Mobility Measurements and Device Preparation
Contents
1. C A Magneto transport characterization using quantitative mobility spectrum analysis QMSA Presented at Workshop on the Physics and Chemistry of Merury Cadmium Telluride and Other IR Materials 4 6 Oct 1994 Alberqueque NM USA ASTM Standard F76 American Society for testing and Materials Philadelphia 1991 13 van der Pauw L J A method of measuring specific resistivity and Hall effect of discs of arbitrary shape Philips Res Reports 13 1 9 1958 14 ASTM Standard F76 American Society for testing and Materials Philadelphia 1991 Volger J Note on the Hall potential across an inhomogeneous conductor Phys Rev 79 1023 24 1950 16 Haeussler J and Lippmann H Hall generatoren mit kleinem Linearisierungsfehler Solid State Electron 11 173 82 1968 7 Jandl S Usadel K D and Fischer G Resistivity measurements with samples in the form of a double cross Rev Sci Inst 19 685 8 1974 be Chwang R Smith B J and Crowell C R Contact size effects in transition metal doped semiconductors with application to Cr doped GaAs J Phys C Solid State Phys 13 2311 23 1974 19 van Daal H J Mobility of charge carriers in silicon carbide Phillips research reports Suppl 3 1 92 1965 2 David J M and Beuhler M G A numerical analysis of various cross sheet resistor test structures Solid State Electron 20 539 43 1977 De Mey G Influence of sample geometry on Ha2. Chichester 1989 Meyer J R Hoffman C A Bartoli F J D A Arnold S Sivananthan and J P Faurie Methods for Magnetotransport Characterization of IR Detector Materials Semicond Sci Technol 1993 805 823 ASTM Standard F76 86 Standard Method for Measuring Hall Mobility and Hall Coefficient in Extrinsic Semiconductor Single Crystals 1991 Annual Book of ASTM Standards Am Soc Test Mat Philadelphia 1991 Look D C et al On Hall scattering factors holes in GaAs J Appl Phys 80 1913 1996 Chwang R Smith B J and Crowell C R Contact size effects on the van der Pauw method for resistivity and Hall coefficient measurement Solid State Electron 17 Dec 1974 1217 1227 Perloff D S Four point probe sheet resistance correction factors for thin rectangular samples Solid State Electron 20 Aug 1977 681 687 David J M and Buehler M G A numerical analysis of various cross sheet resistor test structures Solid State Electron 20 Aug 1977 539 543 Beck W A and Anderson J R Determination of electrical transport properties using a novel magnetic field dependent Hall technique J Appl Phys 62 2 Jul 1987 541 554 a Brugger H and Koser H Variable field Hall technique a new characterization tool for JFET MODFET device wafers lIl Vs Review Vol 8 No 3 1995 41 45 11 Antoszewski J Seymour D J Meyer J R and Hoffman
3. m ART B 0 1 B 0 qm Cm _ Vig o B 0 Vi55 B 0 mem em and Ve 14 LB 0 a Vo 14 LB 0 w m m eT B 0 18 0 Am E Via B 0 Vz14 B 0 wem 1 cm Q cm L B 0 1 B 0 b cm at zero magnetic field If these two values disagree by more than 10 then the sample is too inhomogeneous or anisotropic or has some other problem If they do agree then the average resistivity is given by Pa a Ps 2m 2cm Magnetoresistivity If desired calculate the two magnetoresistivities i I B 1 B LE E Ae b m VB Vssaal B A 8 Vos 8 mom dem po ay TE B LA B Li B Lo B b em and ol B Al er B Vasa b View B wim t m Q E B Lo B Es B 1 B b m Vegsalt B Venal B Vasil B Pa B w cm z cm I5 B I t B I56 B I B b cm If these two values disagree by more than 10 then the sample is too inhomogeneous or anisotropic or has some other problem If they do agree then the average magnetoresistivity is given by p B APEP lo mom A 12 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual Hall Coefficient Calculate the Hall coefficient by fhm ssal B Visa B Peul 8 Val B m c BIT 15 B L B 15 B 1 B Hall Mobility The Hall mobility is given by iz R pav m yA s em YA 57 Pa where p is the magnetoresistivity if it was measured an
4. Hall mobility and sheet density of 2DEG is to measure sheet resistance of the 2DEG Below is the schematic to measure the resistance of the2DEG channel It is advisable to take average by switching current and voltage polarities to check the uniformity of the sample The text shown below has been copied from Appendix Please note that for 2DEG the thickness term t is neglected and hence unit of resistivity for 2DEG is Ohms Mobility Measurements and Device Preparation Page 2 Resistivity Again let V xy indicate a voltage measured across terminals k and with k positive while a positive current flows into terminal and out of terminal j In a similar fashion let R jx indicate a resistance R jk Via l with the voltage measured across terminals k and while a positive current flows into and out of j First calculate the two resistivities 0 Kf t m cm Y aw as Passa mene Q m Q cm In 2 Li d 1 Lo and P T fy t m cm 1 a 21 ae 21 T Fi 523 Ha 23 Q m Q cm In 2 ra La La gt La Geometrical factors f and f are functions of resistance ratios O and O respectively given by E O a E o To O _ f as Fa _ AP ias Ls A F g t r 9 Ki Rosia I 12 I 12 Visia Fa 14 and a Pi 7 T O f Ry 21 Rua _ f Vasa Var fi a La BT a gt 7 7 Pa 23 Ras I 34 I 34 j 41 23 Vaz If either O or Oz is greater than one then use the recipr
5. be measured The scattering factor r depends on the scattering mechanisms in the material and typically lies between 1 and 2 1 Ru A 2 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual Ru Another quantity frequently of interest is the carrier mobility defined as uy Re where uy is the Hall mobility and p is the electrical resistivity at zero magnetic flux density The electrical resistivity can be measured by applying a current between contacts 5 and 6 of the sample shown in Figure A 1 and measuring the voltage between contacts 1 and 3 then using the formula Vis wt B T ET where w is the width and fis the thickness of the Hall bar b is the distance between contacts 1 3 and B is the magnetic flux density at which the measurement is taken The Hall bar is a good geometry for making resistance measurements since about half of the voltage applied across the sample appears between the voltage measurement contacts For this reason Hall bars of similar geometries are commonly used when measuring magnetoresistance or Hall mobility on samples with low resistances Disadvantages of Hall bar geometries include the following A minimum of six contacts to make mobility measurements accuracy of resistivity measurements is sensitive to the geometry of the sample Hall bar width and the distance between the side contacts can be especially difficult to measure accurately The accuracy can be increased by making
6. commonly use two systems of units the Sl system and the so called laboratory system The laboratory system is a hybrid combining elements of the SI emu and esu unit systems Table A 1 lists the most common quantities their symbols their units in both systems and the conversion factor between them Table A 1 Unit Systems and Conversions Quantity Symbol sI Factorx Laboratory Conductivity volume o Magnetic induction 8 tesla vsima 107 gauss To use this table 1 SI unit factor x 1 laboratory unit For example 1 tesla 10 gauss A 3 2 Nomenclature The equations below appear twice once in SI units once in laboratory units In all cases voltages are measured in volts electric currents are measured in amperes and resistances are measured in ohms All other measured quantities appear with their respective unit in brackets For example the width of a sample in SI units appears as wim The equations below indicate voltages and currents as follows VOLTAGE NOMENCLATURE 7 B indicates a voltage difference V V measured between terminals k and Terminal is connected to the excitation current source and terminal j is connected to the current sink gt superscript Indicates the sign of the excitation current supplied by the current source B indicates the sign of the applied magnetic induction B measured in the direction shown on the drawings ample Vig B indicates a voltage
7. difference Y a Ys measured while a negative current was supplied by a current source at terminal 5 and flowed to terminal 6 in the presence of a positive applied magnetic induction CURRENT NOMENCLATURE B indicates a current flowing from terminal to terminal of polarity given by the superscript and with the indicated magnetic field polarity Hall Effect Measurements A 5 Lake Shore 7500 9500 Series Hall System User s Manual A 3 3 Van der Pauw Measurements The van der Pauw structure is probably the most popular Hall measurement geometry primarily because it requires fewer geometrical measurements of the sample In 1958 van der Pauw solved the general problem of the potential in a thin conducting layer of arbitrary shape His solution allowed Hall and resistivity measurements to be made on any sample of uniform thickness provided that the sample was homogeneous and there were no holes in it All that is needed to calculate sheet resistivity or carrier concentration is four point contacts on the edge of the surface or four line contacts on the periphery an additional measurement of sample thickness allows calculation of volume resistivity and carrier concentration These relaxed requirements on sample shape simplify fabrication and measurement in comparison to Hall bar techniques On the other hand the van der Pauw structure is more susceptible to errors caused by the finite size of the contacts than the Hall bar It i
8. field but not to external current This is the one intrinsic error source which can not be eliminated from a Hall voltage measurement by field or current reversal 6 Righi Leduc voltage Vp The Nernst diffusion electrons also experience an Ettingshausen type effect since their spread of velocities result in hot and cold sides and thus again set up a transverse Seebeck voltage known as the Righi Leduc voltage Vr The Righi Leduc voltage is also proportional to magnetic field but not to external current 7 Misalignment voltage Vm The excitation current flowing through the sample produces a voltage gradient parallel to the current flow Even in zero magnetic field a voltage appears between the two contacts used to measure the Hall voltage if they are not electrically opposite each other Voltage contacts are difficult to align exactly The misalignment voltage is frequently the largest spurious contribution to the apparent Hall voltage The apparent Hall voltage Vha measured with a single reading contains all of the above spurious voltages Vha Vh Vo Vs Ve Vn Ve Ve All but the Hall and Ettingshausen voltages can be eliminated by combining measurements as shown in Table B 1 Measurements taken at a single magnetic field polarity still have the misalignment voltage frequently the most significant unwanted contribution to the measurement signal Comparing values of Rh B and Rh B reveals the significance of the misalignme
9. perpendicular to a current and an applied magnetic flux The following is an introduction to the Hall effect and its use in materials characterization A number of other sources are available for further information Hall bar geometry Some common Hall bar geometries are shown in Figure A 1 The Hall voltage developed across an 8 contact Hall bar sample with contacts numbered as in Figure A 1 is RuBl Vu V 4 where V gt is the voltage measured between the opposing contacts numbered 2 and 4 Ry is the Hall coefficient of the material B is the applied magnetic flux density is the current and t is the thickness of the sample in the direction parallel to B This section assumes SI units For a given material increase the Hall voltage by increasing B and and by decreasing sample thickness The relationship between the Hall coefficient and the type and density of charge carriers can be complex but useful insight can be developed by examining the limit B gt when r a p n where ris the Hall scattering factor q is the fundamental electric charge p is the density of positive and n the density of negative charge carriers in the material For the case of a material with one dominant carrier the Hall coefficient is inversely proportional to the carrier density The measurement implication is that the greater the density of dominant charge carriers the smaller the Hall coefficient and the smaller the Hall voltage which must
10. Q em I B Lo B E B Lo B a cm and B Ve ral B Vesa B Vocal B Vasa B wm m Qm I B B Lo B L B b m A Via B Vegya B Vol B Vaal B w em zlem 2 cm I B B 15 B 5 B b cm These two resistivities should agree to within 10 If they do not then the sample is too inhomogeneous or anisotropic or has some other problem If they do then the average magnetoresistivity is given by p B LLP o mc Hall Effect Measurements A 9 Lake Shore 7500 9500 Series Hall System User s Manual Hall Coefficient First calculate the individual Hall coefficients _ t m ar B V56 34 B Vs6 34 B Vacs B Ena BT L B Lo B Lo B E B ee 8 t cm V 55 34 B 7 V 56 34 B T V5634 B ae B 10 cm Cc B gauss I5 B B I5 B B and r m Vos B E Vs B T Vaal B dE LO B mi c7 2 BT Ko B 1 4 B 1 8 1 B 8 t cm Veet B Vico B ae B E a B cm c7 Blgauss 1 B L B I5 B 156 B If R and R do not agree to within 10 then the sample is too inhomogeneous or anisotropic or has some other problem If they do agree then the average Hall Coefficient is given by RER Z we m Cem C7 Hall Mobility R a 2 2 1 1 ayi i EER My m V s cm V gt s gives the Hall mobility where p is the zero field re
11. Series Hall System User s Manual A 1 GENERAL The model Hall effect system consists of a uniform slab of electrically conducting material through which a uniform current density flows in the presence of a perpendicular applied magnetic field The Lorentz force deflects moving charge carriers to one side of the sample and generates an electric field perpendicular to both the current density and the applied magnetic field The Hall coefficient is the ratio of the perpendicular electric field to the product of current density and magnetic field while the resistivity is the ratio of the parallel electric field to the current density Experimental determination of a real material s transport properties requires some significant departures from the ideal model To begin with one cannot directly measure the electric field or current density inside a sample Current density is determined from the total excitation current and the sample s geometry Electric fields are determined by measuring voltage differences between electrical contacts on the sample surface Electrical contacts are made of conductive material and usually have a higher conductivity than the sample material itself Electric current therefore tends to flow through the contacts rather than the sample distorting the current density and electric field in the sample from the ideal Excitation current flowing through the contacts used to measure voltage differences reduces both current density in t
12. This section gives outline of device fabrication and subsequent mobility measurements Thursday September 25 2014 3 31 PM Fabrication of mobility measurement devices This is a simple 3 step process which can be carried out using contact aligner for lithography Hall devices are either four contact van der Pauw or six contact Hall bar devices Supplementary information on Hall bar devices and measurements is provided in Appendix from Lake Shore 7500 9500 Series Hall System User s Manual More information is also available on internet http www nist gov pml div683 hall_intro cfm Fabrication steps are 1 Mesa etch This step is carmied out to isolate 2DEGbetween various devices S Bake 110C for 10 minutes to dry out water Cool down Let wafer cool down before spinning PR PR spinning AZ4210 positive PR Ak rpm 309s Softbake Bs Exposure 135 using MJB 3 contact aligner Development AZADOK DI 1 4 for 7Oseconds Rinse Rinse in Dl water N2gundry sti itsti i lt OS 02 descum BOsintabletopRIE PRIE 5 etching BCI3 SiC14 C12 15 10 2 sccm SOW 10mT He flow 10scem Etch rate 300nm min cen 8 Remove PR with 1165 follow cleaning with AMI 5minuteseach pest sR ESEESE SESAO NESE SEDES ENESESSE SEESE ESEESE ESEESE ESEESE EEEE ESEESE SECIS EE aE SOS EES DES DES ED ESE DES EDS DEES DEES EDS DEES ED ESE SESE EDGES SESE NEDENE EDESSE NEDENE BEEE EDGES ESE ES ESS ES ESSE ESE EES AEDES HERRON CC RR RINT 2 Metal Deposition Use bil
13. Tz where V gt 3 is defined as V2 V3 and l42 indicates the current enters the sample through contact 1 and leaves through contact 2 Two voltage readings are required with the van der Pauw sample whereas the resistivity measurement on a Hall bar requires only one This same requirement applies to Hall coefficient measurement as well so equivalent measurements take twice as long with van der Pauw samples The quantity F is a transcendental function of the ratio R defined as Va hb Rn h2 Vis R2314 E or ky h2 Via R2314 Va Iz Rna whichever is greater and F is found by solving the equation R 1 F exp In 2 F arcosh AAA R 1 In 2 2 F 1 when R 1 which occurs with symmetrical samples like circles or squares when the contacts are equally spaced and symmetrical The best measurement accuracy is also obtained when R 1 Rr Squares and circles are the most common van der Pauw geometries but contact size and placement can significantly effect measurement accuracy A few simple cases were treated by van der Pauw Others have shown that for square samples with sides of length a and square or triangular contacts of size 6 in the four corners if 5 a lt 0 1 then the measurement error is less than 10 The error is reduced by placing the contacts on square samples at the midpoint of the sides rather than in the corners The Greek cross shown in Figure A 2 has arms which serve to isolate the contacts from t
14. a _ E 7 En Be a ee Bo 7 Riz El Rao Laa Laa Vas E V123 A 6 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual If either O or O is greater than one then use the reciprocal instead The relationship between f and O is expressed by the transcendental equation Ll i ney 2 O 1 In2 EH we alle which can be solved numerically The two resistivities O and p should agree to within 10 If they do not then the sample is too inhomogeneous or anisotropic or has some other problem If they do agree the average resistivity is given by Po aa a Q m Q cm Magnetoresistivity If desired calculate the magnetoresistivity as Tf t m al p4 B In 2 HB Ll T als B o V243 B js A B A ale B 13 B Ial B Li 8 h 8 0 B ats cm an al V4 PGs B i we L B B eS i V4 2 1a BV o B V B 41 Calculate factors f and f the same way as at zero magnetic field and the average magnetoresistivity is Pa B Q m Q cm This measurement does not give the true magnetoresistance as defined in terms of the material s conductivity tensor Van der Pauw s calculation of resistivity is invalid in the presence of a magnetic field since the magnetic field alters the current density vector field inside the sample On the other hand magnetoresistance measurements are routinely performed on van der Pauw samples anyway P4 B p B 2 Hall E
15. ayer PR lithography Spin HMDS 4krpm Spin 825 OCG 4krpm Bake 95C 1min Spin SPR 955CM 0 9 4krpm Bake 95C 1min Exposure 18s with MJB 3 contact aligner Develop 40secs 726MIF H20 2 1 Rinse and dry Develop 35secs 726MIF undiluted Rinse and dry Inspect o E gt TFA HDA ST 02 descum 30seconds Quick 1 20 HCI H20 dip and H20 rinse just before loading in metal deposition chamber This is done to remove any oxides Metal deposition in Ebeam 1 chamber 5nm Ni 5 pellets AuGe 375nm 30nmNi 100nmAu Deposit AuGe until all AuGe is evaporated necessary for stoichiometry Soak in 1165 for liftoff It should take 45 minutes Clean using AMI 5 minutes each The bottom image shows a processed four contact van der Pauw device 3 Annealing of Ohmic Metal This is one of the most crucial step in TACIT processing Always save small pieces of the sample to use as calibration for annealing Annealing was done in RTA chamber in cleanroom For the last device the recipe titled 445 50s Forming Gas 5 __ 07 12 2013 rcp was used This means annealing at 445C for 50s in presence of forming gas always start annealing at 430C keeping time fixed and increase temperature by 5C if 430C does not work repeating this on new calibration piece each time Metal will look discolored and rough after annealing Annealing Recipe for 430C New Recipe 1 Purge for 30s Forming gas on 30s Ramp to purge 120C 1min Stay at purge 2min Ramp to prime 250C
16. contact to the sides of the bar at the end of extended arms as shown in Figure A 2 Creating such patterns can be difficult and can result in fragile samples sm Tal ls i 2 1 q T TS By 2 4 b E Fl 4 re l i i i 4 contact 2 2 6 contact 3 1 8 contact 2 2 8 contact 8 contact thin film T Figure A 1 Common Hall Bar Geometries Sample thickness t of a thin film sample diffusion depth or layer thickness Contacts are black numbered according to the standard to mount in Lake Shore sample holders B oriented out of the page a en ec im lt C 0900 circle clover leaf square rectangle cross Figure A 2 Common van der Pauw Sample Geometries The cross appears as a thin film pattern and the others are bulk samples Contacts are black van der Pauw geometry Some disadvantages of Hall bar geometries can be avoided with van der Pauw sample geometries see Figure A 2 Van der Pauw showed how to calculate the resistivity carrier concentration and mobility of an arbitrary flat sample if the following conditions are met Hall Effect Measurements A 3 Lake Shore 7500 9500 Series Hall System User s Manual 1 The contacts are on the circumference of the sample 2 The contacts are sufficiently small 3 The sample is of uniform thickness and 4 The sample is singly connected contains no isolated holes The resistivity of a van der Pauw sample is given by the expression m Va Vis ri In 2 h2
17. d the zero field resistivity if it was not A 4 COMPARISON TO ASTM STANDARD The contact numbering and voltage measurement indexing given above differ in several ways from that given in the ASTM Standard F76 To begin the ASTM contact numbering schemes for the van der Pauw and Hall Bar geometries are incompatible with one another To allow either sample type to be mounted using the same set of contacts Lake Shore s numbering scheme for Hall bar samples differs from the ASTM scheme Second the ASTM standard is inconsistent with the handedness of the van der Pauw contact numbering order with respect to the applied magnetic field Lake Shore numbered the contacts counter clockwise in ascending order when the sample is viewed from above with the magnetic field perpendicular to the sample and pointing toward the observer as shown in Figure A 3 Measuring Resistivity and Hall Coefficient Using a van der Pauw Geometry Finally the ASTM assumes that the direction of the excitation current is to be changed by physically reversing the current connections This technique is not well suited to high resistance samples using a programmable current source like the Keithley Model 220 This current source and others like it has a guarded high current output and an unguarded low current return For proper current source operation the high output lead should be farther from common ground than the low return lead a condition violat
18. ed half of the time when physically reversing the high and low current leads to the sample When this condition is violated leakage current can flow through the voltmeter leading to possibly serious measurement errors To avoid this difficulty Lake Shore reversed the sign of the programmed current source while leaving the contacts alone This requires a more sophisticated notation for voltage measurements V B ij kl In this notation terminal 7 refers to the contact to which the current source output attaches terminal 7 is the current return contact terminal k is the positive voltmeter terminal and terminal is the negative voltmeter terminal The superscript refers to the sign of the programmed current while B refers to the sign of the applied magnetic field relative to the positive direction indicated in the figures Hall Effect Measurements A 13 Lake Shore 7500 9500 Series Hall System User s Manual A 5 SOURCES OF MEASUREMENT ERROR David C Look gives a good treatment of systematic error sources in Hall effect measurements in the first chapter of his book 2 There are two kinds of error sources intrinsic and geometrical A 5 1 Intrinsic Error Sources The apparent Hall voltage Vha measured with a single reading can include several spurious voltages These spurious error sources include the following 1 Voltmeter offset V An improperly zeroed voltmeter adds a voltage V to every measurement The offset does not c
19. erned as a thin film on a thicker substrate Another disadvantage is thatthe gt active area of the cloverleaf is much smaller than the actual sample Figure A 11 Cloverleaf van der Pauw Structure A 16 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual A 4 3 3 Greek Cross Structures The Greek cross is one of the best van der Pauw geometries to minimize finite contact errors Its advantage over sim ler van der Pauw structures is similar to placing Hall bar contacts at the ends of arms David and Beuhler analyzed this structure numerically They found that the deviation of the actual resistivity p from the measured value p obeyed P a E 1 0 59 0 006 exp 6 23 0 02 n C This is a very small error for c c 2a 1 6 where c 2a corresponds to the total dimension of the contact arm E 10 21 22 Hall coefficient results are substantially better De Mey has shown that UU y U tm A m c Ln Hm 24M 1 0450 four contacts Uy Uy Figure A 12 Greek Cross van der Pauw Structure where uy and Uy are the actual and measured Hall mobilities respectively For c c 2a 1 6 this results in Ay Uy 0 04 which is quite respectable References Schroder D K Semiconductor Material and Device Characterization John Wiley amp Sons New York 1990 Look David C Electrical Characterization of GaAs Materials and Devices John Wiley amp Sons
20. ffect Measurements A 7 Lake Shore 7500 9500 Series Hall System User s Manual Hall Coefficient Calculate two values of the Hall coefficient by the following _ tim AB BA Prel B Piel B pa e ke a m C B T I B L B 1 B Li B 8 t cm ave B 7 Void B t Viy B 7 Vase 10 Bleauss B 1 B 1 B 8 and _ t m loy B Voal B T ATG B Sa Vane B 3 8 t cm V p B a Vala B T Vo B z PR B 10 B gauss Tp B p B Ip B 15 B om En These two should agree to within 10 If they do not then the sample is too inhomogeneous or anisotropic or has some other problem If they do then the average Hall coefficient can be calculated by MRS R HD m Ct em Ct Hav 2 Hall Mobility The Hall mobility is given by Ul en m yA s em Vo 57 Pa where p is the magnetoresistivity if it was measured and the zero field resistivity if it was not A 3 4 Hall Bar Measurements Hall bars approximate the ideal geometry for measuring the Hall effect in which a constant current density flows along the long axis of a rectangular solid perpendicular to an applied external magnetic field A 3 4 1 Six contact 1 2 2 1 Hall Bar An ideal six contact 1 2 2 1 Hall bar geometry is symmetrical Contact separations a and b on either side of the sample are equal with contacts located opposite one another Contact pairs are placed symmetrically about the m
21. hange with sample current or magnetic field direction 2 Current meter offset I An improperly zeroed current meter adds a current to every measurement The offset does not change with sample current or magnetic field direction 3 Thermoelectric voltages Vs A temperature gradient across the sample allows two contacts to function as a pair of thermocouple junctions The resulting thermoelectric voltage due to the Seebeck effect is designated V Portions of wiring to the sensor can also produce thermoelectric voltages in response to temperature gradients These thermoelectric voltages are not affected by current or magnetic field to first order 4 Ettingshausen effect voltage Veg Even if no external transverse temperature gradient exists the sample can set up its own The evxB force shunts slow cool and fast hot electrons to the sides in different numbers and causes an internally generated Seebeck effect This phenomenon is known as the Ettingshausen effect Unlike the Seebeck effect Veg is proportional to both current and magnetic field 5 Nernst effect voltage Vj If a longitudinal temperature gradient exists across the sample then electrons tend to diffuse from the hot end of the sample to the cold end and this diffusion current is affected by a magnetic field producing a Hall voltage The phenomenon is known as the Nernst or Nernst Ettingshausen effect The resulting voltage is designated Vy and is proportional to magnetic
22. he active region When using the Greek cross sample geometry with a w gt 1 02 less than 1 error is introduced A cloverleaf shaped structure like the one shown in Figure A 2 is often used for a patternable thin film on a substrate The active area in the center is connected by four pathways to four connection pads around its perimeter This shape makes the measurement much less sensitive to contact size allowing for larger contact areas The contact size affects voltage required to pass a current between two contacts Ideal point contacts would produce no error due to contact size but require an enormous voltage to force the current through the infinitesimal contact area Even with square contacts in the corners of a square sample with d a lt 0 1 the ratio of the output to input voltage V43 V12 is on the order of 1 10 Van der Pauw sample geometries are thus much less efficient at using the available excitation voltage than Hall bars Advantages of van der Pauw samples Only four contacts required No need to measure sample widths or distances between contacts Simple geometries can be used Disadvantages Measurements take about twice as long Errors due to contact size and placement can be significant when using simple geometries Mobility spectra Hall effect measurements are usually performed at just one magnetic flux density although polarity is reversed and the voltage readings averaged to remove some sources or error The resulting
23. he vicinity and the Hall field If a contact extends across the sample in the same direction as the Hall field it can conduct current from one side of the sample to the other shorting out the Hall voltage and leading to an underestimate of the Hall coefficient Finally if pairs of contacts used in a voltage measurement are not aligned properly either perpendicular or parallel to the excitation current density then the voltages measured will not correctly determine the perpendicular or parallel component of the electric field To minimize these geometrical problems one must take care with the size and placement of electrical contacts to the sample There are also many intrinsic physical mechanisms that alter current density and electric field behavior in a real material Most of these relate to the thermoelectric behavior of the material in or out of a magnetic field Some of these effects can be minimized by controlling temperature in the sample s vicinity to minimize thermal gradients across it In addition most errors introduced by intrinsic physical mechanisms can be canceled by reversing either the excitation current or the magnetic field and averaging measurements A 2 HALL EFFECT MEASUREMENT THEORY Hall effect measurements commonly use two sample geometries 1 long narrow Hall bar geometries and 2 nearly square or circular van der Pauw geometries Each has advantages and disadvantages In both types of samples a Hall voltage is developed
24. idpoint of the sample s long axis This geometry allows two equivalent measurement sets to check for sample homogeneity in both resistivity and Hall coefficient However the close location of the Hall voltage contacts to the sample ends may cause the end contacts to short out the Hall voltage leading to an underestimate of the actual Hall coefficient While the 1 2 2 1 Hall bar geometry is included in ASTM Standard F76 the contact ONN A numbering given here differs from the standard A 8 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual Resistivity To calculate resistivity at zero field first calculate _ Ven B 0 Vo B 0 w m m fern 2 IX B 0 1 B 0 alm _ Ver B 0 Vo B 0 w em cm eel 1 B 0 B 0 a cm and A Vera B 0 Vera B 0 w m m B B 0 1 B 0 qm 1 _ Vee 4 B 0 Vo 14 B 0 w cm cm TT B 0 Ta B 0 Aem O These two resistivities should agree to within 10 If they do not then the sample is too inhomogeneous or anisotropic or has some other problem If they do then the average resistivity is given by pp Pile ata Magnetoresistivity Magnetoresistivity is typically used in mobility spectrum calculations but not in Hall mobility calculations To calculate magnetoresistivity first calculate E A B Vs 23 B one B V56 oe B w m t m ADE a O _ V Pi B Vss 23 B gi Ve b Vale B v cm t cm
25. in 1min Stay at prime 1min Ramp to 430C in 40 sec 5sec Hold at 430C for t seconds OA UL ew S Mobility Measurements and Device Preparation Page 1 9 Cool down with FG on for 10min 10 Shut off if T lt 190C Making sure if Ohmic contacts work One way to check if rapid thermal annealing has been successful and good Ohmic contact has been established to the 2DEG is to carry out temperature dependent resistance measurements These measurements can be carried out in any variable temperature cryostat Resistance can be measured between any two contacts or in four contact geometry One should make sure that all the contact pads work on the Hall bar device before beginning Hall measurements The temperature dependence of the resistance of GaAs AlGaAs 2DEG without any back gate looks is monotonic decrease in resistance as temperature is decreased For a device with backgate the resistance temperature dependence is not monotonic The bottom figure shows an example of the Resistance of the channel as a function of temperature 10000 Topto Top Bridge device 8000 6000 ohm cm 4000 Res ch1 2000 0 50 100 150 200 250 Temperature K Mobility Measurements were carried out at MRL s PPMS and Dynacool facilities For mobility measurements sample is mounted on a puck and wire bonded A sample picture is shown below of a hall cross mounted wirebonded to puck pads Greek Cross The first step to measure
26. ity if it was not A 3 4 3 Eight contact 1 3 3 1 Hall Bar The eight contact 1 3 3 1 Hall bar geometry is ideally the most symmetrical of the Hall bars Two sets of three equally spaced contacts lie directly opposite one another on either side of the sample with center contacts numbers 2 and 4 located at the exact center of the sample s length Voltage measurement connections are made to contacts 1 through 4 while current flows from contact 5 to contact 6 Only six of the eight contacts are used in this measuring procedure The remaining two unnumbered contacts are included to keep the sample completely symmetrical Figure A 6 Eight Contact The eight contact Hall bar attempts to combine the homogeneity 1 3 3 1 Hall Bar Geometry checks possible with the 1 2 2 1 six contact geometry and the benefit of measuring the Hall voltage in the center of the sample It allows two resistivity measurements compare for homogeneity but only one Hall voltage measurement Either the 1 2 2 1 or 1 3 1 1 six contact measurements can be made using an eight contact Hall bar simply by moving the electrical connections to the appropriate points The eight contact Hall bar geometry is included in ASTM Standard F76 but the contact numbering given here differs from the standard Hall Effect Measurements A 11 Lake Shore 7500 9500 Series Hall System User s Manual Resistivity First calculate the two resistivities Vs 2 8 0 Vi 23 8 0 w m
27. ll mobility measurements Arch Electron Uebertragungstech 27 309 13 1973 2 De Mey G Potential Calculations for Hall Plates Advances in Electronics and Electron Physics Vol 61 Eds L Marton and C Marton pp 1 61 Academic New York 1983 Hall Effect Measurements A 17 Lake Shore 7500 9500 Series Hall System User s Manual This Page Intentionally Left Blank A 18 Hall Effect Measurements
28. m Ly E Hall Mobility The Hall mobility is given by R Hav 2 1 2 l 1 A m V s eom Veg av Once Hall mobility is calculated sheet density is N Pav eu Where e is electronic charge Below is the list of various 2DEG samples and the measured Hall mobility and sheet density on them 1 Sample 2_1 12 2 The schematic of the sample is shown below There is no backgate in this sample GaAs 60 Alz Ga As 160A 2DEG 100nm below the surface S1 Delta doping Al Ga yAs 100A z Si Delta doping Al Ga y As 680A GaAs 500A GaAs 30A Al Ga As 100A GaAs 6800A aoe ooo oo ooo ooo LD o ooo SS q OO O aul Uuak ul Unk vol Cunt Bui Uae ui Cube Bui U ue ui Uae uh uae uum Gaa uk ue a uk Unt eu UUaUa ul Uae Uk OG aba Ui G ua uk Uune Su oubh bu UuNkeubo uN uM CURSE 80K le 6 1 1 31092E11 240 1 1 9857E5 50K 1e 6 1 1 28386E11 83 29 5 84429E5 5K 1e 6 1 7 90795E10 17 44781 4 52976E6 AAA AAA AAA AAA AAA AAA DARA ARA AAA AAA A A A a A oh ka A a A a i Al a i A a i Ad A AA A A A A A A AD 2 Sample 4 23 13 1 from Princeton This is again a calibration sample without backgate The schematic is Mobility Measurements and Device Preparation Page 4 Si Delta doping GaAs 30A AL Ga As 100A GaAs 6500A GaAs 500A GaAs Substrate This sample was used to optimize dry and wet etch recipes and to compare mobility measurements The comparison of dry vs wet etch in terms of mobility measurements is summa
29. nt voltage relative to the signal voltage A 14 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual A Hall measurement is fundamentally a voltage divided by a current so excitation current errors are equally as important Current offsets are canceled by combining the current measurements then dividing the combined Hall voltage by the combined excitation current Table A 2 Hall effect measurement voltages showing the elimination of all but the Hall and Ettingshausen voltages by combining readings with different current and magnetic field polarities A 5 2 Geometrical Errors in Hall Bar Samples Geometrical error sources in the Hall bar arrangement are caused by deviations of the actual measurement geometry from the ideal of a rectangular solid with constant current density and point like W V voltage contacts 7 The first geometrical consideration with the Hall bar is the tendency of the end contacts to short out the Hall voltage If the aspect ratio of sample length to width w 3 then this error is less than hy NES Therefore it s important w23 The finite size of the contacts affects both the current density and i electric potential in their vicinity and may lead to fairly large errors The errors are larger for a simple rectangular Hall bar than for one Figure A 7 Hall Bar in which the contacts are placed at the end of arms With Finite Voltage Contact
30. ocal instead The relationship between f and O is expressed by the transcendental equation O 1 f l In 2 mA A ee he pis fi w AM RATI which can be solved numerically The two resistivities p and p should agree to within 10 If they do not then the sample is too inhomogeneous or anisotropic or has some other problem If they do agree the average resistivity is given by p fat Pa Es Q m Q cm 2 So after this procedure sheet resistivity for 2DEG is obtained in units of Ohms Measuerments in presence of magnetic field are carried out Again for 2DEG hall coefficient is expression as written below thickness t is neglected so the unit for Hall coefficient is m Ct Mobility Measurements and Device Preparation Page 3 Hall Coefficient Calculate two values of the Hall coefficient by the following BIT B L B L B L B m C7 m Ce gos Lem Vio B Vao B Vira B Vial B cm C7 B gauss E B L B Li B E B and _ m RN B E Fault B as LE a B AT a B m co BET Ti B L B La B 15 B 8 cm Vanl B m Veas B Vas B Var B 10 Blgauss 1 B 1 B L B Li B cm cm Bn These two should agree to within 10 If they do not then the sample is too inhomogeneous or anisotropic or has some other problem If they do then the average Hall coefficient can be calculated by Es Ryc Rup 3 1 3 1 1 E
31. rized in the table below AAA AANA A A AINA NAIA AINA AINA ANIA AMAA AABN MA ANNAN NIAAA N AANA AAA A AMAA ANIMA ANAL N AANA AMMAN AIAN A A NIAAA NIAAA N AANA ANNAN NAAN A A ANNAN MAAN AANA AMAA ANA A A A AANA A AANA A MANNA A A A A NAAN ANAEMIA NAAN AANA AMAA AANA ANNAN AANA ANT A ANIA cnc Etching Method Current pA Magnetic Field T Temperature K Ns p Q Mobility cm V s po Dy oI 82 89 10 5 100 2484 2159x108 o Dy 1 SO 2861x105 32 6806 66 841x10 _ Dy oI 4x88 5 07821 432 13x104 o Wet E802 747105 100 046 22 739x10 Wet i 10 2 654x1015 5 4084 435 41x10 A q O NOS NON NN NN Note Etching method above refers to etching of mesa to define hall devices Dry etching method has been described in the text above Wet etch recipe is i Etch in 1 20 H20 HCI for 30s to get rid of any native oxide This gives better uniformity in etching ii Etch in 1 8 80 H2504 H202 H20 solution Etch rate is 400nm min If want lower etch rate dilute solution Always etch sample upside down 3 Sample 3_26 13 1 This is sample with backgate from which TACIT device was made GaAs cap 10nm Si 9 doping 31m GaAs 60 50 Mobility Measurements and Device Preparation Page 5 A S Resistance of 2DEG Q sq N O gt O O O O ES o 1 6x10 Sheet Density cm 40 50 60 70 80 Temperature K Mobility EF Sheet Density Current 1 uA Field 1T 20 30 40 50 60 70 Temperature K Mobility Mea
32. rrection factor Ap p for a square van der Pauw structure is roughly proportional to c I for both square and triangular contacts At c 1 6 Ao p 2 for identical square contacts and Ap p lt 1 for identical triangular contacts Hall voltage measurement error is much worse unfortunately The correction factor AR Ry is proportional to c and is about 15 for triangular contacts when c 1 6 The correction factor also increases by about 3 at this aspect ratio as the Hall angle increases from tan 0 1 to tan 0 5 Figure A 9 Square van der Pauw Structure with L Either Square or Triangular Contacts ele A 5 3 2 Circular Structures l Circular van der Pauw structures fare slightly better van der Pauw gives a correction factor for circular contacts of Ap l c SS ll Se per contact p 16In2 which results in a correction of Ap p 1 for c 1 6 for four contacts For the Hall coefficient van der Pauw gives the correction gt de HE 20 Figure A 10 e per contact Circular 1 van der Pauw At c Ph 1 6 this results in a correction of 13 for four contacts Structure van Daal reduced these errors considerably by a factor of 10 to 20 for resistivity and 3 to 5 for Hall coefficient by cutting slots to turn the sample into a cloverleaf The clover leaf structure is mechanically weaker than the square and round samples unless it is patt
33. s For a simple rectangle the error in the Hall mobility can be approximated when uB lt lt 1 by E L ai BE ae Dei ay Mr Here All y is the amount 4 must increase to obtain a true value J _ W If w 3 and c w 0 2 then All Uy 0 13 which is certainly a significant error a Reduce the contact size error to acceptable levels by placing i contacts at the ends of contact arms The following aspect ratios l yield small deviations from the ideal p c c lt w 3 1 gt 4w Figure A 8 Hall Bar With Contact Arms Hall Effect Measurements A 15 Lake Shore 7500 9500 Series Hall System User s Manual A 5 3 Geometrical Errors in van der Pauw Structures Van der Pauw s analysis of resistivity and Hall effect in arbitrary structures assumes point like electrical connections to the sample In practice this ideal can be difficult or impossible to achieve especially for small samples The finite contact size corrections depend on the particular sample geometry and for Hall voltages the Hall angle 6 defined by tan6 UB where u is the mobility Look presents the results of both theoretical and experimental determinations of the correction factors for some of the most common geometries We summarize these results here and compare the correction factors for a 1 6 aspect ratio of contact size to sample size A 5 3 1 Square Structures The resistivity co
34. s also impossible to accurately measure magnetoresistance with the van der Pauw geometry so both Hall effect and magnetoresistance i e the whole conductivity tensor measurements must be done with a Hall bar geometry Figure A 3 Measuring Resistivity and Hall Coefficient Using a van der Pauw Geometry In the basic van der Pauw contact arrangement the four contacts made to the sample are numbered counter clockwise in ascending order when the sample is viewed from above with the magnetic field perpendicular to the sample and pointing toward the observer The sample interior should contain no contacts or holes The sample must be homogeneous and of uniform thickness Resistivity Again let Viki indicate a voltage measured across terminals k and with k positive while a positive current flows into terminal and out of terminal j In a similar fashion let Riki indicate a resistance Ria Va lj with the voltage measured across terminals k and while a positive current flows into and out of j First calculate the two resistivities o a a Vizas Vins Vasa In 2 Li Li L La and 22 7 T s t m cm Var Vaa Vos Va Pr Q m Q cm z In 2 I y Geometrical factors f and f are functions of resistance ratios O and Q respectively given by O _ Ea g S B Za E feo L 15 AT nn a 9 R a R314 I o l Vri a Va and O E S
35. single mobility calculated from the measurements is a weighted average of the mobilities of all carriers present in the sample Beck and Anderson developed a technique for interpreting magnetic flux dependent Hall data which generates a mobility spectrum The result is a plot of the carrier concentration of conductivity as a function of the mobility The number of peaks appearing in a mobility spectrum indicates the number of distinct charge carriers active in the material This powerful technique has virtually eliminated the need for destructive testing techniques such as differential profiling An example mobility spectrum analysis performed on a GaAs AlGaAs five quantum well heterostructure is shown in Figure 2 9 of their paper A technique combining mobility spectrum analysis and multi carrier fitting was developed by Brugger and Kosser yielding some improvement The development of quantitative mobility spectrum analysis by Antoszewski et al 2 has produced even greater improvements in capability A 4 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual A 3 SAMPLE GEOMETRIES AND MEASUREMENTS SUPPORTED BY IDEAS HALL SOFTWARE This section describes common sample geometries useful in Lake Shore s 9500 Series Hall Measurement System and formulas used to calculate resistivities Hall coefficients carrier concentrations and mobilities A 3 1 System of Units Hall effect and magnetoresistance measurements
36. sistivity Pa A 3 4 2 Six contact 1 3 1 1 Hall Bar The ideal 1 3 1 1 Hall bar geometry places contacts 2 and 4 directly across from one another in the exact middle of the sample s length and contacts 1 and 3 symmetrically on either side of contact 2 This geometry allows no homogeneity checks but measuring the Hall voltage in the exact center of the sample s length helps minimize the shorting of the Hall voltage via the end contacts The 1 3 1 1 Hall bar is not included in ASTM Standard F76 Resistivity Figure A 5 Six Contact 1 3 1 1 Hall Bar Geometry Calculate the resistivity at zero field by _ Vee13 B 0 Ve 13 B 0 w m zm P k Am Ve B 0 V B 0 w em t cm B 0 1J8 0 qe O A 10 Hall Effect Measurements Lake Shore 7500 9500 Series Hall System User s Manual Magnetoresistivity If desired calculate the magnetoresistivity by By Usual B Vseaal B Vseas 8 a 8 wm em I B 1 B 1 gt B E B am Q m E Ven B Vilt B Vail B Vog 3 B wem tem Ta BJ lalt A B B N Hall Coefficient Calculate the Hall coefficient by BT L B 1 B 1 B K B t em en Gs B V 56 04 B T V 56 04 B gt Vics B 3 B gauss I5 B L B Lo B 15 B Hall Mobility Ru The Hall mobility is given by 4 2 m V 5 cm V s7 P where p is the magnetoresistivity if it was measured and the zero field resistiv
37. surements and Device Preparation Page 6 10 Mobility cm V s Lake Shore 7500 9500 Series Hall System User s Manual APPENDIX A HALL EFFECT MEASUREMENTS Contents A 1 GENERA runa red dle A coda A 2 A 2 HALL EFFECT MEASUREMENT THEORY a a A 2 A 3 SAMPLE GEOMETRIES amp MEASUREMENTS SUPPORTED BY HALL SOFTWARE A 5 A 3 1 SVS ENTOT UNS a bial eee ened ee dente A 5 A 3 2 NomenciatUTSs ei de ed do Do edad e o craton ance dhe es E A 5 A 3 3 Van der Pauw MeasureMentsS ccceccccececccececcececcaeuceneaceneaeueeneaeeueaeaeeneaeentaesneneaeentaneneatanes A 6 A 3 4 HalBar Measurenments a a cid A 8 A 3 4 1 SIX CONlACE We2 7 21 Aall al secs o ones A 8 A 3 4 2 SIX CONTACE t3 T A Ao de A 10 A 3 4 3 Elgntcontact 1 3 3 1 Hal Bar tad A 11 A 4 COMPARISON TO ASTM STANDARD 0 ccc ccccececececcccccccecececcccaeacacececeneauauatetenenecuaeaeenenes A 13 A 5 SOURCES OF MEASUREMENT ERROR Q 0 cccccececececececcucececececeucececeneneuaeacstenenennananenenes A 14 A 5 1 IEPINSIC TETTOR SOUIlCOS dd costes A 14 A 5 2 Geometrical Errors in Hall Bar Samples oocccoccccocncccnccccnccncnonannnonnnonononnnncnncnnnanonannnnnnnnos A 15 A 5 3 Geometrical Errors in van der Pauw Structures c cccccccccececececececcaeacecsceceneneaeaneteneneneas A 16 A 5 3 1 Square MU US od A 16 A 5 3 2 A A deena te eerie A 16 A 4 3 3 reek Cross AU ds A 17 Hall Effect Measurements A 1 Lake Shore 7500 9500
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