light emitting devices. Appl Phys Lett 2013, 103:063703.

8. Cao YW, Jin R, Mirkin CA: DNA-modified core-shell Ag/Au nanoparticles.J Amer Chem Soc 2001, 123:7961.

9. Nair AS, Suryannarayanan V, Pradeep T, Thomas J, Anija M, Philip R:AuxAgy@ZrO2 core - shell nanoparticles: synthesis, characterization,reactivity and optical limiting. Mater Sci Eng B 2005, 117:173.

10. Ispasoiu RG, Balogh L, Varnavski OP, Tomalia DA, Goodson T: Large opticallimiting from novel metal-dendrimer nanocomposite materials. J AmChem Soc 2000, 122:11005.

11. Sun WF, Dai Q, Worden JG, Huo Q: Optical limiting of a covalentlybonded gold nanoparticle/polylysine hybrid material. J Phys Chem B2005, 109:20854.

12. Wang G, Sun WF: Optical limiting of gold nanoparticle aggregatesinduced by electrolytes. J Phys Chem B 2006, 110:20901.

13. Francois L, Mostafavi M, Belloni J, Delouis JF, Delaire J, Feneyrou P: Opticallimitation induced by gold clusters. J Phys Chem B 2000, 104:6133.

14. Philip R, Kumar GR, Sandhyarani N, Pradeep T: Picosecond opticalnonlinearity in monolayer-protected gold, silver, and gold-silver alloynanoclusters. Phys Rev B 2000, 62:13160.

15. Yeh YH, Yeh MS, Lee YP, Yeh CS: Formation of Cu nanoparticles from CuOpowder by laser ablation in 2-propanol. Chem Lett 1998, 11:1183.

16. Gaskell DR: Introduction to the Thermodynamics of Materials. 5th edition.New York: Taylor & Francis; 2008:249.

17. Theiss W: Optische Eigenschaften Inhomogener Materialien. Dissertation, RWTHAachen; 1989.

18. Templeton AC, Pietron JJ, Murray RW, Mulvaney P: Solvent refractive indexand core charge influences on the surface plasmon absorbance ofalkanethiolate monolayer-protected gold clusters. J Phys Chem B 2000,104:564.

19. Fuchs R: Theory of the optical properties of ionic crystal cubes. Phys Rev B1975, 11:1732.

20. Prasad PN: Nanophotonics. Hoboken: Wiley; 2004:130.21. Fox M: Optical Properties of Solids. London: Oxford University Press; 2001:151.22. Kim MJ, Na HJ, Lee KC, Yoo EA, Lee M: Preparation and characterization of

Au-Ag and Au-Cu alloy nanoparticles in chloroform. J Mater Chem 2003,13:1789.

doi:10.1186/1556-276X-8-408Cite this article as: Su and Wang: Surface plasmon resonance of Au-Cubimetallic nanoparticles predicted by a quasi-chemical model. NanoscaleResearch Letters 2013 8:408.

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