Shockley-Queisser limit

The Shockley-Queisser limit is the upper theoretical limit for the efficiency of p-n junction solar cells. It plays an important role to estimate how far an experimental solar cell efficiency is from the maximum achievable theoretical efficiency.
◼
source: William Shockley and Hans J.Queisser, J.Appl.Phys.32, 510 (1961); doi : 10.1063/1.1736034

Physical model

◼
  • Single cut-off frequency: for E ≥ hν effect otherwise no effect
    • ◼
  • p-n junction solar cell is at
    T
    C
    = 0, surrounded by a blackbody at T =
    T
    s.
    • ◼
  • Introduce finite
    T
    C
    and replace surrounding body at
    T
    S
    by radiation coming from the sun at a small solid angle
    ω
    s
    .

    • Ultimate efficiency hypothesis

    ◼
  • EachphotonwithE>
    hν
    g
    produces1
    q
    e
    atavoltageof
    V
    g
    =
    hν
    g
    q
    e

    • Formula check for incident power

    In[20]:=
    Remove[h,ν,Ts,κ,c]
    In[21]:=
    (2πh)
    ∞
    ∫
    0
    3
    ν
    exp
    hν
    κTs
    -1
    ν
    2
    c
    Out[21]=
    ConditionalExpression
    2
    5
    π
    4
    Ts
    4
    κ
    15
    2
    c
    3
    h
    ,Re
    h
    Tsκ
    >0

    Physical constants

    ◼
  • Planck’s constant
    • In[22]:=
    h=QuantityMagnitude@UnitConvert[Quantity[1,"PlanckConstant"],"SIBase"]
    Out[22]=
    6.626070×
    -34
    10
    ◼
  • speed of light
    • In[23]:=
    c=QuantityMagnitude@UnitConvert[Quantity[1,"SpeedOfLight"],"SIBase"]
    Out[23]=
    299792458
    ◼
  • Boltzmann constant
    • In[24]:=
    κ=QuantityMagnitude@UnitConvert[Quantity[1,"BoltzmannConstant"],"SIBase"]
    Out[24]=
    1.38065×
    -23
    10
    In[25]:=
    qe=QuantityMagnitude@UnitConvert[Quantity[1,"ElementaryCharge"],"SIBase"]
    Out[25]=
    1.6021766×
    -19
    10

    Black-body Radiation

    Built - in

    In[26]:=
    <<BlackBodyRadiation`​​BlackBodyProfile[4000Kelvin,5000Kelvin,6000Kelvin,PlotRange{{0,2.0*10^-6},{0,1.1*10^14}},ImageSize400]
    Out[27]=

    User - made

    Bν (T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.
    In[28]:=
    Remove[T,B]​​B[T_]:=
    2h
    3
    ν
    2
    c
    1
    Exp
    hν
    κT
    -1
    ;(*T=300;*)​​(*Plot[
    B
    ν
    ,{ν,1,10000},PlotRangeAll]*)​​Plot[{B[300],B[400]},{ν,0
    13
    10
    ,10
    13
    10
    },AxesLabel{"f(Hz)","Intensity (W/
    3
    m
    ​)"}]
    Out[30]=

    useful parameters for the calculation

    E
    g
    =
    hν
    g
    =q
    V
    g
    ​​
    x
    g
    =
    E
    g
    κTs
    ​​
    x
    c
    =
    Tc
    Ts

    Surrounding temperature

    In[31]:=
    Ts=6000;

    Calculation & Result

    Output power from the solar cell

    ◼
  • Qs: # of frequency quanta greater than
    v
    g
    incident per unit time per unit area for black body radiation
    • In[173]:=
    Qs=
    2π
    2
    c
    ∞
    ∫
    v
    g
    2
    ν
    exp
    hν
    κTs
    -1
    ν
    Putting
    hν
    κTs
    =x;
    In[32]:=
    Qs[xg_]:=
    2π
    2
    c
    3
    κTs
    h
    ∞
    ∫
    xg
    2
    x
    Exp[x]-1
    x
    Outputpower=h
    ν
    g
    AQ
    s
    ,fromUltimateefficiencyhypothesis

    Incident Power

    Ps=
    (2πh)
    ∞
    ∫
    0
    3
    ν
    exp
    (hν)
    κTs
    -1
    ν
    2
    c
    after variable changed to x
    In[33]:=
    Ps=
    2π
    2
    c
    3
    h
    4
    (κTs)
    ∞
    ∫
    0
    3
    x
    Exp[x]-1
    x
    Out[33]=
    7.3488×
    7
    10
    ◼
  • Ultimate efficiency function
    • In[34]:=
    u[xg_]:=κTsxgQs[xg]/Ps(*
    Qs[xg]
    Ps
    *);
    ◼
  • ~44% maximum efficiency achievable
    • In[35]:=
    u[2.2]//AbsoluteTiming
    Out[35]=
    {5.85332,0.438691}

    As a function of xg

    In[37]:=
    maxeffdata=Transpose[{​​Range[0,10,.5],​​100*u[#]&/@Range[0,10,.5]​​}];
    In[38]:=
    ListLinePlot[maxeffdata,PlotRangeAll,AxesLabel{"xg","% efficiency"}]
    Out[38]=

    As a function of bandgap

    ◼
  • At the bandgap of 1.1 eV ~44% maximum efficiency achievable
    • ◼
    Radiative recombination & lifetime consideration.