Temperature Effect on Intermetallic Compound Growth Kinetics of Cu Pillar/Sn Bumps

The in situ intermetallic compound (IMC) growth in Cu pillar/Sn bumps was investigated by isothermal annealing at 120°C, 150°C, and 180°C using an in situ scanning electron microscope. Only the Cu₆Sn₅ phase formed at the interface between the Cu pillar and Sn during the reflow process. The Cu₃Sn phase formed and grew at the interfaces between the Cu pillar and Cu₆Sn₅ with increased annealing time. Total (Cu₆Sn₅ + Cu₃Sn) IMC thickness increased linearly with the square root of annealing time. The growth slopes of total IMC decreased after 240 h at 150°C and 60 h at 180°C, due to the fact that the Cu₆Sn₅ phase transforms to the Cu₃Sn phase when all of the remaining Sn phase in the Cu pillar bump is completely exhausted. The complete consumption time of the Sn phase at 180°C was shorter than that at 150°C. The apparent activation energy for total IMC growth was determined to be 0.57 eV.     

Study of the Au/In Reaction for Transient Liquid-Phase Bonding and 3D Chip Stacking

The latest three-dimensional (3D) chip-stacking technology requires the repeated stacking of additional layers without remelting the joints that have been formed at lower levels of the stack. This can be achieved by transient liquid-phase (TLP) bonding whereby intermetallic joints can be formed at a lower temperature and withstand subsequent higher-temperature processes. In order to develop a robust low-temperature Au/In TLP bonding process during which all solder is transformed into intermetallic compounds, we studied the Au/In reaction at different temperatures. It was shown that the formation kinetics of intermetallic compounds is diffusion controlled, and that the activation energy of Au/In reaction is temperature dependent, being 0.46 eV and 0.23 eV for temperatures above and below 150°C, respectively. Moreover, a thin Ti layer between Au and In was found to be an effective diffusion barrier at low temperature, while it did not inhibit joint formation at elevated temperatures during flip-chip bonding. This allowed us to control the intermetallic formation during the distinct stages of the TLP bonding process. In addition, a minimal indium thickness of 0.5 μm is required in order to enable TLP bonding. Finally, Au/In TLP joints of ∅40 μm to 60 μm were successfully fabricated at 180°C with very small solder volume (1 μm thickness).     

Interaction Between Ni and Cu Across 95Pb-5Sn High-Lead Layer

Ni/95Pb-5Sn/Cu ternary diffusion couples were used to investigate the cross-interaction between Ni and Cu across a layer of 95Pb-5Sn solder. High-lead solder layers with a thickness of 100 μm or 400 μm were electroplated over Cu foils. A pure Ni layer (20 μm) was then deposited over the as-deposited high-lead solder surface. The diffusion couples were then aged at 150°C to 250°C for different periods of time. With this technique, the diffusion couples were assembled without experiencing any high-temperature process such as reflow, which would have accelerated the interaction and caused difficulties in analysis. This study revealed that massive spalling also occurred during aging even though reflow was not used. The massive spalling began with the formation of microvoids. When the microvoids had congregated into large enough voids, intermetallic compounds (Cu₃Sn) started to spall from the interface. This spalling phenomenon occurred sooner with increasing temperature and decreasing solder volume.     

Electromigration-Induced Interfacial Reactions in Cu/Sn/Electroless Ni-P Solder Interconnects

The effect of electromigration (EM) on the interfacial reaction in a line-type Cu/Sn/Ni-P/Al/Ni-P/Sn/Cu interconnect was investigated at 150°C under 5.0 × 10³ A/cm². When Cu atoms were under downwind diffusion, EM enhanced the cross-solder diffusion of Cu atoms to the opposite Ni-P/Sn (anode) interface compared with the aging case, resulting in the transformation of interfacial intermetallic compound (IMC) from Ni₃Sn₄ into (Cu,Ni)₆Sn₅. However, at the Sn/Cu (cathode) interface, the interfacial IMCs remained as Cu₆Sn₅ (containing less than 0.2 wt.% Ni) and Cu₃Sn. When Ni atoms were under downwind diffusion, only a very small quantity of Ni atoms diffused to the opposite Cu/Sn (anode) interface and the interfacial IMCs remained as Cu₆Sn₅ (containing less than 0.6 wt.% Ni) and Cu₃Sn. EM significantly accelerated the dissolution of Ni atoms from the Ni-P and the interfacial Ni₃Sn₄ compared with the aging case, resulting in fast growth of Ni₃P and Ni₂SnP, disappearance of interfacial Ni₃Sn₄, and congregation of large (Ni,Cu)₃Sn₄ particles in the Sn solder matrix. The growth kinetics of Ni₃P and Ni₂SnP were significantly accelerated after the interfacial Ni₃Sn₄ IMC completely dissolved into the solder, but still followed the t ^1/2 law.     

Phase identification and growth kinetics of the intermetallic compounds formed during in-49Sn/Cu soldering reactions

For the application of In-49Sn solder in bonding recycled-sputtering targets to Cu back plates, the intermetallic compounds formed at the In-49Sn/Cu interface are investigated. Scanning electron microscopy (SEM) observations show that the interfacial intermetallics consist of a planar layer preceded by an elongated scalloped structure. Electron-probe microanalyzer analyses indicate that the chemical compositions of the planar layer and the scalloped structure are Cu_74.8In_12.2Sn_13.0 and Cu_56.2In_20.1Sn_23.7, respectively, which correspond to the ε-Cu₃(In,Sn) and η-Cu₆(In,Sn)₅ phases. Kinetics analyses show that the growth of both intermetallic compounds is diffusion controlled. The activation energies for the growth of η- and ε-intermetallics are calculated to be 28.9 kJ/mol and 186.1 kJ/mol. Furthermore, the formation mechanism of intermetallic compounds during the In-49Sn/Cu soldering reaction is clarified by marking the original interface with a Ta-thin film. Wetting tests are also performed, which reveal that the contact angles of liquid In-49Sn drops on Cu substrates decline to an equilibrium value of 25°C.     

金丝键合工艺技术研究

金丝键合是芯片组装的关键工序。分析了金丝键合的工艺控制要点：键合时间和键合功率,通过工艺实验总结出了键合时间和键合功率对键合强度的影响规律：（1）在小超声功率条件下,键合强度对键合时间敏感,键合强度随时间增加迅速增大;在大超声功率条件下,键合强度对键合时间的敏感性下降。（2）超声功率过小不能形成足够的键合强度,超声功率过大使得键合成功后的键合强度被破坏,即过高的超声功率将不利于键合强度的提高。     

Cu wire bond microstructure analysis and failure mechanism

In this study, copper wire bonding samples were aged at 205 °C in air from 0 h to 2000 h. It was found that the bonding of a Cu wire and an Al pad formed Cu₉Al₄, CuAl, and CuAl₂ intermetallic compounds, and an initial crack was formed by the ultrasonic squeeze effect during thermosonic wire bonding. The cracks grew towards the ball bond center with an increase in the aging time, and the Cl ions diffused through the crack into the ball center. This diffusion caused a corrosion reaction between the Cl ions and the Cu-Al intermetallic phases, which in turn caused copper wire bonding damage.     

Growth behavior of Cu/Al intermetallic compounds and cracks in copper ball bonds during isothermal aging

Copper wires are increasingly used in place of gold wires for making bonded interconnections in microelectronics. There are many potential benefits for use of copper in these applications, including better electrical and mechanical properties, and lower cost. Usually, wires are bonded to aluminum contact pads. However, the growth of Cu/Al intermetallic compounds (IMC) at the wire/pad interfaces is poorly understood, and if excessive would increase the contact resistance and degrade the bond reliability.To study the Cu/Al IMC growth in Cu ball bonds, high temperature aging at 250 °C for up to 196 h has been used to accelerate the aging process of the bonds. The Cu/Al IMCs growth behavior was then recorded and the IMC formation rate of 6.2 ± 1.7 × 10⁻¹⁴ cm²/s was obtained. In addition to the conventional yz-plane cross-section perpendicular to the bonding interface, a xy-plane cross-section parallel through the interfacial layers is reported. Three IMC layers were distinguished at the Cu/Al interfaces by their different colors under optical microscopy on the xy-plane cross-sections of ball bonds. The results of micro-XRD analysis confirmed that Cu₉Al₄, and CuAl₂ were the main IMC products, while a third phase is found which possibly is CuAl. During the aging process, IMC film growth starts from the periphery of the bond and propagates inward towards the centre area. Subsequently, with increased aging time, cavities are observed to develop between the IMC layer and the Cu ball surface, also starting at the bond periphery. The cavitation eventually links up and progresses toward the centre area leading to a nearly complete fracture between the ball and the intermetallic layer, as observed after 81 h.     

Kinetics of Solid-State Reactive Diffusion in the (Pd-Ni)/Sn System

The growth of compounds during energization heating at the interconnection between a Sn-based solder and a multilayer Pd/Ni/Cu conductor may be inhibited by the alloying of Pd with Ni. To examine such influence of Ni on the compound growth, the kinetics of solid-state reactive diffusion in the (Pd-Ni)/Sn system was experimentally determined in the present study. Experiments were conducted using Sn/(Pd-Ni)/Sn diffusion couples with Ni mol fractions of y = 0.257, 0.505, and 0.746 which were prepared by a diffusion bonding technique. The diffusion couples were isothermally annealed in the temperature range of 433 K to 473 K for various times up to 771 h. During annealing, different compounds are formed as rather uniform layers at the interface in the diffusion couple. In all the annealed diffusion couples, (Pd,Ni)Sn₄ was observed clearly. Furthermore, (Pd,Ni)Sn₃ and (Pd,Ni)Sn₂ were recognized for y = 0.257, and Ni₃Sn₄ was discerned for y = 0.746. However, no other compounds except (Pd,Ni)Sn₄ were detected for y = 0.505. The total thickness of the compound layers is proportional to a power function of the annealing time. The exponent of the power function is rather close to 0.5 for y = 0.257 and 0.505 but smaller than 0.5 for y = 0.746. Thus, volume diffusion is the rate-controlling process of the compound growth for y = 0.257 and 0.505, but boundary diffusion contributes to the rate-controlling process for y = 0.746. At the experimental annealing times, the overall growth rate of the compound layers is insensitive to y at y < 0.5 but decreases monotonically with increasing value of y at y > 0.5. Consequently, the compound growth is actually decelerated by the addition of Ni into Pd with y > 0.5 in the multilayer Pd/Ni/Cu conductor.     

The law of ultrasonic energy conversion in thermosonic flip chip bonding interfaces

In this paper, the vibration characteristics during the flip chip (FC) bonding process were observed by using a laser Doppler vibrometer (LDV), and the atom diffusion features in vertical section of the FC bonding interfaces were inspected by using a high resolution transmission electron microscope (HRTEM). Results show that the vibration velocity of a die was about 500 mm/s during the traditional FC bonding process, and that of a substrate was only about 180 mm/s. It led to the difference of atom diffusion in the FC interfaces. For the given variables, the thickness of atom diffusion at an up-interface (i.e. Au/Al interface) of the FC bonding was about 500 nm where was an inter-metallic compound (i.e. AuAl₂), and that of atom diffusion at a down-interface (i.e. Au/Ag interface) was about 200 nm. Furthermore, the law of ultrasonic energy conversion was found that the ratio of the up-interface to the down-interface in the FC bonding was statistically about 2.21:1. According to this principle, different bonding processes are suggested to improve the performance of two interfaces. The experimental evaluation confirms the effectiveness of the suggested processes on minimizing the inter-metallic compound layer and equilibrating the thickness of atom diffusion at two interfaces.     

Characterization of intermetallic compounds in Cu–Al ball bonds: Mechanical properties,interface delamination and thermal conductivity

In high power automotive electronics copper wire bonding is regarded as the most promising alternative for gold wire bonding in 1st level interconnects and therefore subjected to severe functional requirements. In the Cu–Al ball bond interface the growth of intermetallic compounds may deteriorate the physical and mechanical properties. The layer growth and properties of these intermetallic compounds are crucial in the prediction of the long term behavior. To mimic the growth of intermetallic compounds during and after copper ball bonding, diffusion couples of aluminium and copper were annealed at 225–500 °C and chemically analyzed by SEM/EDS. Also five separate intermetallic compounds were melted together from the pure elements and aged in evacuated quartz ampoules for 240 h at 500 °C. In this work values for the indentation Young’s modulus, load independent hardness, indentation fracture toughness, volumetric densities, interface delamination and thermal conductivity are presented. It can be concluded that the Cu-rich intermetallics Cu₉Al₄ and Cu₃Al₂ are less sensitive to fracture and have lower average densities than the other intermetallic compounds. The volumetric decrease during formation causes internal stress. Interfacial delamination initiates in the Al-rich intermetallics (CuAl, CuAl₂) and propagates easily into other intermetallic layers. The Cu₉Al₄–Cu s.s. interface is also found to be susceptible for delamination fracture. The thermal conductivity for 3 intermetallics is much lower than for pure copper or aluminium and in the range of 26–87 W/m⁻¹ K⁻¹, where Cu₃Al₂ layer has the lowest thermal conductivity (26–33 W/m⁻¹ K⁻¹).     

Solid-state interfacial reactions at the solder joints employing Au/Pd/Ni and Au/Ni as the surface finish metallizations

A tri-layer of nickel/palladium/gold (Au/Pd/Ni) is a promising candidate to replace the conventional Au/Ni bi-layer as the surface finish metallization for lead-free packaging. A surface finish metallization (Au/Pd/Ni or Au/Ni) and a Sn layer are sequentially deposited on a Cu substrate and then are subjected to thermal aging at 150 and 200 °C to investigate the interfacial reactions in the stacking multilayer structure made by low-temperature solid-state bonding. Because of the absence of the reflow process, the Pd and Au layers do not dissolve in the Sn matrix but remain at the interface and participate in the interfacial reaction to form the (Pd,Ni,Au)Sn₄ and (Au,Ni)Sn₄ phases at the Au/Pd/Ni- and Au/Ni-based interfaces, respectively. Though the Pd layer was only 0.4 μm, its resulting (Pd,Ni,Au)Sn₄ phase is much thicker than the (Au,Ni)Sn₄ phase. These two intermetallic compounds exhibit very different microstructural evolution which significantly affects the interfacial microstructures and growth rate of other intermetallic compound formed at the same interfaces.     

Electromigration-Induced Void Formation at the Cu<Subscript>5</Subscript>Zn<Subscript>8</Subscript>/Solder Interface in a Cu/Sn-9Zn/Cu Sandwich

The electromigration that occurs in a Cu/Sn-9Zn/Cu sandwich was investigated for void formation at room temperature with 10³ A/cm². A focused ion beam revealed that voids nucleated at the intermetallic compound (IMC)/solder interface regardless of the electron flow direction. The needle-like voids initiated at the cathode Cu₅Zn₈/solder interface due to the outward diffusion of Zn atoms in the Zn-rich phase and expanded as a result of the surface diffusion of Sn atoms upon current stressing.     

A comparative study of the kinetics of interfacial reaction between eutectic solders and Cu/Ni/Pd metallization

A comparative study of the kinetics of interfacial reaction between the eutectic solders (Sn-3.5Ag, Sn-57Bi, and Sn-38Pb) and electroplated Ni/Pd on Cu substrate (Cu/Ni/NiPd/Ni/Pd) was performed. The interfacial microstructure was characterized by imaging and energy dispersive x-ray analysis in scanning electron microscope (SEM). For a Pd-layer thickness of less than 75 nm, the presence or the absence of Pd-bearing intermetallic was found to be dependent on the reaction temperature. In the case of Sn-3.5Ag solder, we did not observe any Pd-bearing intermetallic after reaction even at 230°C. In the case of Sn-57Bi solder the PdSn₄ intermetallic was observed after reaction at 150°C and 180°C, while in the case of Sn-38Pb solder the PdSn₄ intermetallic was observed after reaction only at 200°C. The PdSn₄ grains were always dispersed in the bulk solder within about 10 μm from the solder/substrate interface. At higher reaction temperatures, there was no Pd-bearing intermetallic due to increased solubility in the liquid solder. The presence or absence of Pd-bearing intermetallic was correlated with the diffusion path in the calculated Pd-Sn-X (X=Ag, Bi, Pb) isothermal sections. In the presence of unconsumed Ni, only Ni₃Sn₄ intermetallic was observed at the solder-substrate interface by SEM. The presence of Ni₃Sn₄ intermetallic was consistent with the expected diffusion path based on the calculated Ni-Sn-X (X=Ag, Bi, Pb) isothermal sections. Selective etching of solders revealed that Ni₃Sn₄ had a faceted scallop morphology. Both the radial growth and the thickening kinetics of Ni₃Sn₄ intermetallic were studied. In the thickness regime of 0.14 μm to 1.2 μm, the growth kinetics always yielded a time exponent n >3 for liquid-state reaction. The temporal law for coarsening also yielded time exponent m >3. The apparent activation energies for thickening were: 16936J/mol for the Sn-3.5Ag solder, 17804 J/mol for the Sn-57Bi solder, and 25749 J/mol for the Sn-38Pb solder during liquid-state reaction. The corresponding activation energies for coarsening were very similar. However, an apparent activation energy of 37599 J/mol was obtained for the growth of Ni₃Sn₄ intermetallic layer during solid-state aging of the Sn-57Bi/substrate diffusion couples. The kinetic parameters associated with thickening and radial growth were discussed in terms of current theories.     

Evaluation of Diffusion Barrier Between Lead-Free Solder Systems and Thermoelectric Materials

The intermetallic compound SnTe rapidly formed at interfaces between p-type bismuth telluride (Bi_0.5Sb_1.5Te₃) thermoelectric materials and lead-free solders. The intermetallic compound influences the mechanical properties of the joints and the reliability of the thermoelectric modules. Various lead-free solder alloys, Sn-3.5Ag, Sn-3Ag-0.5Cu, Sn-0.7Cu, and Sn-2.5Ag-2Ni, were used to investigate the interfacial reactions. The results thus obtained show that Ag and Cu preferentially diffused into the Te-rich phase in Bi_0.5Sb_1.5Te₃, so layers of Ag-Te and Cu-Te compounds could not form an effective diffusion barrier. Electroless nickel-phosphorus was plated at the interfaces to serve as a diffusion barrier, and the (Cu,Ni)₆Sn₅ compound formed instead of SnTe. Furthermore, the intermetallic compound NiTe formed between nickel- phosphorus and Bi_0.5Sb_1.5Te₃ and also served as a diffusion barrier. A plot of thickness as a function of annealing time yielded the growth kinetics of the intermetallic compounds in the thermoelectric material systems. The activation energy for the growth of the NiTe intermetallic compound is 111 kJ/mol.     

Ambient Temperature Ultrasonic Bonding of Si-Dice Using Sn-3.5wt.%Ag

Ultrasonic bonding of Si-dice to type FR-4 printed circuit boards (PCB) with Sn-3.5wt.%Ag solder at ambient temperature was investigated. The under-bump metallization (UBM) on the Si-dice comprised Cu/Ni/Al from top to bottom with thicknesses of 0.4 μm, 0.4 μm, and 0.3 μm, respectively. The pads on the PCBs consisted of Au/Ni/Cu with thicknesses of 0.05/5/18 μm, sequentially from top to bottom. Solder was supplied as Sn-3.5wt.%Ag foil rolled to 100 μm thickness, and inserted in the joints. The ultrasonic bonding time was varied from 0.5 s to 3.0 s, and the ultrasonic power was 1400 W. The experimental results showed that reliable joints could be produced between the Si-dice and the PCBs with Sn-3.5wt.%Ag solder. The joint breaking force of “Si-die/solder/FR-4” increased with bonding times up to 2.5 s with a maximum value of 65 N. A bonding time of 3.0 s proved to be excessive, and resulted in cracks along the intermetallic compound between the UBM and solder, which caused a decrease in the bond strength. The intermetallic compound produced by ultrasonic bonding between the UBM and solder was confirmed to be (Cu, Ni)₆Sn₅. An erratum to this article can be found at     

Interfacial Reactions in Sn/Fe-xNi Couples

The interfacial reactions between Sn and several Fe-xNi alloys were investigated in this study. Two different FeSn₂ phase types formed at the Sn/Fe-40.8 at.%Ni (Sn/alloy 42) interface. When the Ni content of the Fe-Ni alloy was less than 80 at.%, only the FeSn₂ phase with layer structure could be formed at the interface. When the Ni content was increased to 80 at.% to 90 at.%, both FeSn₂ and Ni₃Sn₄ phases formed in the Sn/Fe-xNi couples. When the Ni content was larger than 95 at.%, only the Ni₃Sn₄ phase was formed at the interface. A fast reaction rate and the thickest intermetallic compound (IMC) layer could be observed in the Sn/Fe-90 at.%Ni couple. The reaction paths of each reaction couple at 270°C were as follows: L/FeSn₂/Fe-40.8 at.%Ni (alloy 42), L/FeSn₂/Ni₃Sn₄/Fe-80 at.% Ni, and L/Ni₃Sn₄/Fe-95 at.%Ni, respectively.     

Interfacial Reaction Effect on Electrical Reliability of Cu Pillar/Sn Bumps

Thermal annealing and electromigration (EM) tests were performed with Cu pillar/Sn bumps to understand the growth mechanism of intermetallic compounds (IMCs). Annealing tests were carried out at both 100°C and 150°C. At 150°C, EM tests were performed using a current density of 3.5 × 10⁴ A/cm². The electrical failure mechanism of the Cu pillar/Sn bumps was also investigated. Cu₃Sn formed and grew at the Cu pillar/Cu₆Sn₅ interface with increasing annealing and current-stressing times. The growth mechanism of the total (Cu₆Sn₅ + Cu₃Sn) IMC changed when the Sn phase in the Cu pillar/Sn bump was exhausted. The time required for complete consumption of the Sn phase was shorter during the EM test than in the annealing test. Both IMC growth and phase transition from Cu₆Sn₅ to Cu₃Sn had little impact on the electrical resistance of the whole interconnect system during current stressing. Electrical open failure in the Al interconnect near the chip-side Cu pillar edge implies that the Cu pillar/Sn bump has excellent electrical reliability compared with the conventional solder bump.     

Effect of Isothermal Aging and Thermal Cycling on Interfacial IMC Growth and Fracture Behavior of SnAgCu/Cu Joints

The growth behavior of interfacial intermetallic compounds (IMCs) of SnAgCu/Cu soldered joints was investigated during the reflow process, isothermal aging, and thermal cycling with a focus on the influence of these parameters on growth kinetics. The SnAgCu/Cu soldered joints were isothermally aged at 125°C, 150°C, and 175°C while the thermal cycling was performed within the temperature ranges from −25°C to 125°C and −40°C to 125°C. It was observed that a Cu₆Sn₅ layer formed, followed by rapid coarsening at the solder/Cu interface during reflowing. The grain size of the interfacial Cu₆Sn₅ was found to increase with aging time, and the morphology evolved from scallop-like to needle-like to rod-like and finally to particles. The rod-like Ag₃Sn phase was formed on the solder side in front of the previously formed Cu₆Sn₅ layer. However, when subject to an increase of the aging time, the Cu₃Sn phase was formed at the interface of the Cu₆Sn₅ layer and Cu substrate. The IMC growth rate increased with aging temperature for isothermally aged joints. During thermal cycling, the thickness of the IMC layer was found to increase with the number of thermal cycles, although the growth rate was slower than that for isothermal aging. The dwell time at the high-temperature end of the thermal cycles was found to significantly influence the growth rate of the IMCs. The growth of the IMCs, for both isothermal aging and thermal cycling, was found to be Arrhenius with aging temperature, and the corresponding diffusion factor and activation energy were obtained by data fitting. The tensile strength of the soldered joints decreased with increasing aging time. Consequently, the fracture site of the soldered joints migrated from the solder matrix to the interfacial Cu₆Sn₅ layer. Finally, the shear strength of the joints was found to decrease with both an increase in the number of thermal cycles and a decrease in the dwell temperature at the low end of the thermal cycle.     

The Feasibility of Au Ball Bonding on Sn-Plated Cu

The feasibility of thermosonic gold wire bonding on Cu coupons with Sn/Cu metallizations was studied by evaluating shear strength and microstructure of balls bonded on different Sn metallization samples. The 0.85 ± 0.08 μm and 5.34 ± 0.21 μm thick metallizations were produced by dipping the Cu coupon in 250°C molten Sn solder for 1 s (sample 1) and 30 s (sample 2), respectively. Cu₆Sn₅ intermetallic compounds are formed during dipping. After wire bonding, Au-Cu-Sn layers are found on the ball-coupon interface of both samples. The highest ball shear force observed was 40 gf (1 gf = 9.81 mN) and was achieved on sample 1 using 520 mW and 40 gf of ultrasonic power and bonding force, respectively. The shear fracture goes through the Au ball. The Sn is squeezed out of the contact zone during wire bonding and forms flashes that extend 5 μm and 25 μm beyond the contact zone for samples 1 and 2, respectively.