2 Stress, Strain, Piezoresistivity, and Piezoelectricity

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1 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity 2.1 STRAIN TENSOR Strin in crystls is creted by deformtion nd is defined s reltive lttice displcement. For simplicity, we use 2D lttice model in Fig. 2.1 to illustrte this concept, but discuss the generl concept in 3D cses. As shown in Fig. 2.1, we my use two unit vectors ˆ, ŷ to represent the unstrined lttice, nd in simple squre lttice, they correspond to the lttice bsis vectors. Under smll uniform deformtion of the lttice, the two vectors re distorted in both orienttion nd length, which is shown in Fig. 2.1b. The new vectors ˆ nd ŷ my be written in terms of the old vectors: nd in the 3D cse, we lso hve ˆ =(1+ε )ˆ + ε y ŷ + ε z ẑ, (2.1) ŷ = ε yˆ +(1+ε yy )ŷ + ε yz ẑ, (2.2) ẑ = ε zˆ + ε zy ŷ +(1+ε zz )ẑ. (2.3) The strin coefficients ε αβ define the deformtion of the lttice nd re dimensionless. The 3 3mtri ε = ε ε y ε z ε y ε yy ε yz (2.4) ε z ε zy ε zz is clled the strin tensor. A tensor is mthemticl nottion, usully represented by n rry, to describe liner reltion between two physicl quntities. A tensor cn be sclr, vector, or mtri. A sclr is zerornk tensor, vector is first-rnk tensor, nd mtri is second-rnk tensor, nd so on. The strin tensor is second-rnk tensor, which in this book is lbeled with two brs over the hed. However, in plces without confusion, we usully neglect the brs. Suppose lttice point is originlly locted Y. Sun et l., Strin Effect in Semiconductors: 9 Theory nd Device Applictions, DOI / , c Springer Science+Business Medi, LLC 2010

2 10 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity y b y^ y' ^ ^ Undeformed lttice ' ^ Deformed lttice Fig Digrm for () n undeformed lttice nd (b) deformed lttice t r = ˆ + yŷ + zẑ, then with uniform deformtion this point will be t r = ˆ + yŷ + zẑ. For generl vrying strin, the strin tensor my be written s ε α,β = u α, u α = u,u y,u z, β =, y, z, (2.5) β where u α is the displcement of lttice point under study long α.astrin tensor (2.4) is symmetric, i.e., ε αβ = ε βα = 1 ( uα + u ) β. (2.6) 2 β α The ntisymmetric prt of tensor (2.4) represents rottion of the entire body. Usully people work with the other set of strin components, which re defined s e = ε ; e yy = ε yy ; e zz = ε zz, (2.7) which describe infinitesiml distortions ssocited with chnge in volume, nd the other strin components e y, e yz,nde z redefinedintermsof chnges of ngle between the bsis vectors. Neglecting the terms of order ε 2 in the smll strin pproimtion, they re e y = ˆ ŷ = ε y + ε y, e yz = ŷ ẑ = ε yz + ε zy, e z = ẑ ˆ = ε z + ε z. (2.8) These si coefficients completely define the strin. We cn write these si strin coefficients in the form of n rry s e = {e,e yy,e zz,e yz,e z,e y }. The introduction of this set of nottion for the strin components is merely for the convenience of describing the reltions between strin nd the other strinrelted physicl quntities. The reltion between two second-rnk tensors

3 2.2 Stress Tensor 11 must be described by fourth-rnk tensor, which is very complicted; while fter trnsforming the second-rnk tensors to first-rnk, only second-rnk tensor is required. The crystl diltion under deformtion cn be evluted through clculting the volume defined by ˆ, ŷ,ndẑ, nd the diltion δ then is given by V = ˆ ŷ ẑ =1+e + e yy + e zz, (2.9) δ = δv V = e + e yy + e zz, (2.10) which is the trce of the strin tensor. The diltion is negtive for hydrosttic pressure. 2.2 STRESS TENSOR Crystl deformtions cn be induced by eternlly pplied forces, or in other words, solid resists deformtions, thus deformtions will generte forces. Stress is defined s the force in response to strin in unit re. Stress hs nine components nd is second-rnk tensor, which we write s τ αβ, α, β =, y, z. On the surfce of n infinitesiml volume cube, the stress distribution is illustrted in Fig. 2.2,whereτ represents force pplied in the direction to unit re of the plne whose outwrd-drwn norml lies in the direction, nd τ y represents force pplied in the direction to unit re of the z t zz t z t yz tzy t z t y t y t yy t y Fig Illustrtion for stress components on the surfces of n infinitesiml cube

4 12 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity z F z F y A F y Fig Illustrtion of the forces pplied on one surfce with re of A of the cube showninfig.2.2 plne whose outwrd-drwn norml lies in the y direction. The stress tensor is symmetric just s the strin tensor. The ntisymmetric prt of the stress tensor represents torque, nd in stte of equilibrium, ll torques must vnish inside solid. The stress nd force reltion is better illustrted in Fig. 2.3 where we show force pplied on n infinitesiml plne whose norml is long nd hs n re A. In such cse, we resolve the force into components long the coordinte es, i.e., F, F y,ndf z. The stress components in this plne re τ = F A,τ y = F y A,τ z = F z A. (2.11) We now study some simple stress cses to determine the stress tensors. 1. Hydrosttic pressure: Under hydrosttic pressure P, ll sher stress is zero. Stress long ny principle direction is P,nmely, τ = P P 0. (2.12) 0 0 P Here the sign convention is tht tensile stress is positive nd compressive stress is negtive. 2. Uniil stress T long the [001] direction: For uniil stress T long the [001] direction, ll stress components but τ zz re zero, nd τ zz = T.So τ = (2.13) 00T

5 2.2 Stress Tensor Uniil stress T long the [110] direction: The cse for uniil stress long the [110] direction is little more complicted. Generlly when we tlk bout stress T long the 110 direction, it refers to the force eerted long the 110 direction divided by the cross section of the (110) surfce, but not necessrily equl to ny of the stress tensor elements. To find the stress elements, we cn use two methods. First is to resolve the force into three coordinte es. For [110] uniil stress T s shown in Fig. 2.4, the force long the [110] direction is F = T 2. Its component long [001] is zero. Along both nd y direction, the force is F/ 2. However, the cross re for the force long [110] shown in Fig. 2.4 is 2 nd is 2 2 for the forces long the nd y direction. Thus, the stress long both nd y is F/2 2 = T/2. The sher stress on both [100] nd [010] plnes is lso T/2. The second method to obtin the stress components is through the coordinte trnsformtion method. Suppose in n unprimed coordinte system, stress T is long the direction, nd thus τ = T, nd ll the other stress components re zero. We cn rotte the nd y es 45 clockwise, nd then n originl [100] uniil stress tht only hs one nonvnishing component τ = T now corresponds to [110] uniil stress in primed coordinte system, s shown in Fig. 2.4b. The stress elements in the primed coordinte system re given by the trnsformtion, τ ij = i j τ mn, (2.14) mn m n where i m, etc. represent the directionl cosines of the trnsformed es mde to the originl es. This eqution results from the generl tensor trnsformtion of S to S under n orthogonl coordinte trnsformtion A, S = ASA T, (2.15) F y y b y y F 110 F 0 45 Fig () The decomposition of force long the [110] direction long the nd y directions, nd their stress reltions. Plese note tht in this figure, the nd y directions re long the digonls of the surfces insted of long the edges. (b) The coordinte systems before nd fter 45 rottion clockwise. The unprimed nd primed systems re the coordinte systems before nd fter the rottion

6 14 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity where A T is the trnspose of mtri A. The stress tensor under the [110] uniil stress found using both methods is τ = T (2.16) Becuse stress tensor is symmetric, similr to the strin tensor cse, the si coefficients, τ, τ yy, τ zz, τ yz, τ z,ndτ y completely define the stress. Similr to strin tensor, second-rnk stress tensor cn be reduced to 1D rry form. 2.3 ELASTIC COMPLIANCE AND STIFFNESS CONSTANTS In the liner elstic theory, Hooke s lw is justified nd stress is proportionl to strin τ ij = C ijαβ e αβ, i,j,α,β =, y, z, (2.17) αβ where the coefficients C ijαβ re clled elstic stiffness constnts. Elstic stiffness constnts re fourth-rnk tensor. Becuse of the symmetry of both the strin tensor nd the stress tensor, we hve C ijαβ = C jiαβ = C ijβα, (2.18) so we my write both strin nd stress tensor s si-component rry s e =(e,e yy,e zz,e yz,e z,e y ) (2.19) nd τ =(τ,τ yy,τ zz,τ yz,τ z,τ y ) (2.20) nd reduce the elstic stiffness tensor to 6 6mtri τ i = m C im e m. (2.21) This 6 6 mtri hs very simple form in cubic crystls due to the high symmetry. It hs only three independent components nd hs the form C 11 C 12 C C 12 C 11 C C ij = C 12 C 12 C C (2.22) C C 44

7 2.3 Elstic Complince nd Stiffness Constnts 15 This is esy to understnd by inspecting (2.21) nd considering the trnsformtion of this eqution under symmetry opertions. First, the elstic stiffness tensor must be symmetric. Second, since for cubic crystls, the three es re equivlent, therefore we must hve C 11 = C 22 = C 33,ndC 44 = C 55 = C 66. Third, sher strin cnnot cuse norml stress, so terms like C 14 =0. And sher strin long one is cnnot induce forces cusing sher long nother is, so terms like C 45 = 0. Finlly in the view of force long one is, the other two es re equivlent, nd thus we hve C 12 = C 13,etc. These results cn lso be obtined by investigting the trnsformtion of the components in (2.17) under symmetry opertions using n eqution similr to (2.14) C lkγδ = l k δ C ijαβ. (2.23) i j α β ijαβ For emple, it is esy to verify tht under reflection nd thus, C yzz = C yzz,sointhe6 6mtri,C 63 =0. In mny cses it is convenient to work with the inverse of the elstic stiffness tensor, which is defined through the reltion between strin nd stress γ ε αβ = ij S αβij τ ij. (2.24) The fourth-rnk tensor S αβij, clled the complince tensor, cn lso be reduced into 6 6 mtri. Under cubic symmetry, it hs the sme form s the stiffness tensor S 11 S 12 S S 12 S 11 S S 12 S 12 S S , (2.25) S S 44 nd the strin stress reltion cn be written s e m = i S mi τ i. (2.26) Becuse the elstic stiffness tensor nd complince tensor re inverse to ech other, so it is esy to work out the reltions between the components s S 11 = S 12 = C 11 + C 12 (C 11 C 12 )(C 11 +2C 12 ), C 12 (C 11 C 12 )(C 11 +2C 12 ), S 44 = 1 C 44. (2.27)

8 16 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity In mechnicl engineering, Young s modulus Y nd Poisson rtion ν re commonly used. For homogeneous, isotropic mteril, strin is relted to stress through ε = 1 Y (τ ν(τ yy + τ zz )), ε yy = 1 Y (τ yy ν(τ zz + τ )), ε zz = 1 Y (τ zz ν(τ + τ yy )). (2.28) In cubic systems Young s modulus nd Poisson rtion ν re relted to the complince constnts by Y = 1,ν= S 12. (2.29) S 11 S EXAMPLES OF STRESS STRAIN RELATIONS Now we use two emples to illustrte how to determine the strin tensor from stress using the reltions we hve discussed erlier. 1. Biil stress: A semiconductor lyer pseudomorphiclly grown on (001)-oriented ltticemismtched substrte is schemticlly shown in Fig Inthiscse,the top lyer is biilly strined, nd the strin components e nd e yy re e = e yy = 0. (2.30) z b Unstrined Strined Fig Illustrtion of biil stress (strin). () Two different mteril lyers hve different lttice constnt before growth; (b) After pseudomorphic film growth, the lttice constnt of the top lyer conforms to tht of the bottom lyer nd is under biil stress (strin)

9 2.4 Emples of Stress Strin Reltions 17 The strin is tensile in the -y plne. To obtin the strin in the z direction, we use the strin stress reltion (2.26), i.e., e S 11 S 12 S τ e yy S 12 S 11 S τ yy e zz e z = S 12 S 12 S τ zz S τ z. (2.31) e yz S 44 0 τ yz e y S 44 τ y In the current cse, τ = τ yy = T, τ zz =0,ndτ z = τ yz = τ y =0. Therefore, we hve e = e yy =(S 11 + S 12 )T, e zz =2S 12 T. (2.32) Thus, e zz = 2S 12 e. (2.33) S 11 + S 12 Strin tensor in this cse is e = e e 0. (2.34) 0 0 e zz 2. [110] uniil stress: The -y plne of cubic crystl under [110] uniil stress is illustrted in Fig The stress tensor is lredy obtined in Eq. (2.16), i.e., τ = τ yy = τ y = T/2, nd τ zz = τ z = τ yz = 0. Substituting into (2.31), we obtin y T T Fig Illustrtion of the [110] uniil compressive stress (strin)

10 18 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity e = e yy = S 11 + S 12 T, 2 e y = S 44 2 T, e zz = S 12 T. (2.35) The strin tensor in this cse then is ε = e e y /2 0 e y /2 e 0. (2.36) 0 0 e zz Hydrosttic nd Sher Strin An rbitrry strin tensor cn be decomposed into three seprte tensors s following: ε ε y ε z ε y ε yy ε yz = 1 ε + ε yy + ε zz ε + ε yy + ε zz 0 3 ε z ε zy ε zz 0 0 ε + ε yy + ε zz + 1 2ε (ε yy + ε zz ) ε yy (ε zz + ε ) ε zz (ε + ε yy ) + 0 ε y ε z ε y 0 ε yz, ε z ε zy 0 (2.37) where the first constnt tensor whose digonl element is one-third of the trce of the originl strin tensor ccounts for the volume chnge [see (2.10)], nd the ltter two trceless tensors ccount for the shpe chnge of n infinitesiml cube. Correspondingly, the first tensor describes the effect of hydrosttic strin, nd the ltter two tensors describe the effect of sher strin. Among the two sher strin tensors, the digonl one is relted to the chnge of lengths long the three es nd the other one with digonl elements being zero is relted to the rottion of the es of n infinitesiml cube. For cubic crystl, the first type of sher occurs when uniil stress is pplied long ny of the three 100 es, nd the second type of sher is nonzero only when stresses re pplied long 110 or 111 es. Obviously, for cube under the hydrosttic strin, the shpe does not chnge, while under n rbitrry first type of sher, the shpe of the cube will become orthorhombic, nd under n rbitrry second type of sher, the shpe of the cube will become triclinic. A cubic crystl under biil stress becomes tetrgonl, nd it becomes orthorhombic under uniil stress long 110.

11 2.5 Piezoresistivity 19 For first look, pplying compressive uniil stress long [001] nd biil tensile stress in the -y plne to cubic crystl seems identicl. Indeed, if for both cses the stress is T, nd we decompose the resulting strin tensor into the hydrosttic nd sher prts, the sher strin coincides. However, the hydrosttic strin differs in sign nd fctor of 2 in mgnitude. 2.5 PIEZORESISTIVITY Piezoresistivity is n effect of stress-induced resistivity chnge of mteril. The piezoresistnce coefficients (π coefficients) tht relte the piezoresistivity nd stress re defined by π = ΔR/R, (2.38) T where R is the originl resistnce tht is relted to semiconductor smple dimension by R = ρ l wh, ΔR signifies the chnge of resistnce, nd T is the pplied mechnicl stress. The rtio of ΔR to R cnbeepressedinterms of reltive chnge of the smple length Δl/l, widthδw/w, heightδh/h, nd resistivity Δρ/ρ s ΔR R = Δl l Δw w Δh h + Δρ ρ, (2.39) where resistivity ρ is inversely proportionl to the conductivity. The first three terms of the RHS of (2.39) depict the geometricl chnge of the smple under stress, nd the lst term Δρ/ρ is the resistivity dependence on stress. For most semiconductors, the stress-induced resistivity chnge is severl orders of mgnitude lrger thn the geometricl chnge-induced resistnce chnge, so the resistivity chnge by stress is the determinnt fctor of the piezorestivity. In generl conditions, resistivity ρ = 1/σ is second-rnd tensor, nd stress T is lso second-rnk tensor. The resistivity chnge, Δρ ij, is connected to stress by fourth-rnk tensor π ijkl, the piezoresistnce tensor. Under rbitrry stress in liner response regime, Δρ ij ρ = Δσ ij σ = k,l π ijkl τ kl, (2.40) where summtion is over, y, ndz. Following the sme discussion for complince nd stiffness tensor, nd writing Δρ ij tovectorformδρ i,wherei =1, 2,...,6, s we did for stress nd strin, we cn rewrite (2.40) s Δρ i 6 ρ = π ik τ k, (2.41) k=1

12 20 2 Stress, Strin, Piezoresistivity, nd Piezoelectricity where π ik is 6 6 mtri. For cubic structures, it hs only three independent elements due to the cubic symmetry, π 11 π 12 π π 12 π 11 π π ik = π 12 π 12 π π (2.42) π π 44 Among the three independent π-coefficients, π 11 depicts the piezoresistive effect long one principl crystl is for stress long this principl crystl is (longitudinl piezoresistive effect), π 12 depicts the piezoresistive effect long one principl crystl is for stress directed long one perpendiculr crystl is (trnsverse piezoresistive effect), nd π 44 describes the piezoresistive effect on n out-of-plne electric field by the chnge of the in-plne current induced by in-plne sher stress. The detiled discussion of semiconductor piezoresistivity will be covered in Chp PIEZOELECTRICITY Different from the piezoresistive effect, the piezoelectric effect rises from stress-induced chrge polriztion in crystl tht lcks center of inversion. Thus, the piezoelectric effect does not eist in Si, Ge, etc. elementry semiconductors. The zinc-blende semiconductors re the simplest crystls with this property. The polriztion is relted to stress through the piezoelectric tensor ē, P =[e]e strin, (2.43) where P is the polriztion vector nd e strin is the strin written s sicomponent vector. Thus the piezoelectric tensor is 3 6mtri.Forzincblende semiconductors, the piezoelectric tensor only hs one nonvnishing tensor element, e 14, nd the polriztion induced by strin is then given by P P y = 000e e 14 0 P z e 14 e e yy e zz e yz e z e y. (2.44) Becuse of the specil form of the piezoelectric tensor, only the sher strin genertes the piezoelectricity. For zinc-blende semiconductors such s GAs grown on (001) direction, the biil strin does not generte piezoelectricity.

13 2.6 Piezoelectricity 21 The piezoelectric effect is lrgest long the 111 es, since the nions nd ctions re stcked in the (111) plnes, thus strin cretes reltive displcement between them. The piezoelectric constnts of GAs were mesured nd theoreticlly clculted (Adchi, 1994). The commonly dopted vlue is e 14 = 0.16 C/m 2. (2.45) On the other hnd, the pplied electric field cross the piezoelectric mteril cn generte strin. The piezoelectric strin tensor d hs the sme form s the piezoelectric tensor nd lso hs only one nonvnishing component, d 14, for zinc-blende semiconductors. It is relted to e 14 by d 14 = S 44 e 14. (2.46) The commonly dopted vlue for d 14 for GAs is m/v. The sign of e 14 or d 14 is negtive for III V semiconductors. If the crystl is epnded long the 111 direction, the A-fces (ction fces) becomes negtively chrged. This is different from the II V semiconductors, where e 14 is positive. For the other semiconductors lcking inversion symmetry, the piezoelectric tensor my hve more thn one nonvnishing component. In wurtzite semiconductors such s GN, there re three nonvnishing components, e 13, e 33,nde 15. Piezoelectric effect my ply n importnt role in semiconductor trnsport. In n AlGN/GN heterostructure, the spontneous polriztion nd the piezoelectric effect cn induce lrge density of electrons even when there is no doping (Bernrdini nd Fiorentini, 1997; Jogi, 1998; Scconi et l, 2001). In GAs/InGAs superlttices grown in the 111 direction, piezoelecticity induced bnd bending cn gretly chnge the potentil profile, nd thus lter the chrge distribution nd trnsport properties (Smith nd Milhiot, 1988; Kim, 2001).

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