{
"metadata": {
"name": "",
"signature": "sha256:89b26be9b34fcc4abdf434215c6ed45e9bed850a39f0e736d3ded9e4d9969c4c"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"\n",
"Chapter2:ELECTRONS IN SEMICONDUCTORS"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.1:pg-55"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"h=1.05*10**-34\n",
"mo = 9.1*10**-31\n",
"E = 0.1*1.6*10**(-19)\n",
"m=0.067*mo\n",
"k = sqrt(2*m*E)/h\n",
"print\"The k-value for an electron in the conduction band of GaAs is ,k = \",\"{:.1e}\".format(k),\"m**-1\"\n",
"ko = 1.625*10**9\n",
"print\"the two value are quite difference since the k value represent effective momentum\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The k-value for an electron in the conduction band of GaAs is ,k = 4.2e+08 m**-1\n",
"the two value are quite difference since the k value represent effective momentum\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.2:pg-56"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"mo = 9.1*10**-31\n",
"ml = 0.98*mo\n",
"mt = 0.19*mo\n",
"mhh =0.49*mo\n",
"mlh = 0.16*mo\n",
"mdos = (((6)**(2.0/3))*((ml)*((mt)**2))**(1.0/3))\n",
"print\"The conduction band density of states mass is ,mdos* =\",\"{:.2e}\".format(mdos),\"kg\"\n",
"mdos1 = (((mhh)**(3/2)+(mlh)**(3/2))**(2.0/3))\n",
"print\"The Valence band density of states mass is ,mdos1*=\",\"{:.2e}\".format(mdos1),\"kg\"\n",
"# In the book ,the answer is given in the form of mo\n",
" \n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The conduction band density of states mass is ,mdos* = 9.86e-31 kg\n",
"The Valence band density of states mass is ,mdos1*= 7.05e-21 kg\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.3:pg-59"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"h=1.05*10**-34\n",
"mo = 9.1*10**-31\n",
"mhh =0.5*mo\n",
"k = 0.1*10**10\n",
"Ev = 0\n",
"e = 1.6*10**-19\n",
"#(we have assumed the valence band energy Ev=0eV as it is not provided in the book)\n",
"Ee= Ev-(((h**2)*(k**2))/(2*mhh))\n",
"print\"The electron energy in the valence band is ,Ee=\",\"{:.2e}\".format(Ee),\"J\"\n",
"Ee1= Ee/e\n",
"print\"The electron energy in the valence band is ,Ee= Ee/e=\",\"{:.2e}\".format(Ee1),\"eV\"\n",
"Eh= Ev+((((h**2)*(k**2))/(2*mhh))/e)\n",
"print\"The hole energy in the valence band is ,Eh=\",\"{:.2e}\".format(Eh),\"eV\"\n",
"\n",
" \n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The electron energy in the valence band is ,Ee= -1.21e-20 J\n",
"The electron energy in the valence band is ,Ee= Ee/e= -7.57e-02 eV\n",
"The hole energy in the valence band is ,Eh= 7.57e-02 eV\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.4:pg-62"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"h=1.06*10**-34\n",
"mo = 9.1*10**-31\n",
"m = 0.067*mo\n",
"print\"m = \",\"{:.2e}\".format(m),\"kg\"\n",
"E = 0.5*1.6*10**-19\n",
"print\"E = \",\"{:.2e}\".format(E),\"J\" #initializing value of electron energy measured from the bandedge\n",
"# Effective momentum of electron in the conduction band of GaAs\n",
"hk = sqrt(2*m*E)\n",
"print\"The effetive momentum of an electron in the conduction band of GaAs is ,hk = \"\"{:.2e}\".format(hk),\"m**-1\"#calculation\n",
"k = hk/h\n",
"print\"the corresponding wavevector is,k = \",\"{:.1e}\".format(k),\"m**-1\"\n",
"#Effective momentum of free electron in the space with same energy\n",
"p = sqrt(2*mo*E)\n",
"print\"The effetive momentum of an electron in the space is ,p = \",\"{:.1e}\".format(p),\"kgms**-1\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"m = 6.10e-32 kg\n",
"E = 8.00e-20 J\n",
"The effetive momentum of an electron in the conduction band of GaAs is ,hk = 9.88e-26 m**-1\n",
"the corresponding wavevector is,k = 9.3e+08 m**-1\n",
"The effetive momentum of an electron in the space is ,p = 3.8e-25 kgms**-1\n"
]
}
],
"prompt_number": 12
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.5:pg-63"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"h=1.05*10**-34 #initializing value of reduced plancks constant or dirac constant or h-bar\n",
"mo = 9.1*10**-31 #initializing value of mass of electron\n",
"ml = 0.98*mo #initializing value of longitudinal mass\n",
"mt = 0.19*mo #initializing value of transverse mass\n",
"a = 5.43*10**-10 #initializing value of latice constant\n",
"kx = ((2*math.pi*0.95)/a) #initializing value of given k-value in x direction \n",
"ky = ((2*math.pi*0.1)/a) #initializing value of given k-value in y direction \n",
"kz = ((2*math.pi*0.0)/a) #initializing value of given k-value in z direction \n",
"kxo = ((2*math.pi*0.85)/a) #initializing value of k-value for Si occupies the (100) valley in x direction \n",
"kyo = ((2*math.pi*0.0)/a) #initializing value of k-value for Si occupies the (100) valley in y direction \n",
"kzo = ((2*math.pi*0.0)/a) #initializing value of k-value for Si occupies the (100) valley in z direction \n",
"kl = kx-kxo\n",
"print\"the change in k vector in x direction is,kl = kx-kxo = \",\"{:.3e}\".format(kl),\"m**-1\"\n",
"kt = ky-kyo\n",
"print\"the change in k vector in y direction is,kt = ky-kyo = \",\"{:.3e}\".format(kt),\"m**-1\"\n",
"E= (((h**2)*(kl**2))/(2*ml))+(((h**2)*(kt**2))/(2*mt))\n",
"print\"The electron energy measured from the conduction bandege is ,E= \",\"{:.3e}\".format(E),\"J\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the change in k vector in x direction is,kl = kx-kxo = 1.157e+09 m**-1\n",
"the change in k vector in y direction is,kt = ky-kyo = 1.157e+09 m**-1\n",
"The electron energy measured from the conduction bandege is ,E= 5.097e-20 J\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.9:pg-70"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"h=1.05*10**-34 #initializing value of reduced plancks constant or dirac constant or h-bar\n",
"mo = 9.1*10**-31 #initializing value of mass of electron\n",
"me = 0.067*mo #initializing value of effective mass of GaAs\n",
"kbT = 4.16*10**-21 #initializing value of kbT at 300K\n",
"Nc=2*(((me*kbT)/(2*math.pi*(h**2)))**(3/2))\n",
"print\"for GaAs conduction band case effective density of states is ,Nc= \",\"{:.2e}\".format(Nc),\"m**-3\"\n",
"ml = 0.98*mo #initializing value of longitudinal mass\n",
"mt = 0.19*mo #initializing value of transverse mass\n",
"mdos = (((6)**(2.0/3))*((ml)*((mt)**2))**(1.0/3))\n",
"Nc1 = 2*((mdos*kbT)/(2*(math.pi)*(h**2)))**(3/2)\n",
"print\"for silicon conduction band case effective density of states is ,Nc = \",\"{:.2e}\".format(Nc1),\"m**-3\"\n",
"\n",
"# Note : due to different precisions taken by me and the author ... my answer differ \n",
"\n",
"print\"for silicon\"\n",
"mhh =0.5*mo #initializing value of heavy hole mass for silicon\n",
"mlh = 0.15*mo #initializing value of light hole mass for silicon\n",
"Nv1 =((kbT/(2*(math.pi)*(h**2)))**(3/2))*2*(mhh**(3/2)+mlh**(3/2))\n",
"print\"for silicon valence band case effective density of states is ,Nv = \",\"{:.2e}\".format(Nv1),\"m**-3\"\n",
"print\"for GaAs \"\n",
"mhh1 =0.45*mo #initializing value of heavy hole mass\n",
"mlh1 = 0.08*mo #initializing value of light hole mass\n",
"Nv = 2*(mhh1**(3/2)+mlh1**(3/2))*((kbT/(2*(math.pi)*(h**2)))**(3/2))\n",
"print\"for GaAs valence band case effective density of states is ,Nv= \",\"{:.2e}\".format(Nv),\"m**-3\"\n",
"\n",
"# Answer given in the book for valence band case is wrong\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"for GaAs conduction band case effective density of states is ,Nc= 7.32e+15 m**-3\n",
"for silicon conduction band case effective density of states is ,Nc = 1.18e+17 m**-3\n",
"for silicon\n",
"for silicon valence band case effective density of states is ,Nv = 7.10e+16 m**-3\n",
"for GaAs \n",
"for GaAs valence band case effective density of states is ,Nv= 5.79e+16 m**-3\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.10:pg-70"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"mo = 9.1*10**-31 #initializing value of mass of electron\n",
"me = 0.067*mo #initializing value of effective mass of GaAs\n",
"kbT = 0.026 #initializing value of kbT at 300K\n",
"ml = 0.98*mo #initializing value of longitudinal mass\n",
"mt = 0.19*mo #initializing value of transverse mass\n",
"mh = 0.55*mo #initializing value of density of state mass for the valence band \n",
"#let\n",
"Eg = 0.0 #initializing value of valence bandedge energy\n",
"mdos = (((6)**(2/3))*((ml)*((mt)**2))**(1.0/3))\n",
"print\"The desity of states of effective mass of the combined six valleys of silicon is mdos* = \",\"{:.2e}\".format(mdos),\"kg\"\n",
"Efi = (Eg/2)+((3.0/4)*kbT*log(mh/mdos))\n",
"print\"The intrinsic fermi level is given by Efi= \",\"{:.2e}\".format(Efi),\"eV\"\n",
"# -ve sign show that fermi level is below the centre of mid-bandgap\n",
"# In this question the answer is provided in the book is in terms of Eg and i have assumed value of Eg = 0 V \n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The desity of states of effective mass of the combined six valleys of silicon is mdos* = 2.99e-31 kg\n",
"The intrinsic fermi level is given by Efi= 1.01e-02 eV\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.11:pg-71"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"mo = 9.1*10**-31 #initializing value of mass of electron\n",
"me = 0.027*mo #initializing value of effective mass of GaAs\n",
"kbT = 0.026 #initializing value of kbT at 300K\n",
"mh = 0.4*mo #initializing value of longitudinal mass\n",
"h=1.05*10**-34 #initializing value of plank constant.\n",
"Eg = 0.35 #initializing value of valence bandedge energy\n",
"ni =2*(((kbT*1.6*10**-19)/(2*(math.pi)*h**2))**(3/2))*((me*mh)**(3/4))*(exp(-Eg/(2*kbT)))\n",
"print\"ni =2*(kbT/(2*(math.pi)*h**2))**(3/2)*((me*mh)**(3/4))*(exp(-Eg/(2*kbT)))= \",\"{:.2e}\".format(ni),\"m**-3\"\n",
"kbT = 0.05175\n",
"print\"kbT = \",\"{:.2e}\".format(kbT),\"eV\" #initializing value of kbT at 600K\n",
"ni =2*(((kbT*1.6*10**-19)/(2*(math.pi)*h**2))**(3/2))*((me*mh)**(3/4))*(exp(-Eg/(2*kbT)))\n",
"print\"ni =2*(kbT/(2*(math.pi)*h**2))**(3/2)*((me*mh)**(3/4))*(exp(-Eg/(2*kbT)))= \",\"{:.2e}\".format(ni),\"m**-3\"\n",
"#Note: In the textbook wrong answer is given for intrinsic carrier concentration at 600K\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"ni =2*(kbT/(2*(math.pi)*h**2))**(3/2)*((me*mh)**(3/4))*(exp(-Eg/(2*kbT)))= 1.43e+44 m**-3\n",
"kbT = 5.17e-02 eV\n",
"ni =2*(kbT/(2*(math.pi)*h**2))**(3/2)*((me*mh)**(3/4))*(exp(-Eg/(2*kbT)))= 8.13e+45 m**-3\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.12:pg-75"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"mo = 9.1*10**-31 #initializing value of mass of electron\n",
"m_star=0.067*mo #initializing value of appropriate mass in the conduction band for GaAs\n",
"apsilen = 13.2*8.85*10**-14 #initializing value of relative permitivity for GaAs\n",
"apsilen_not = 8.85*10**-14 #initializing value of permitivity\n",
"ml = 0.98*mo #initializing value of longitudinal mass\n",
"mt = 0.2*mo #initializing value of transverse mass\n",
"m_sigma_star = (3)/((1.0/ml)+(2.0/mt))\n",
"print\"The conductivity mass for silicon is ,m_sigma_star = (3*mo)/((1/ml)+(2/mt))= \",\"{:.2e}\".format(m_sigma_star),\"Kg\"\n",
"print\"The shallow level energies are given by,Ed = Ec-(13.6(eV)*((m_star/mo)/(apsilen/apsilen_not)**2))\"\n",
"#Let Ec = 0 V and taking positive answer, \n",
"Ed_GaAs = (13.6*((m_star/mo)/(apsilen/apsilen_not)**2))\n",
"print\"The donor level energy in GaAs is ,Ed_GaAs = Ed= \",\"{:.2e}\".format(Ed_GaAs),\"eV\"\n",
"m_dot_GaAs=0.45*mo\n",
"print\"m_dot_GaAs=0.45*mo = \",\"{:.2e}\".format(m_dot_GaAs),\"kg\" #initializing value of heavy hole mass for GaAs\n",
"Ea_GaAs = (13.6*((m_dot_GaAs/mo)/(apsilen/apsilen_not)**2))\n",
"print\"The acceptor level energy in GaAs is ,Ea_GaAs = \",\"{:.2e}\".format(Ea_GaAs),\"eV\"\n",
"apsilen = 11.9*8.85*10**-14 #initializing value of relative permitivity for GaAs\n",
"m_dot_Si=0.5*mo #initializing value of heavy hole mass for GaAs\n",
"Ea_Si = (13.6*((m_dot_Si/mo)/(apsilen/apsilen_not)**2))\n",
"print\"The acceptor level energy in Si is ,Ea_Si = (13.6*((m_dot_Si/mo)/(apsilen/apsilen_not)**2))= \",\"{:.2e}\".format(Ea_Si),\"eV\"\n",
"Ed_Si = (13.6*((m_sigma_star/mo)/(apsilen/apsilen_not)**2))\n",
"print\"The donor level energy in Si is ,Ed_Si = (13.6*((m_sigma_star/mo)/(apsilen/apsilen_not)**2))= \",\"{:.2e}\".format(Ed_Si),\"eV\"\n",
"# Note : due to different precisions taken by me and the author ... my answer differ \n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The conductivity mass for silicon is ,m_sigma_star = (3*mo)/((1/ml)+(2/mt))= 2.48e-31 Kg\n",
"The shallow level energies are given by,Ed = Ec-(13.6(eV)*((m_star/mo)/(apsilen/apsilen_not)**2))\n",
"The donor level energy in GaAs is ,Ed_GaAs = Ed= 5.23e-03 eV\n",
"m_dot_GaAs=0.45*mo = 4.10e-31 kg\n",
"The acceptor level energy in GaAs is ,Ea_GaAs = 3.51e-02 eV\n",
"The acceptor level energy in Si is ,Ea_Si = (13.6*((m_dot_Si/mo)/(apsilen/apsilen_not)**2))= 4.80e-02 eV\n",
"The donor level energy in Si is ,Ed_Si = (13.6*((m_sigma_star/mo)/(apsilen/apsilen_not)**2))= 2.61e-02 eV\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.13:pg-77"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"n = 10**17 #initializing value of free density of electron of GaAs\n",
"kBT=0.026 #initializing value of multiplication of boltzmann constant and temperature \n",
"Nc = 4.45*10**17 #initializing value of effective density of electron\n",
"#(we have assumed the valence band energy Ev=0eV as it is not provided in the book)\n",
"E1= kBT*((log(n/Nc)))\n",
"print\"Ef(B)=\",\"{:.1e}\".format(E1),\"eV\"\n",
"E2= kBT*((log(n/Nc))+(1/sqrt(8))*(n/Nc))\n",
"print\"E(J)=\",\"{:.1e}\".format(E2),\"eV\"\n",
"#for Boltzmann approximation the carrier concentration and fermi level are related as : Ef = Ec+E1\n",
"#for joyce dixon approximation the carrier concentration and fermi level are related as : Ef = Ec+E2\n",
"e=E1-E2\n",
"print\"The error produced by using boltzmann approx. is e=\"\"{:.2e}\".format(e),\"eV\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Ef(B)= -3.9e-02 eV\n",
"E(J)= -3.7e-02 eV\n",
"The error produced by using boltzmann approx. is e=-2.07e-03 eV\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.14:pg-77"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"print\"In the Boltzmann approximation, the carrier density is simply\"\n",
"print\"n = Nc = 2.78*10**19 cm**-3\"\n",
"N=2.78*10**19 #initializing value of carrier density\n",
"#In joyce dixon approximation the carrier density is obtained from the solution of the equation\n",
"print\"Ef = 0 = kBT *(log(n/Nc)+(n/(sqrt8*Nc)))\"\n",
"#solving by trial and error , we get\n",
"#n/Nc= 0.76\n",
"n=0.76*N\n",
"print\"electron carrier concentration is n=0.76*Nc= \",\"{:.2e}\".format(n),\" cm**-3\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"In the Boltzmann approximation, the carrier density is simply\n",
"n = Nc = 2.78*10**19 cm**-3\n",
"Ef = 0 = kBT *(log(n/Nc)+(n/(sqrt8*Nc)))\n",
"electron carrier concentration is n=0.76*Nc= 2.11e+19 cm**-3\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.16:pg-80"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Nc = 2.8*10**19 #initializing value of effective density of electron\n",
"Nd = 10**16 #initializing value of donor atom\n",
"Ec_minus_Ed = 45*10**-3 #initializing value of donor binding energy\n",
"kBT=0.026 #initializing value of multiplication of boltzmann constant and temperature \n",
"\n",
"#let fraction of ionised donor are represented as Fd = (nd/(n+nd))\n",
"Fd= (1/(((Nc/(2*Nd))*exp(-(Ec_minus_Ed/kBT)))+1))*100\n",
"print\"fraction of ionised donor is Fd=\",round(Fd,2),\"%\"\n",
"Nd = 10**18\n",
"print\"Nd = \",\"{:.2e}\".format(Nd),\"cm**-3\"\n",
"Fd= (1.0/(((Nc/(2*Nd))*exp(-(Ec_minus_Ed/kBT)))+1))*100\n",
"print\"fraction of ionised donor is Fd=\",round(Fd,2),\"%\"\n",
"# Note : due to different precisions taken by me and the author ... my answer differ \n",
"\n",
"\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"fraction of ionised donor is Fd= 0.4 %\n",
"Nd = 1.00e+18 cm**-3\n",
"fraction of ionised donor is Fd= 28.74 %\n"
]
}
],
"prompt_number": 10
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Ex2.17:pg-80"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Nc_Si = 2.78*10**19 #initializing value of effective density of electron for silicon\n",
"Nc_GaAs = 4.45*10**17 #initializing value of effective density of electron for GaAs\n",
"print\"for joyce dixon approximation the carrier concentration and fermi level are related as : Ef -Ec = kBT*(log(n/Nc)+(n/(sqrt8*Nc))\"\n",
"print(\"using Ef-Ec = 3* kBT\") \n",
"print(\"solving above equation by hit and trial method for n/Nc,we get n/Nc = 4.4\") \n",
"n_by_Nc = 4.4\n",
"n_Si = n_by_Nc*Nc_Si\n",
"print\"carrier density for silicon is n=\"\"{:.2e}\".format(n_Si),\"cm**-3\"\n",
"n_GaAs = n_by_Nc*Nc_GaAs\n",
"print\"carrier density for GaAs is n=\",\"{:.2e}\".format(n_GaAs),\"cm**-3\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"for joyce dixon approximation the carrier concentration and fermi level are related as : Ef -Ec = kBT*(log(n/Nc)+(n/(sqrt8*Nc))\n",
"using Ef-Ec = 3* kBT\n",
"solving above equation by hit and trial method for n/Nc,we get n/Nc = 4.4\n",
"carrier density for silicon is n=1.22e+20 cm**-3\n",
"carrier density for GaAs is n= 1.96e+18 cm**-3\n"
]
}
],
"prompt_number": 12
}
],
"metadata": {}
}
]
}