ProblemsandSolutionsto PhysicsofSemiconductorDevices

1 Problems and Solutions toPhysics of Semiconductor Lavrov Contents1 Properties of Semiconductors .. schottky Diode .. Idealp-nJunction .. Nonidealp-nJunction .. Solar Cells .. Bipolar Transistor .. MIS/MOS Capacitor and MOSFET .. Low-dimensional Structures .. LEDs and Lasers .. 82 Literature93 Tables104 Answers and Properties of Semiconductors .. schottky Diode .. Idealp-nJunction .. Nonidealp-nJunction .. Solar Cells .. Bipolar Transistor .. MIS/MOS Capacitor and MOSFET .. Low-dimensional Structures .. LEDs and Lasers .. 23 IAP/HLP, Tel: 33637, Physikgeb aude C305, Properties of of the following semiconductors are transparent, partially transparent, non-transparent for visible light ( = m): Si, GaAs, GaP, and GaN?

2 Gap of Si depends on the temperature asEg= eV 10 4T2T+ a concentration of electrons in the conduction band of intrinsic(undoped) Si atT= 77 K if at 300 Kni= 1010cm mobility in Si is 1400 cm2V 1s 1. Calculate the mean free time in scat-tering (Relaxationszeit) of electrons. Effective mass ism e/m0= thermal velocity of electrons and holes in GaAs at room masses arem e/m0= andm h/m0= mobility in Ge at room temperature is 1900 cm2V 1s 1. Find the dielectric relaxation time inp-type Ge at room temperature. Assume thatall acceptors are 1015cm 3, = 16, p= 1900 cm2V 1s dielectric relaxation time in intrinsic Si at 300 K. = 12, n= 1400cm2V 1s 1, n= Debye length inp-type Ge at 300 K ifNa= 1014cm 3.

3 Assume that allacceptors are ionized, = the ambipolar diffusion coefficient of intrinsic (undoped) Ge at 300 K. n/ p= , n= 3900 cm2V 1s are injected inton-type Ge so that at the sample surface p0= 1014cm pat the distance of 4 mm from the surface if p= 10 3s andDp= 49 cm2 schottky a hight of the potential barrier for a Au-n-Ge schottky contact at room tem-perature (T= 293 K) if = 1 cm, Au= eV, and Ge= eV. Electronmobility in Ge is 3900 cm2V 1s 1, density of the states in the conduction band isNc= 1015 T3/2cm the depletion width for a Pt-n-Si schottky diode (T= 300 K) atV= 0,+ , and 2 V. Concentration of doping impurity in Si equals 4 1016cm 3. Workfunction of Pt is eV, electron affinity of Si is eV, Si= , density of thestates in the conduction band isNc= 1015 T3/2cm a schottky contact Au-GaAs calculate the maximum electric fieldwithin thespace charge region atV= 0, + , and 100 1016cm 3, GaAs= eV, GaAs= Work function of Au is eV,T= 300K, density of the states in theconduction band isNc= 1013 T3/2cm is the electric fieldEfor a schottky diode Au-n-Si atV= 5 V at the distanceof m from the interface at room temperature if = 10 cm, n= 1400 cm2V 1s 1,Nc= 1015 T3/2cm current densitiesjat room temperature for a schottky diode Pt-n-GaAs atV= + and 5 V if = 50 cm.

4 N= 8800 cm2V 1s 1,mn/m0= , workfunction of Pt is eV, GaAs= eV,Nc= 1013 T3/2cm 3. Applythermionic-emission capacitance of a Au-n-GaAs schottky diode is given by the relation 1/C2= 1015 1015V, whereCis expressed in F andVis in Volts. Taking thediode area to be cm2, calculate the barrier height and the dopant comparison of the de Broglie wavelength of electron with the depletion widthof a contact metal-n-Si, estimate the electron concentration at which schottky diodeloses its rectifying characteristics. For the estimate, assume that the height of thepotential barrier a the contact is half the value of the band gap at room temperature(Eg= eV),m e=m0,T= 300 K, and Si= the built-in potential for ap-nSi junction at room temperature if the bulkresistivity of Si is 1 cm.

5 Electron mobility in Si at RT is 1400 cm2V 1s 1; n/ p= ;ni= 1010cm thep-nSi junction from the previous problem calculate the width of the spacecharge region for the applied voltagesV= 10, 0, and + V. Si= the parameters given in the previous problem find the maximum electric fieldwithin the space charge region. Compare these values with the electric field within ashallow donor:E e/ Sia2B, whereaBis the Bohr radius of a shallow donor,aB= Si~2/m ee2andm e/m0= the capacity of thep-njunction from the problem 2 if the area of thejunction is of ap-nSi junction has a resistivity of 1 cm. What should be the resistivityofp-Si so that 99 % of the total width of the space charge region wouldbe located inn-Si (p+-njunction)?

6 For the parameters needed see problem room temperature under the forward bias of V the currentthrough ap-njunction is mA. What will be the current through the junction under reverse bias? ap+-nSi junction the reverse current at room temperature is the minority-carrier lifetime ifNd= 1015cm 3,ni= 1010cm 3, and p= 450 cm2V 1s does the reverse current of a Sip-njunction change if the temperature raisesfrom 20 to 50 C? The same for a Gep-njunction. Band gaps of Si and Ge are eV, temperatures at whichp-njunctions made of Ge, Si, and GaN lose theirrectifying characteristics. In all casesNa=Nd= 1015cm 3. Assume thatEgareindependent of the temperature and are , , and eV forGe, Si, and GaN,respectively.

7 Intrinsic carrier concentrations at room temperature arenGei= 2 1013,nSii= 1010, andnGaNi= 10 9cm withNd= 7 1015cm 3additionally containsNt= 1015cm 3generation-recombination centers located at the intrinsic Fermi level with n= p= 10 15cm2andvt= 107cm/s. Calculate generation rate, low as compared to the equilibrium value2. onlypis below the equilibrium Si,ni= 1010cm ofn-type Si (Nd= 1016cm 3) generates 1021cm 3/s electron-holepairs. Si hasNt= 1015cm 3generation-recombination centers with n= p=10 16cm2. Calculate equilibrium concentration of electrons and holes ifEt=Ei, whereEiis the Fermi level of intrinsic Si, andvt= 107 +-nSi junction (ni= 1010cm 3, = ) is formed in ann-type substratewithNd= 1015cm 3.

8 If the junction contains 1015cm 3generation-recombinationcenters located at the intrinsic Fermi level with n= p= 10 15cm2(vt= 107cm/s),calculate generation current density at a reverse bias of 10 ap-nSi junction with thep-side doped to 1017cm 3, then-side doped to1019cm 3(n+-pjunction), and a reverse bias of 2 V, calculate the generation currentdensity at room temperature, assuming that the effective lifetime is10 ap-nGaAs junction at room temperature find the donor/acceptor concentrationat which de Broglie wavelength ( = 2 ~/ 2m E) of electrons/holes is equal to thewidth of the space charge region. AssumehEi= 3kT/2,m e/m0= ,m h/m0= ,and GaAs= ,nGaAsi= 106cm 3, andNa= a siliconp+-njunction is reverse-biased to 30 V, the depletion-layer ca-pacitance is nF/cm2.

9 If the maximum electric field at avalanche breakdown is3 105V/cm, find the breakdown voltage. Si= ap+-nSi junction withNd= 1016cm 3, the breakdown voltage is 32 the maximum electric field at the breakdown. Si= Solar spectrum of Sun could be reasonably well modelled by that of theblack bodywithT 5800 K. In this case, the number of photons and power per unit energy couldbe approximated asdN =g( )d 2d e~ /kT 1,dE =~ g( )d 3d e~ /kT the maximum flux density and power per photonenergycoming to Earth from Sun(find maxima ofg( ) and g( )). What are the corresponding maxima inwavelength?Hint: use the relation = 2 c/ in g( ). a Sip-njunction solar sell of area 2 cm2. If the dopings of the solar cellareNa= 1016cm 3andNd= 5 1019cm 3, and given n= 10 s, p= s,Dn= cm2/s,Dp= cm2/s, andIL= 95 mA, (i) calculate the open-circuit voltage,and (ii) determine the maximum output power of the solar cell at room temperature, an ideal solar cell has a short-circuit current of 3 A and anopen-circuit voltage of V.

10 Calculate and sketch its power outputas a function ofoperation voltage and find its fill factor from this power happens to the short-circuit current, the open-circuit voltage, and the max-imum output power of the solar cell from the previous problem if it is employed as apower supply for the Mars Pathfinder mission? Mean distance from the Mars to theSun is approximately a factor of longer than that of between theEarth and the that in both cases the solar cell operates at room room temperature, an ideal solar cell has a short-circuit current of 2 A and anopen-circuit voltage of V. How does the open-circuit voltage change if the short-circuit current drops by a factor of 2, 5, or 10? 300 K, an ideal Sip-njunction solar cell has a short-circuit current of 2 A andan open-circuit voltage of V.